#  $X_{0}(2900)$ production in the $B^{+}$ decay process

Zhuo Yu1 Qi Wu2 Dian-Yong Chen1,3111Corresponding author<EMAIL_ADDRESS>1
School of Physics, Southeast University, Nanjing 210094, China  2Institute of
Particle and Nuclear Physics, Henan Normal University, Xinxiang 453007, China
3 Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou 730000,
China

###### Abstract

In the present work, we investigate the $X_{0}(2900)$ production process
$B^{+}\to D^{+}X_{0}(2900)$ in the molecular frame, where $X_{0}(2900)$ is
considered as a $\bar{D}^{\ast}K^{\ast}$ molecular state. We infer that the
initial $B^{+}$ meson and the final $D^{+}X_{0}(2900)$ are connected through
$D_{s1}^{+}(2460)\bar{D}^{\ast 0}$ mesons by exchanging a $K^{\ast 0}$ meson.
The contributions from such a meson loop to the process $B^{+}\to
D^{+}X_{0}(2900)$ are evaluated and our estimations indicate that the measured
branching fraction of $B^{+}\to D^{+}X_{0}(2900)$ could be well reproduced in
a reasonable model parameter range, which indicates that $X_{0}(2900)$ could
be interpreted as a $\bar{D}^{\ast}K^{\ast}$ molecular state.

##  I introduction

Over the past two decades, an increasing number of new hadron states have been
reported by Belle, BESIII, and LHCb collaborations (More details can be found
in the reviews Guo:2017jvc ; Olsen:2017bmm ; Brambilla:2019esw ; Ali:2017jda ;
Guo:2019twa ; Liu:2013waa ; Dong:2017gaw ; Liu:2019zoy ). Among these exotic
candidates, most of them were observed in hidden charm channels, such as
$X(3872)$ Belle:2003nnu , $Z_{c}(3900)/Z_{c}(4020)$BESIII:2013ouc ;
BESIII:2013mhi , which are named charmonium-like state. In addition, some new
hadron states with open flavor have also been observed, the typical examples
are $D_{s0}^{\ast}(2317)$ and $D_{s1}(2460)$ BaBar:2003oey ; CLEO:2003ggt .
The $J^{P}$ quantum numbers of these two states are consistent with those of
$P$-wave charmed-strange meson, $c\bar{s}(3P_{0})$ and $c\bar{s}(P_{1})$,
respectively, thus the $P$-wave charmed-strange meson interpretations for
$D_{s0}^{\ast}(2317)$ and $D_{s1}(2460)$ have been proposed Xie:2010zza ;
Zhang:2006ix ; Bicudo:2004dx ; Faessler:2007gv ; Faessler:2007us ;
Cleven:2014oka ; Datta:2003re ; Lutz:2008zz ; Hwang:2004cd . However, the
observed masses of $D_{s0}^{\ast}(2317)$ and $D_{s1}(2460)$ are much smaller
than the quark model expectations for the $P$ wave charmed-strange mesons
Godfrey:1986wj ; Zeng:1994vj ; Gupta:1994mw ; Ebert:1997nk ; Lahde:1999ih ;
DiPierro:2001dwf ; Liu:2020ruo . Moreover, the observed masses of
$D_{s0}^{\ast}(2317)$ and $D_{s1}(2460)$ are about 40 MeV below the thresholds
of $DK$ and $D^{\ast}K$, which lead to the populations of the molecular
interpretations to $D_{s0}^{\ast}(2317)$ and $D_{s1}(2460)$ Guo:2006fu ;
Gamermann:2006nm ; Altenbuchinger:2013vwa ; Yao:2015qia ; Guo:2015dha ;
Du:2017ttu ; Bali:2017pdv ; Hu:2020mxp ; Xiao:2016hoa .

In the year 2020, the LHCb collaboration reported the observation of two new
narrow peaks, named $X_{0}(2900)$ and $X_{1}(2900)$, in the $D^{-}K^{+}$
invariant mass distributions of the decay process $B^{+}\to D^{+}D^{-}K^{+}$
LHCb:2020pxc ; LHCb:2020bls . The helicity amplitudes analysis indicated that
the spins of these two peaks were 0 and 1, respectively. Thus, the $J^{P}$
quantum numbers of $X_{0}(2900)$ and $X_{1}(2900)$ are $0^{+}$ and $1^{+}$,
respectively. The measured mass and width of $X_{0}(2900)$ were

$\begin{split}m_{0}&=(2866\pm 7\pm 2)\,{\rm MeV},\\\ \Gamma_{0}&=(57\pm 12\pm
4)\,{\rm MeV},\end{split}$ (1)

while the ones of $X_{1}(2900)$ were reported to be,

$\begin{split}m_{1}&=(2904\pm 5\pm 1)\,{\rm MeV},\\\ \Gamma_{1}&=(110\pm 11\pm
4)\,{\rm MeV}.\end{split}$ (2)

From the observed channel of $X_{0,1}(2900)$, it is evident that the quark
components of $X_{0,1}(2900)$ are $ud\bar{s}\bar{c}$, which indicates that
$X_{0,1}(2900)$ are fully open flavor exotic states. Thus, the investigations
of their properties will significantly improve our understanding on the non-
perturbative behaviors of the strong interaction in QCD. Inspired by the
observations of $X_{0,1}(2900)$, the tetraquark interpretations have been
proposed in various framework. The estimations in the framework of the
improved chromo-magnetic interaction model indicated that $X_{0}(2900)$ could
be a compact tetraquark state composed of $ud\bar{s}\bar{c}$ Guo:2021mja . The
mass of the diquark-antidiquark states were estimated by using the QCD sum
rule and the authors in Refs. Zhang:2020oze ; Wang:2020xyc  found that
$X_{0}(2900)$ could be assigned as an axialvector-diquark-axialvector-
antidiquark type tetraquark states. By considering the SU(3) flavor symmetry,
the authors in Ref. He:2020jna  interpreted the $X_{0}(2900)$ as the radial
excited tetraquark state with $J^{P}=0^{+}$. In Ref. Wang:2020prk , the
compact $\bar{c}\bar{s}qq$ tetraquark states have been systematically
estimated in the quark model with the Coulomb, the linear confinement, and the
hyperfine interaction, and the estimated mass and width of the ground state
with $I(J^{P})=1(0^{+})$ are consistent with that of $X_{0}(2900)$. However,
there are exist some estimations which do not support the tetraquark
interpretations of $X_{0}(2900)$, for example, the estimations in the extended
relativized quark model indicate the mass of ground state of compact
$\bar{c}\bar{s}qq$ tetraquark states with $J^{P}=0^{+}$ is about 100 MeV below
the one of $X_{0}(2900)$ Lu:2020qmp , the spectroscopic parameters of either
axial-vector and scalar diquark-antidiquark computed by employing the QCD two-
point sum rule method is inconsistent with those of $X_{0}(2900)$ either
Agaev:2022eeh .

Moreover, the observed mass of $X_{0,1}(2900)$ are very close to the threshold
of $D^{\ast}K^{\ast}$, which indicates that these two states could be good
candidates of the molecular states composed of $\bar{D}^{\ast}{K}^{\ast}$.
Considering the $J^{P}$ quantum numbers of $X_{0,1}(2900)$, one can find
$X_{0}(2900)$ and $X_{1}(2900)$ could be assigned as a $S-$wave and $P-$wave
$\bar{D}^{\ast}K^{\ast}$ molecular state, respectively. The estimations in
Bethe-Salpeter framework indicate that $X_{0}(2900)$ could be assigned as a
$\bar{D}^{\ast}K^{\ast}$ molecular state with $I(J^{P})=0(0^{+})$ Ke:2022ocs ;
Kong:2021ohg . While the investigations in the chiral effective field theory
also supported the $\bar{D}^{\ast}K^{\ast}$ molecular interpretations of
$X_{0}(2900)$, and $X_{1}(2900)$ was considered as the $P$-wave excitation of
the ground-state $X_{0}(2900)$ Wang:2021lwy . By using the QCD sum rule
methods, the mass spectroscopy, decay constants of the
$\bar{D}^{\ast}K^{\ast}$ molecular state had been investigated Chen:2021erj ;
Agaev:2020nrc ; Chen:2020aos ; Mutuk:2020igv , which indicate that
$X_{0}(2900)$ could be a $\bar{D}^{\ast}K^{\ast}$ molecule with $J^{P}=0^{+}$,
while the $X_{1}(2900)$ could be considered as the $P$ wave $\bar{c}\bar{s}qq$
tetraquark with $J^{P}=1^{-}$ Chen:2020aos . In Ref. Liu:2020nil , the
$D^{\ast}$ and $\bar{K}^{\ast}$ interaction have been investigated in the one-
boson-exchange model, and the mass of $X_{0}(2900)$ could be reproduced,
however, the $X_{1}(2900)$ could not be considered as a
$\bar{D}^{\ast}K^{\ast}$ molecular state with $J^{P}=1^{-}$ since
$\bar{D}^{\ast}K^{\ast}$ potential is repulsive in the case of $P$-wave. In
addition, by using the effective Lagrangian approach, the authors in Refs.
Xiao:2020ltm ; Huang:2020ptc  estimated the decay width of $X_{0}(2900)$,
which is in agreement with the experimental value within the uncertainty of
the model. Besides the tetraquark and molecular interpretation, the
kinematical cusp effect Swanson:2021fqw ; Burns:2020xne , could also provide
insights into understanding these states.

Besides the resonance parameters and decay properties, the production
processes are also crucial in decoding the nature of $X_{0}(2900)$, which
sparked numerous theoretical predictions using various approaches. For
example, the cross sections for $K^{+}p\to\Sigma_{c}^{++}X_{0}(2900)$
Lin:2022eau  and decay widths of
$\Lambda_{b}\to\Sigma_{c}^{0,(++)}X_{0,1}^{{}^{\prime}0,(--)}$ Hsiao:2021tyq
have been investigated. On the experimental side, the fit fraction of
$X_{0}(2900)$ in the process $B^{+}\to D^{+}D^{-}K^{+}$ is measured to be
$(5.6\pm 1.4\pm 0.5)\%$. The branching faction of $B^{+}\to D^{+}D^{-}K^{+}$
is $(2.2\pm 0.7)\times 10^{-4}$, thus the branching fraction of the two body
cascade decay process $B^{+}\to D^{+}X_{0}(2900)\to D^{+}(D^{-}K^{+})$ is
measured to be $(1.2\pm 0.5)\times 10^{-5}$. The estimations in Refs.
Xiao:2020ltm ; Huang:2020ptc  indicated that the $X_{0}(2900)$ is dominantly
decay into $\bar{D}K$ channel, where $X_{0}(2900)$ are considered as a
$\bar{D}^{\ast}K^{\ast}$ molecular state. Considering the isospin symmetry,
one can find the branching fraction of $X_{0}(2900)\to D^{-}K^{+}$ is about
$50\%$, thus, one can approximately conclude,

$\displaystyle\mathcal{B}[B^{+}\to D^{+}X_{0}(2900)]\approx(2.4\pm 1.0)\times
10^{-5}.$ (3)

The investigations of the production mechanism of $X_{0}(2900)$ in the $B^{+}$
decay would go a long way towards revealing its nature.

Figure 1: (Color online.) Diagram contributing to $B^{+}\to D^{+}X_{0}(2900)$
at the quark level.

For the first step, we can analysis the process $B^{+}\to D^{+}X_{0}(2900)$ at
the quark level as shown in Fig. 1. The anti-bottom quark transits into an
anti-charm quark by emitting a $W^{+}$ boson with the up quark as a spectator,
and then $W^{+}$ boson couples to $c\bar{s}$ quark pair, this weak transition
process is phenomenalogically described by the operator $H_{W}$. With the
$d\bar{d}$ quark created from the vacuum, $c$ quark and the $\bar{d}$ quark
hadronize as a $D^{+}$ meson, while the rest quark hadronize as the
$X_{0}(2900)$, here we adopt an operator $H_{T}$ to phenomenalogically
describe this hadronization process. Then, the process $B^{+}\to
D^{+}X_{0}(2900)$ could be described by the product of $\mathcal{H}_{W}$ and
$\mathcal{H}_{T}$, i.e., $\mathcal{H}=\mathcal{H}_{W}\mathcal{H}_{T}$. The
direct estimations of the diagram in Fig. 1 are rather difficult, thus, in the
present work, we simply the estimations by inserting a complete basis formed
by two mesons, then the decay amplitude becomes,

$\displaystyle\langle D^{+}X_{0}(2900)\left|\mathcal{H}\right|B^{+}\rangle$
$\displaystyle\qquad=\langle
D^{+}X_{0}(2900)\left|\mathcal{H}_{W}\left|M_{1}M_{2}\rangle\langle
M_{1}M_{2}\right|\mathcal{H}_{T}\right|B^{+}\rangle.\ \ \ $ (4)

In principle, all the possible $|M_{1}M_{2}\rangle$ meson pair which could
bridge the initial $B^{+}$ and final $D^{+}X_{0}(2900)$ should be involved. In
practice, only the dominant contributions would be considered at the hadron
level. For example, in Ref. Chen:2020eyu , the authors investigate the process
$B^{+}\to D^{+}X_{0}(2900)$ by assuming the loop in Fig. 2-(a) as the dominant
contributions based on the fact that the branching fraction of $B^{+}\to
D_{s}^{\ast+}\bar{D}^{0}$ is $(7.6\pm 1.6)\times 10^{-3}$, which is about 2
order larger than the one of $B^{+}\to D^{+}X_{0}(2900)$, and $X_{0}(2900)$
dominantly decay into $DK$ final state222In Ref. Chen:2020eyu , the authors
investigated the process $B^{-}\to D^{-}\bar{X}_{0}(2900)$. The mechanism
working in $B^{-}\to D^{-}\bar{X}_{0}(2900)$ should be the same as $B^{+}\to
D^{+}X_{0}(2900)$..

|
---|---
$(a)$ | $(b)$
Figure 2: (Color online.) Diagram contributing to $B^{+}\to D^{+}X_{0}(2900)$
at the quark level. Diagram (a) is the one considered in Ref. Chen:2020eyu ,
while diagram (b) will be considered in the present estimations.

In the molecular frame, $X_{0}(2900)$ is composed of $\bar{D}^{\ast}$ and
$K^{\ast}$, which indicates the strong coupling between $X_{0}(2900)$ and its
components $\bar{D}^{\ast}K^{\ast}$. In addition, the branching fractions of
$B^{+}\to D_{s1}(2460)\bar{D}^{\ast}$ is measured to be $(1.2\pm 0.3)\%$,
which is about 3 order larger than the one of $B^{+}\to D^{+}X_{0}(2900)$, and
more importantly, the charmed-strange meson $D_{s1}(2460)$ couples to
$D^{+}K^{\ast}$ via $S$-wave. Thus, we expect the digram in Fig. 2-(b) should
be essential in $B^{+}\to D^{+}X_{0}(2900)$ if one assigns the $X_{0}(2900)$
as a $\bar{D}^{\ast}K^{\ast}$ molecular state. In the present work, we
estimate the contributions of Fig. 2-(b) to $B^{+}\to D^{+}X_{0}(2900)$ and
compare our estimations with the experimental data in Eq. (3), which could be
a crucial test to the $\bar{D}^{\ast}K^{\ast}$ molecular interpretations to
the $X_{0}(2900)$.

This paper is organized as follows. After introduction, we present our
estimations of the branching fractions of the process $B^{+}\to
X_{0}(2900)D^{+}$ in Section II, where the loops in Fig. 2-(b) plays the
dominant role. The numerical results and related discussions are presented in
Section III, and the last section is devoted to a short summary.

##  II $X_{0}(2900)$ Production in the $B$ decay process

###  II.1 Effective Lagrangian

As discussed above, the contribution of the loop in Fig. 2-(b) is expected to
be dominant for the process $B^{+}\to D^{+}X_{0}(2900)$ when one considers
$X_{0}(2900)$ as the $\bar{D}^{\ast}K^{\ast}$ molecular state. In the present
work, all the hadron interactions are described by the effective Lagrangian.
As for the effective Lagrangian describe the coupling between $X_{0}(2900)$
and its component, we employ the simple $S$-wave coupling, which is
Xiao:2020ltm ,

$\mathcal{L}_{X_{0}D^{*}K^{*}}=g_{X_{0}}X_{0}\left[{D}^{*-\mu}K^{*+}_{\mu}-\bar{D}^{*0\mu}K^{*0}_{\mu}\right],$
(5)

where $g_{X_{0}}$ is the effective coupling constant, and its value will be
discussed later. As for the weak interaction vertex $B^{+}\to
D_{s1}^{+}(2460)\bar{D}^{\ast 0}$, we utilize the parametrized hadronic matrix
elements obtained by the effective Hamiltonian at the quark level, which are
Cheng:2003sm ; Soni:2021fky ,

$\displaystyle\langle
0|J_{\mu}|D_{s1}(p_{1},\epsilon)\rangle=f_{D_{s1}}\epsilon_{\mu}m_{D_{s1}}\;,$
$\displaystyle\langle D^{\ast}(p_{2},\epsilon)|J_{\mu}|B(p)\rangle$
$\displaystyle\qquad=\frac{i\epsilon^{\nu}}{m_{B}+m_{D^{\ast}}}\Bigg{\\{}i\varepsilon_{\mu\nu\alpha\beta}P^{\alpha}q^{\beta}A_{V}(q^{2})$
$\displaystyle\qquad+\left(m_{B}+m_{D^{\ast}}\right)^{2}g_{\mu\nu}A_{1}(q^{2})-P_{\mu}P_{\nu}A_{2}(q^{2})$
$\displaystyle\qquad-2m_{D^{\ast}}\left(m_{B}+m_{D^{\ast}}\right)\frac{P_{\nu}q_{\mu}}{q^{2}}\left[A_{3}(q^{2})-A_{0}(q^{2})\right]\Bigg{\\}},\qquad\qquad\;$
(6)

with $J_{\mu}=\bar{q}_{1}\gamma_{\mu}(1-\gamma_{5})q_{2}$,
$P_{\mu}=p_{\mu}+p_{2\mu}$ and $q_{\mu}=p_{\mu}-p_{2\mu}$. $A_{0,1,2}(q^{2})$
and $A_{V}(q^{2})$ are the weak transition form factors, while $A_{3}(q^{2})$
is the linear combination of form factors $A_{1}(q^{2})$ and $A_{2}(q^{2})$,
which is Cheng:2003sm ,

$A_{3}(q^{2})=\frac{m_{B}+m_{D^{\ast}}}{2m_{D^{\ast}}}A_{1}(q^{2})-\frac{m_{B}-m_{D^{\ast}}}{2m_{D^{\ast}}}A_{2}(q^{2}).$
(7)

With the above hadronic matrix elements, we can get the amplitude of $B^{+}\to
D_{s1}^{+}\bar{D}^{*}$, which is,

$\displaystyle\mathcal{A}(B^{+}\to D_{s1}^{+}\bar{D}^{*})$
$\displaystyle\qquad=\frac{G_{F}}{\sqrt{2}}V_{cb}V_{cs}a_{1}\left\langle
D_{s1}^{+}\left|J_{\mu}\right|0\right\rangle\left\langle\bar{D}^{*}\left|J_{\mu}\right|B^{+}\right\rangle$
(8)

where $G_{F}$ is the Fermi constant, $V_{bc}$ and $V_{cs}$ are the CKM matrix
elements. $a_{1}=c_{1}^{eff}+c_{2}^{eff}/N_{c}$ with $c_{1,2}^{eff}$ being the
effective Wilson coefficients, obtained by the factorization approach
Bauer:1986bm . In the present work, we adopt $G_{F}=1.166\times 10^{-5}\,{\rm
GeV^{-2}}$, $V_{cb}=0.041$, $V_{cs}=0.987$ and $a_{1}=1.05$ as in Refs.
ParticleDataGroup:2020ssz ; Ali:1998eb ; Ivanov:2006ni .

For the $D_{s1}DK^{*}$ coupling, the effective Lagrangian could be constructed
based on the heavy quark limit and chiral symmetry Casalbuoni:1996pg ;
Casalbuoni:1992gi ; Casalbuoni:1992dx , which reads,

$\displaystyle\mathcal{L}_{\mathcal{D}_{1}\mathcal{D}\mathcal{V}}=g_{\mathcal{D}_{1}\mathcal{D}\mathcal{V}}\mathcal{D}_{1b}^{\mu}\mathcal{V}_{\mu
ba}\mathcal{D}_{a}^{\dagger}$ (9)

where the $\mathcal{D}$ is charmed meson triplets $(D^{0},D^{+},D_{s}^{+})$
and $\mathcal{V}$ is matrix form of the light vector mesons with the concrete
form as,

$\displaystyle\mathcal{V}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&\rho^{+}&K^{*+}\\\
\rho^{-}&-\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&K^{*0}\\\
K^{*-}&{\bar{K}}^{*0}&\phi\\\ \end{array}\right).$ (13)

It is worth noting that the only the $S-$wave coupling is considered in Eq.
(9).

###  II.2 Decay Amplitude

With all the above effective Lagrangians, we can now obtain the amplitude of
$B^{+}\to X_{0}(2900)D^{+}$ corresponding to Fig. 2-(b), which is,

$\displaystyle\mathcal{M}$ $\displaystyle=$ $\displaystyle
i^{3}\int\frac{d^{4}q}{(2\pi)^{4}}\left[\mathcal{A}_{\mu\nu}^{B\to
D_{s1}\bar{D}^{*}}(p_{1},p_{2})\right]\left[ig_{D_{1}DV}g^{\alpha\beta}\right]\left[ig_{X_{0}}g^{\phi\theta}\right]$
(14)
$\displaystyle\times\left[\frac{-g^{\mu\alpha}+p_{1}^{\mu}p_{1}^{\alpha}/m_{1}^{2}}{p_{1}^{2}-m_{1}^{2}}\right]\left[\frac{-g^{\nu\theta}+p_{2}^{\nu}p_{2}^{\theta}/m_{2}^{2}}{p_{2}^{2}-m_{2}^{2}}\right]$
$\displaystyle\times\left[\frac{-g^{\beta\phi}+q^{\beta}q^{\phi}/m_{q}^{2}}{q^{2}-m_{q}^{2}}\right]\mathcal{F}\left(q^{2},m_{q}^{2}\right).$

In the above amplitude, we introduce a form factor $\mathcal{F}(q^{2},m^{2})$
in monopole form to compensate for the off-shell effect and avoid the
ultraviolet divergence in the loop integral. Its concrete form is,

$\mathcal{F}(q^{2},m^{2})=\frac{m^{2}-\Lambda^{2}}{q^{2}-\Lambda^{2}},$ (15)

where ${\rm\Lambda}=m+\alpha{\rm\Lambda_{QCD}}$ Cheng:2004ru  is the cut-off
parameter, with ${\rm\Lambda_{QCD}}$ = 220 MeV and $\alpha$ to be the model
parameter.

Then, the partial width of $B^{+}\to D^{+}X_{0}(2900)$ can be obtained by,

$\Gamma_{B^{+}\to
D^{+}X_{0}}=\frac{1}{8\pi}\frac{|\vec{p}\,|}{m_{B}^{2}}\overline{\left|\mathcal{M}_{B^{+}\to
D^{+}X_{0}}\right|^{2}}.$ (16)

Then we can obtain the branching fraction of $B^{+}\to D^{+}X_{0}(2900)$
depending on the model parameter $\alpha$.

##  III Numerical Results and Discussions

###  III.1 Coupling Constants

Before we estimate the branching fraction of $B^{+}\to D^{+}X_{0}(2900)$, some
relevant coupling constants should be further clarified. The coupling constant
of $X_{0}(2900)\bar{D}^{\ast}K^{\ast}$ could be estimated by compositeness
conditions in the molecular frame. In Ref. Xiao:2020ltm , the author estimated
the decay properties of $X_{0}(2900)$ in the molecular frame and found the
measured width of $X_{0}(2900)$ could be well reproduced. In the considered
model parameter range, the coupling constant $g_{X_{0}}$ was estimated to be
about 9.0.

As for the weak transition form factor, the estimations in Ref. Cheng:2003sm ;
Soni:2021fky  parametrized them in the form,

$F(Q^{2})=\frac{F(0)}{1-a\zeta+b\zeta^{2}}$ (17)

with $\zeta=Q^{2}/m_{B}^{2}$. The relevant parameters $F(0)$, $a$, and $b$ for
each form factor are collected in Table 1. In order to simplify the
estimations, we further parametrize Eq. (17) in the form,

$F(Q^{2})=F(0)\frac{\Lambda_{1}^{2}}{Q^{2}-\Lambda_{1}^{2}}\frac{\Lambda_{2}^{2}}{Q^{2}-\Lambda_{2}^{2}}.$
(18)

By fitting Eq. (18) with Eq. (17), we can obtain the values of $\Lambda_{1}$
and $\Lambda_{2}$ for each form factor, which are listed in Table 2.

Considering the heavy quark limit and chiral symmetry, the coupling constants
of $D_{1}DV$ in Eq. (9) are given as Ding:2008gr ,

$\displaystyle g_{D_{1}DV}$ $\displaystyle=-\sqrt{2}\zeta
g_{V}\sqrt{M_{D}M_{D_{1}}}\ \ ,$ (19)

with $g_{V}\simeq 5.8$ by imposing the Kawarabayashi-Suzuki-Riazuddin-
Fayyazuddin relations Ding:2008gr  and $\zeta=0.1$ Casalbuoni:1996pg .

Table 1: The values of the parameters $F(0)$, $a$ and $b$ in the form factors Verma:2011yw  Parameter |  $F(0)$ |  $a$ |  $b$
---|---|---|---
$A_{0}$ |  0.68 |  1.21 |  0.36
$A_{1}$ |  0.65 |  0.60 |  0.00
$A_{2}$ |  0.61 |  1.12 |  0.31
$V$ |  0.77 |  1.25 |  0.38
Table 2: Values of the parameters $\Lambda_{1}$ and $\Lambda_{2}$ obtained by fitting Eq. (18) with Eq. (17) Parameter |  $A_{0}$ |  $A_{1}$ |  $A_{2}$ |  $V$
---|---|---|---|---
$\Lambda_{1}$ |  6.5 |  8.95 |  6.65 |  6.30
$\Lambda_{2}$ |  7.1 |  8.75 |  7.55 |  7.10
Figure 3: (Color online.) The branching fraction of $B^{+}\to
X_{0}(2900)D^{+}$ (in unit of $10^{-5}$) depending on the model parameter
$\alpha$. The red horizontal line with cyan band refers to the measured
branching fraction as shown in Eq. (3).

###  III.2 Branching Fraction

With above preparations, all the relevant coupling constants have been fixed
except the model parameter $\alpha$. This phenomenological model parameter
cannot be determined by first principle methods. In our experience, $\alpha$
should be of order unity. In the present estimation, we vary $\alpha$ from $1$
to $4$ to check the parameter dependence of the branching fraction. By
reproducing the experimental data in Eq. (3), we can determine the value of
parameter $\alpha$ and check the rationality of the parameter range, which can
further test the $\bar{D}^{\ast}K^{\ast}$ molecular interpretations to the
$X_{0}(2900)$.

The branching fraction of $B^{+}\to D^{+}X_{0}(2900)$ depending on the model
parameter $\alpha$ is presented in Fig. 3, where the experimental measurement
from the LHCb Collaboration is also given for comparison. Our estimations
indicate that the branching fraction increases with the increasing of model
parameter $\alpha$. In the range $1.3<\alpha<3.2$, our estimation could well
reproduce the experimental data. From this parameter range, we can conclude
that the $\bar{D}^{\ast}K^{\ast}$ molecular interpretations to the
$X_{0}(2900)$ should be reasonable.

##  IV SUMMARY

Since the observation of $X_{0,1}(2900)$ by the LHCb collaboration, some
interpretation have been proposed to investigate their internal structure. In
particular, the tetraquark interpretations were proposed inspired by the fully
open flavor properties of $X_{0,1}(2900)$. In addition, the observed mass of
$X_{0,1}(2900)$ is close the threshold of $\bar{D}^{\ast}K^{\ast}$, and the
$J^{P}$ quantum numbers of the $S-$wave $\bar{D}^{\ast}K^{\ast}$ molecule
could be consistent with those of $X_{0}(2900)$, thus, some molecular
interpretations have been proposed for the $X_{0}(2900)$. The mass
spectroscopy and decay width of $X_{0}(2900)$ have been reproduced in the
molecular frame. To further test the rationality of the molecular
interpretation, we investigate the $X_{0}(2900)$ production process $B^{+}\to
D^{+}X_{0}(2900)$ in the molecular frame in present work.

In the molecular frame, $X_{0}(2900)$ is composed of $\bar{D}^{\ast}$ and
$K^{\ast}$, and we also find that the branching fraction of $B^{+}\to
D_{s1}^{+}(2460)\bar{D}^{\ast 0}$ is measured to be order of $10^{-2}$, which
is approximately three orders of magnitude larger than that of $B^{+}\to
D^{+}X_{0}(2900)$. Additionally, it is worth noting that $D_{s1}^{+}(2460)$
couples to $D^{+}K^{\ast}$ through an $S$-wave interaction. Thus, we can infer
that the initial $B^{+}$ meson and the final $D^{+}X_{0}(2900)$ might be
connected through $D_{s1}^{+}(2460)\bar{D}^{\ast 0}$ meson pair by exchanging
a $K^{\ast 0}$ meson. In the present work, we estimated such a meson loop
contribution to the process $B^{+}\to D^{+}X_{0}(2900)$ in the hadronic level.
In oder to compensate for the off-shell effect and avoid the ultraviolet
divergence in the loop integrals, we incorporate a form factor in the monopole
form in the amplitude. This form factor is characterized by a model parameter
$\alpha$, which should be of order unity. By comparing the experimental
measurement from LHCb Collaboration, we find the branching fraction of
$B^{+}\to D^{+}X_{0}(2900)$ could be well reproduce in the parameter range
$1.3<\alpha<3.2$, which indicate that the $\bar{D}^{\ast}K^{\ast}$ molecular
interpretations for the $X_{0}(2900)$ should be reasonable.

## ACKNOWLEDGMENTS

The authors would like to thank Prof. Fu-Sheng Yu for useful discussions. This
work is supported by the National Natural Science Foundation of China under
the Grant 11775050 and 12335001.

## References

  * (1) S. L. Olsen, T. Skwarnicki and D. Zieminska, Rev. Mod. Phys. 90 (2018) no.1, 015003 doi:10.1103/RevModPhys.90.015003 [arXiv:1708.04012 [hep-ph]].
  * (2) N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C. P. Shen, C. E. Thomas, A. Vairo and C. Z. Yuan, Phys. Rept. 873 (2020), 1-154 doi:10.1016/j.physrep.2020.05.001 [arXiv:1907.07583 [hep-ex]].
  * (3) A. Ali, J. S. Lange and S. Stone, Prog. Part. Nucl. Phys. 97 (2017), 123-198 doi:10.1016/j.ppnp.2017.08.003 [arXiv:1706.00610 [hep-ph]].
  * (4) Y. Dong, A. Faessler and V. E. Lyubovitskij, Prog. Part. Nucl. Phys. 94 (2017), 282-310 doi:10.1016/j.ppnp.2017.01.002
  * (5) Y. R. Liu, H. X. Chen, W. Chen, X. Liu and S. L. Zhu, Prog. Part. Nucl. Phys. 107, 237-320 (2019) doi:10.1016/j.ppnp.2019.04.003 [arXiv:1903.11976 [hep-ph]].
  * (6) F. K. Guo, X. H. Liu and S. Sakai, Prog. Part. Nucl. Phys. 112 (2020), 103757 doi:10.1016/j.ppnp.2020.103757 [arXiv:1912.07030 [hep-ph]].
  * (7) X. Liu, Chin. Sci. Bull. 59 (2014), 3815-3830 doi:10.1007/s11434-014-0407-2 [arXiv:1312.7408 [hep-ph]].
  * (8) F. K. Guo, C. Hanhart, U. G. Meißner, Q. Wang, Q. Zhao and B. S. Zou, Rev. Mod. Phys. 90, no.1, 015004 (2018) [erratum: Rev. Mod. Phys. 94, no.2, 029901 (2022)] doi:10.1103/RevModPhys.90.015004 [arXiv:1705.00141 [hep-ph]].
  * (9) S. K. Choi et al. [Belle], Phys. Rev. Lett. 91, 262001 (2003) doi:10.1103/PhysRevLett.91.262001 [arXiv:hep-ex/0309032 [hep-ex]].
  * (10) M. Ablikim et al. [BESIII], Phys. Rev. Lett. 111, no.24, 242001 (2013) doi:10.1103/PhysRevLett.111.242001 [arXiv:1309.1896 [hep-ex]].
  * (11) M. Ablikim et al. [BESIII], Phys. Rev. Lett. 112, no.13, 132001 (2014) doi:10.1103/PhysRevLett.112.132001 [arXiv:1308.2760 [hep-ex]].
  * (12) B. Aubert et al. [BaBar], Phys. Rev. Lett. 90, 242001 (2003) doi:10.1103/PhysRevLett.90.242001 [arXiv:hep-ex/0304021 [hep-ex]].
  * (13) D. Besson et al. [CLEO], Phys. Rev. D 68, 032002 (2003) [erratum: Phys. Rev. D 75, 119908 (2007)] doi:10.1103/PhysRevD.68.032002 [arXiv:hep-ex/0305100 [hep-ex]].
  * (14) Z. X. Xie, G. Q. Feng and X. H. Guo, Analyzing $D_{s0}^{*}(2317)^{+}$ in the $DK$ molecule picture in the Beth-Salpeter approach, Phys. Rev. D 81, 036014 (2010).
  * (15) Y. J. Zhang, H. C. Chiang, P. N. Shen and B. S. Zou, Possible $S$-wave bound-states of two pseudoscalar mesons, Phys. Rev. D 74, 014013 (2006)
  * (16) P. Bicudo, The Family of strange multiquarks as kaonic molecules bound by hard core attraction, Nucl. Phys. A 748, 537 (2005).
  * (17) A. Faessler, T. Gutsche, V. E. Lyubovitskij and Y. L. Ma, Strong and radiative decays of the $D_{s0}^{*}(2317)$ meson in the $DK$-molecule picture, Phys. Rev. D 76, 014005 (2007).
  * (18) A. Faessler, T. Gutsche, V. E. Lyubovitskij and Y. L. Ma, $D^{*}K$ molecular structure of the $D_{s1}(2460)$ meson, Phys. Rev. D 76, 114008 (2007).
  * (19) M. Cleven, H. W. Grießhammer, F. K. Guo, C. Hanhart and U. G. Meißner, Strong and radiative decays of the $D^{*}_{s0}(2317)$ and $D_{s1}(2460)$, Eur. Phys. J. A 50, no. 9, 149 (2014).
  * (20) A. Datta and P. J. O’donnell, Understanding the nature of $D_{s}(2317)$ and $D_{s}(2460)$ through nonleptonic B decays, Phys. Lett. B 572 , 164 (2003).
  * (21) M. F. M. Lutz and M. Soyeur, Open-charm meson systems in the hadrogenesis conjecture, Prog. Part. Nucl. Phys. 61, 155 (2008).
  * (22) D. S. Hwang and D. W. Kim, Mass of $D^{*}_{sJ}(2317)$ and coupled channel effect, Phys. Lett. B 601 , 137 (2004).
  * (23) S. Godfrey and R. Kokoski, Phys. Rev. D 43, 1679-1687 (1991) doi:10.1103/PhysRevD.43.1679
  * (24) J. Zeng, J. W. Van Orden and W. Roberts, Phys. Rev. D 52, 5229-5241 (1995) doi:10.1103/PhysRevD.52.5229 [arXiv:hep-ph/9412269 [hep-ph]].
  * (25) S. N. Gupta and J. M. Johnson, Phys. Rev. D 51, 168-175 (1995) doi:10.1103/PhysRevD.51.168 [arXiv:hep-ph/9409432 [hep-ph]].
  * (26) D. Ebert, V. O. Galkin and R. N. Faustov, Phys. Rev. D 57, 5663-5669 (1998) [erratum: Phys. Rev. D 59, 019902 (1999)] doi:10.1103/PhysRevD.59.019902 [arXiv:hep-ph/9712318 [hep-ph]].
  * (27) T. A. Lahde, C. J. Nyfalt and D. O. Riska, Nucl. Phys. A 674, 141-167 (2000) doi:10.1016/S0375-9474(00)00154-8 [arXiv:hep-ph/9908485 [hep-ph]].
  * (28) M. Di Pierro and E. Eichten, Phys. Rev. D 64, 114004 (2001) doi:10.1103/PhysRevD.64.114004 [arXiv:hep-ph/0104208 [hep-ph]].
  * (29) J. Liu, Q. Wu, J. He, D. Y. Chen and T. Matsuki, Phys. Rev. D 101, no.1, 014003 (2020) doi:10.1103/PhysRevD.101.014003 [arXiv:2001.00212 [hep-ph]].
  * (30) C. J. Xiao, D. Y. Chen and Y. L. Ma, Radiative and pionic transitions from the $D_{s1}(2460)$ to the $D_{s0}^{\ast}(2317)$, Phys. Rev. D 93, no. 9, 094011 (2016)
  * (31) F. K. Guo, P. N. Shen, H. C. Chiang, R. G. Ping and B. S. Zou, Phys. Lett. B 641, 278-285 (2006) doi:10.1016/j.physletb.2006.08.064 [arXiv:hep-ph/0603072 [hep-ph]].
  * (32) D. Gamermann, E. Oset, D. Strottman and M. J. Vicente Vacas, Phys. Rev. D 76, 074016 (2007) doi:10.1103/PhysRevD.76.074016 [arXiv:hep-ph/0612179 [hep-ph]].
  * (33) M. Altenbuchinger, L. S. Geng and W. Weise, Phys. Rev. D 89, no.1, 014026 (2014) doi:10.1103/PhysRevD.89.014026 [arXiv:1309.4743 [hep-ph]].
  * (34) D. L. Yao, M. L. Du, F. K. Guo and U. G. Meißner, JHEP 11, 058 (2015) doi:10.1007/JHEP11(2015)058 [arXiv:1502.05981 [hep-ph]].
  * (35) Z. H. Guo, U. G. Meißner and D. L. Yao, Phys. Rev. D 92, no.9, 094008 (2015) doi:10.1103/PhysRevD.92.094008 [arXiv:1507.03123 [hep-ph]].
  * (36) M. L. Du, F. K. Guo, U. G. Meißner and D. L. Yao, Eur. Phys. J. C 77, no.11, 728 (2017) doi:10.1140/epjc/s10052-017-5287-6 [arXiv:1703.10836 [hep-ph]].
  * (37) G. S. Bali, S. Collins, A. Cox and A. Schäfer, Phys. Rev. D 96, no.7, 074501 (2017) doi:10.1103/PhysRevD.96.074501 [arXiv:1706.01247 [hep-lat]].
  * (38) M. W. Hu, X. Y. Lao, P. Ling and Q. Wang, Chin. Phys. C 45, no.2, 021003 (2021) doi:10.1088/1674-1137/abcfaa [arXiv:2008.06894 [hep-ph]].
  * (39) R. Aaij et al. [LHCb], Phys. Rev. D 102, 112003 (2020) doi:10.1103/PhysRevD.102.112003 [arXiv:2009.00026 [hep-ex]].
  * (40) R. Aaij et al. [LHCb], Phys. Rev. Lett. 125, 242001 (2020) doi:10.1103/PhysRevLett.125.242001 [arXiv:2009.00025 [hep-ex]].
  * (41) T. Guo, J. Li, J. Zhao and L. He, Phys. Rev. D 105, no.5, 054018 (2022) doi:10.1103/PhysRevD.105.054018 [arXiv:2108.06222 [hep-ph]].
  * (42) J. R. Zhang, Phys. Rev. D 103, no.5, 054019 (2021) doi:10.1103/PhysRevD.103.054019 [arXiv:2008.07295 [hep-ph]].
  * (43) Z. G. Wang, Int. J. Mod. Phys. A 35, no.30, 2050187 (2020) doi:10.1142/S0217751X20501870 [arXiv:2008.07833 [hep-ph]].
  * (44) X. G. He, W. Wang and R. Zhu, Eur. Phys. J. C 80, no.11, 1026 (2020) doi:10.1140/epjc/s10052-020-08597-1 [arXiv:2008.07145 [hep-ph]].
  * (45) G. J. Wang, L. Meng, L. Y. Xiao, M. Oka and S. L. Zhu, Eur. Phys. J. C 81, no.2, 188 (2021) doi:10.1140/epjc/s10052-021-08978-0 [arXiv:2010.09395 [hep-ph]].
  * (46) Q. F. Lü, D. Y. Chen and Y. B. Dong, Phys. Rev. D 102, no.7, 074021 (2020) doi:10.1103/PhysRevD.102.074021 [arXiv:2008.07340 [hep-ph]].
  * (47) S. S. Agaev, K. Azizi and H. Sundu, Phys. Rev. D 106, no.1, 014019 (2022) doi:10.1103/PhysRevD.106.014019 [arXiv:2204.08498 [hep-ph]].
  * (48) H. W. Ke, Y. F. Shi, X. H. Liu and X. Q. Li, Phys. Rev. D 106, no.11, 114032 (2022) doi:10.1103/PhysRevD.106.114032 [arXiv:2210.06215 [hep-ph]].
  * (49) S. Y. Kong, J. T. Zhu, D. Song and J. He, Phys. Rev. D 104, no.9, 094012 (2021) doi:10.1103/PhysRevD.104.094012 [arXiv:2106.07272 [hep-ph]].
  * (50) B. Wang and S. L. Zhu, Eur. Phys. J. C 82, no.5, 419 (2022) doi:10.1140/epjc/s10052-022-10396-9 [arXiv:2107.09275 [hep-ph]].
  * (51) H. X. Chen, Phys. Rev. D 105, no.9, 094003 (2022) doi:10.1103/PhysRevD.105.094003 [arXiv:2103.08586 [hep-ph]].
  * (52) S. S. Agaev, K. Azizi and H. Sundu, J. Phys. G 48, no.8, 085012 (2021) doi:10.1088/1361-6471/ac0b31 [arXiv:2008.13027 [hep-ph]].
  * (53) H. X. Chen, W. Chen, R. R. Dong and N. Su, Chin. Phys. Lett. 37, no.10, 101201 (2020) doi:10.1088/0256-307X/37/10/101201 [arXiv:2008.07516 [hep-ph]].
  * (54) H. Mutuk, J. Phys. G 48, no.5, 055007 (2021) doi:10.1088/1361-6471/abeb7f [arXiv:2009.02492 [hep-ph]].
  * (55) M. Z. Liu, J. J. Xie and L. S. Geng, Phys. Rev. D 102, no.9, 091502 (2020) doi:10.1103/PhysRevD.102.091502 [arXiv:2008.07389 [hep-ph]].
  * (56) C. J. Xiao, D. Y. Chen, Y. B. Dong and G. W. Meng, Phys. Rev. D 103, no.3, 034004 (2021) doi:10.1103/PhysRevD.103.034004 [arXiv:2009.14538 [hep-ph]].
  * (57) Y. Huang, J. X. Lu, J. J. Xie and L. S. Geng, Eur. Phys. J. C 80, no.10, 973 (2020) doi:10.1140/epjc/s10052-020-08516-4 [arXiv:2008.07959 [hep-ph]].
  * (58) E. S. Swanson, Nucl. Part. Phys. Proc. 312-317, 130-134 (2021) doi:10.1016/j.nuclphysbps.2021.05.033
  * (59) T. J. Burns and E. S. Swanson, Phys. Rev. D 103, no.1, 014004 (2021) doi:10.1103/PhysRevD.103.014004 [arXiv:2009.05352 [hep-ph]].
  * (60) Q. Y. Lin and X. Y. Wang, Eur. Phys. J. C 82, no.11, 1017 (2022) doi:10.1140/epjc/s10052-022-10995-6 [arXiv:2209.06062 [hep-ph]].
  * (61) Y. K. Hsiao and Y. Yu, Phys. Rev. D 104, no.3, 034008 (2021) doi:10.1103/PhysRevD.104.034008 [arXiv:2104.01296 [hep-ph]].
  * (62) Y. K. Chen, J. J. Han, Q. F. Lü, J. P. Wang and F. S. Yu, Eur. Phys. J. C 81, no.1, 71 (2021) doi:10.1140/epjc/s10052-021-08857-8 [arXiv:2009.01182 [hep-ph]].
  * (63) H. Y. Cheng, C. K. Chua and C. W. Hwang, Phys. Rev. D 69, 074025 (2004) doi:10.1103/PhysRevD.69.074025 [arXiv:hep-ph/0310359 [hep-ph]].
  * (64) N. R. Soni, A. Issadykov, A. N. Gadaria, Z. Tyulemissov, J. J. Patel and J. N. Pandya, Eur. Phys. J. Plus 138, no.2, 163 (2023) doi:10.1140/epjp/s13360-023-03779-8 [arXiv:2110.12740 [hep-ph]].
  * (65) M. Bauer, B. Stech and M. Wirbel, Z. Phys. C 34, 103 (1987) doi:10.1007/BF01561122
  * (66) P. A. Zyla et al. [Particle Data Group], PTEP 2020, no.8, 083C01 (2020) doi:10.1093/ptep/ptaa104
  * (67) A. Ali, G. Kramer and C. D. Lu, Phys. Rev. D 58, 094009 (1998) doi:10.1103/PhysRevD.58.094009 [arXiv:hep-ph/9804363 [hep-ph]].
  * (68) M. A. Ivanov, J. G. Korner and P. Santorelli, Phys. Rev. D 73, 054024 (2006) doi:10.1103/PhysRevD.73.054024 [arXiv:hep-ph/0602050 [hep-ph]].
  * (69) R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio and G. Nardulli, Phys. Rept. 281, 145-238 (1997) doi:10.1016/S0370-1573(96)00027-0 [arXiv:hep-ph/9605342 [hep-ph]].
  * (70) R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio and G. Nardulli, Phys. Lett. B 292, 371-376 (1992) doi:10.1016/0370-2693(92)91189-G [arXiv:hep-ph/9209248 [hep-ph]].
  * (71) R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio and G. Nardulli, Phys. Lett. B 299, 139-150 (1993) doi:10.1016/0370-2693(93)90895-O [arXiv:hep-ph/9211248 [hep-ph]].
  * (72) H. Y. Cheng, C. K. Chua and A. Soni, Phys. Rev. D 71, 014030 (2005) doi:10.1103/PhysRevD.71.014030 [arXiv:hep-ph/0409317 [hep-ph]].
  * (73) R. C. Verma, J. Phys. G 39, 025005 (2012) doi:10.1088/0954-3899/39/2/025005 [arXiv:1103.2973 [hep-ph]].
  * (74) G. J. Ding, Phys. Rev. D 79, 014001 (2009) doi:10.1103/PhysRevD.79.014001 [arXiv:0809.4818 [hep-ph]].