#  Unified description of the productions of $\bar{D}^{*}D$ and
$\bar{D}^{*}D^{*}$ molecules in $B$ decays

Qi Wu  School of Physics and Center of High Energy Physics, Peking University,
Beijing 100871, China  Ming-Zhu Liu  Corresponding author:
<EMAIL_ADDRESS>School of Space and Environment, Beihang University,
Beijing 102206, China  School of Physics, Beihang University, Beijing 102206,
China  Li-Sheng Geng  Corresponding author<EMAIL_ADDRESS>School of
Physics, Beihang University, Beijing 102206, China  Peng Huanwu Collaborative
Center for Research and Education, Beihang University, Beijing 100191, China
Beijing Key Laboratory of Advanced Nuclear Materials and Physics, Beihang
University, Beijing, 102206, China  Southern Center for Nuclear-Science Theory
(SCNT), Institute of Modern Physics, Chinese Academy of Sciences, Huizhou
516000, Guangdong Province, China

###### Abstract

The exotic states $X(3872)$ and $Z_{c}(3900)$ have long been conjectured as
isoscalar and isovector $\bar{D}^{*}D$ molecules, respectively. In this
letter, we propose a unified framework to understand the productions of
$\bar{D}^{*}D$ molecules as well as their heavy quark spin symmetry partners,
$\bar{D}^{*}D^{*}$ molecules, in $B$ decays. We show that the large isospin
breaking of the ratio $\mathcal{B}[B^{+}\to X(3872)K^{+}]/\mathcal{B}[B^{0}\to
X(3872)K^{0}]$ can be attributed to the isospin breaking of the $\bar{D}^{*}D$
neutral and charged components. Because of this, the branching fractions of
$Z_{c}(3900)$ in $B$ decays are smaller than the corresponding ones of
$X(3872)$ by at least one order of magnitude, which naturally explains the
non-observation of $Z_{c}(3900)$ in $B$ decays. Furthermore, we predict a
hierarchy for the productions fractions of all the $\bar{D}^{*}D$ and
$\bar{D}^{*}D^{*}$ molecules in $B$ decays, which are consistent with all the
existing data and can help elucidate the internal structure of the $XZ$ states
around the $\bar{D}^{*}D$ and $\bar{D}^{*}D^{*}$ mass thresholds, if confirmed
by future experiments.

Introduction.— The $X(3872)$ was discovered in 2003 by the Belle Collaboration
Choi  _et al._ (2003) and later confirmed in many other experiments Aubert
_et al._ (2004); Acosta  _et al._ (2004); Abazov  _et al._ (2004); Chatrchyan
_et al._ (2013); Aaij  _et al._ (2012); Ablikim  _et al._ (2014); Aaij  _et
al._ (2013). Its mass, $3871.69\pm 0.17$ MeV, is lower than the prediction of
the Goldfrey-Isgur model Godfrey and Isgur (1985) by almost 90 MeV. In
addition, the ratio $\mathcal{B}[X(3872)\to
J/\psi\pi^{+}\pi^{-}\pi^{0}]/\mathcal{B}[X(3872)\to J/\psi\pi^{+}\pi^{-}]$ Abe
_et al._ (2005); del Amo Sanchez  _et al._ (2010); Ablikim  _et al._ (2019)
shows large isospin breaking effects, difficult to understand for a
conventional charmonium. However, both its low mass Swanson (2004); Voloshin
(2004); AlFiky  _et al._ (2006); Liu  _et al._ (2008); Sun  _et al._ (2011);
Nieves and Valderrama (2012); Guo  _et al._ (2013); Karliner and Rosner (2015)
and the large isospin breaking in its decays Braaten and Kusunoki (2005a);
Gamermann and Oset (2009); Ortega  _et al._ (2010); Hanhart  _et al._ (2012);
Li and Zhu (2012); Takeuchi  _et al._ (2014); Zhou and Xiao (2018); Wu  _et
al._ (2021) can be well described if $X(3872)$ contains a large $\bar{D}^{*}D$
molecule component. In 2013, the BESIII Collaboration and Belle Collaboration
observed a charged charmonium-like state $Z_{c}(3900)$ in the
$J/\psi\pi^{\pm}$ mass distribution of $e^{+}e^{-}\to J/\psi\pi^{+}\pi^{-}$
Ablikim  _et al._ (2013a); Liu  _et al._ (2013), which is above the mass
threshold of $\bar{D}^{*}D$ and has been explained as a $\bar{D}^{*}D$
resonant state and the isospin partner of $X(3872)$ Albaladejo  _et al._
(2016); Karliner and Rosner (2015). Recently, the molecular picture received
renewed attention and further support Yang  _et al._ (2021); Meng  _et al._
(2020); Baru  _et al._ (2022); Yan  _et al._ (2021); Du  _et al._ (2022) after
its SU(3)-flavor partner, $Z_{cs}(3985)$, was discovered Ablikim  _et al._
(2021).

Figure 1:  Triangle diagrams accounting for (a-c)$B^{+}({B}^{0})\to
D_{s}^{(*)+}\bar{D}^{(*)0}(D^{(\ast)+})\to X(3872)K^{+}(K^{0})$ and
(d-e)$B^{+}({B}^{0})\to D_{s}^{(*)+}\bar{D}^{(*)0}(D^{(\ast)+})\to
X_{2}(4013)K^{+}(K^{0})$.

Treating $X(3872)$ and $Z_{c}(3900)$ as $\bar{D}^{*}D$ molecules, heavy quark
spin symmetry (HQSS) implies the existence of two $\bar{D}^{*}D^{*}$
molecules, i.e., a $J^{PC}=2^{++}$ bound state Nieves and Valderrama (2012);
Guo  _et al._ (2013); Baru  _et al._ (2016) and a $J^{PC}=1^{+-}$ resonant
state Yang  _et al._ (2021); Meng  _et al._ (2020); Baru  _et al._ (2022); Yan
_et al._ (2021); Du  _et al._ (2022). The former may correspond to the
$X(4014)$ state newly discovered in the $\gamma\psi(2S)$ mass distribution of
$\gamma\gamma\to\gamma\psi(2S)$ by the Belle Collaboration Wang  _et al._
(2022), and the latter may correspond to the resonant state $Z_{c}(4020)$
discovered in the $\pi^{\pm}h_{c}$ mass distribution of $e^{+}e^{-}\to
h_{c}\pi^{+}\pi^{-}$ by the BESIII Collaboration Ablikim  _et al._ (2013b). It
is natural to expect that all the four states, as $\bar{D}^{*}D^{(*)}$
molecules, should share many common features. That is to say, not only their
masses and decays, but also their productions can be understood in a unified
framework.

The production mechanism of $X(3872)$ in $B$ decays was firstly proposed by
Braaten et al. Braaten  _et al._ (2004); Braaten and Kusunoki (2005b), where
the $B$ meson first decays into $\bar{D}^{\ast}DK$ and then the charmed mesons
rescatter and dynamically generate the $X(3872)$. The predicted ratio
$\mathcal{B}[B^{0}\to X(3872)K^{0}]/\mathcal{B}[B^{+}\to X(3872)K^{+}]$
depends on two parameters and the natural value is one order of magnitude
smaller than its experimental counterpart. This ratio is reasonably described
in Ref. Wang  _et al._ (2023), but not the absolute branching fractions. It is
important to note that up to now, a complete understanding of the ratio
$\mathcal{B}[B^{0}\to X(3872)K^{0}]/\mathcal{B}[B^{+}\to X(3872)K^{+}]$ and
the absolute branching fractions in a unified framework is still missing.
Another puzzle related to the four $\bar{D}^{*}D^{(*)}$ molecules is that the
$X(3872)$ has been observed in multiple channels of $B$ decays, but the other
three have not, which calls for an explanation. Furthermore, for planning
future experiments, it is imperative to know the branching fractions of the
other three states in $B$ decays, given the fact that $B$ decays have served
as important discovery channels for many exotic states and more data can be
expected from $e^{+}e^{-}$ colliders.

In this letter, we propose to test the molecular nature of these states in $B$
decays in a unified framework. We first demonstrate that this framework can
predict both the absolute branching fractions of $\mathcal{B}[B^{+}\to
X(3872)K^{+}]$ and $\mathcal{B}[B^{0}\to X(3872)K^{0}]$ and their ratio
without any free parameters and in nice agreement with the existing data. In
particular, we show how the interesting interplay between the neutral and
charged $\bar{D}^{*}D$ channels generates the large isospin breaking effect.
The same interplay is responsible for the non-observation of the
$Z_{c}(3900)$. We then further predict the branching fractions of the two
$\bar{D}^{*}D^{*}$ molecules.

Production mechanism.— In this letter, we propose the triangle mechanism to
account for the productions of the $\bar{D}^{*}D$ and $\bar{D}^{*}D^{\ast}$
molecules in $B$ decays. In this mechanism, the $B$ meson first weakly decays
into a pair of charmed mesons $D_{s}^{(\ast)}\bar{D}^{(\ast)}$, which proceed
via the $W$-emission mechanism at the quark level as shown in Fig. 3 in the
Supplemental Material. We only consider the $W$-emission mechanism because it
is usually the dominant one Chau (1983); Chau and Cheng (1987); Molina  _et
al._ (2020). As shown later, our results corroborate this assumption. Next,
the charmed-strange mesons $D_{s}^{(\ast)}$ decay into a charmed meson
$D^{(\ast)}$ and a kaon. Finally the $\bar{D}D^{*}$ and $\bar{D}^{*}D^{*}$
molecules are dynamically generated via the final-state interactions of
$\bar{D}^{(*)}D^{*}$ as shown in Fig. 1.

Here the isoscalar $\bar{D}D^{*}$ and $\bar{D}^{*}D^{*}$ molecules refer to
$X(3872)$ and $X_{2}(4013)$, and the isovector counterparts are $Z_{c}(3900)$
and $Z_{c}(4020)$. We do not explicitly present the triangle diagrams for the
$Z_{c}(3900)$ and $Z_{c}(4020)$, which can be obtained by replacing the
$X(3872)$ of Fig. 1 and $X_{2}(4013)$ of Fig. 2 with $Z_{c}(3900)^{0}$ and
$Z_{c}(4020)^{0}$, respectively. We note that recently a similar mechanism was
applied to investigate the branching fractions of the exited charmed strange
mesons $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ in $B$ decays, which indicated
that the molecular components account for a large portion of the
$D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ wave functions Liu  _et al._ (2022).

In the following, we employ the effective Lagrangian approach to calculate the
Feynman diagrams of Fig. 1. The Lagrangians and the determination of the
relevant couplings either by fitting to data or relying on symmetries are
presented in the Supplemental Material. By solving the Lippmann-Schwinger
equation, we obtain the pole positions of the $\bar{D}^{*}D$ molecules, and
then derive the couplings from the residues of the poles Xie  _et al._
(2022a), where the $\bar{D}^{*}D$ interactions in the contact-range effective
field theory are determined by dynamically generating $X(3872)$ and
$Z_{c}(3900)$ (see the Supplemental Material for details). With HQSS, we
predict the $\bar{D}^{*}D^{*}$ molecules, $X_{2}(4013)$ and $Z_{c}(4020)$, and
then obtain the corresponding couplings in the same approach. The couplings of
the four molecules to their constituents are listed in Table 3 of the
Supplemental Material.

Now it is straightforward to calculate the Feynman diagrams of Fig. 1 and
obtain the following amplitudes

$\displaystyle\mathcal{A}_{a}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}q_{3}}{(2\pi)^{4}}\frac{{\rm
i}\mathcal{A}(B^{+}\to
D_{s}^{+}\bar{D}^{0})\mathcal{A}\left(D_{s}^{+}\to{D}^{\ast
0}K^{+}\right)\mathcal{A}\left({D}^{\ast 0}\bar{D}^{0}\to
X(3872)\right)}{\left(q_{1}^{2}-m_{D_{s}^{+}}^{2}\right)\left(q_{2}^{2}-m_{\bar{D}^{0}}^{2}\right)\left(q_{3}^{2}-m_{{D}^{\ast
0}}^{2}\right)},$ (1) $\displaystyle\mathcal{A}_{b}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}q_{3}}{(2\pi)^{4}}\frac{{\rm
i}\mathcal{A}(B^{+}\to
D_{s}^{\ast+}\bar{D}^{0})\mathcal{A}\left(D_{s}^{\ast+}\to{D}^{\ast
0}K^{+}\right)\mathcal{A}\left({D}^{\ast 0}\bar{D}^{0}\to
X(3872)\right)}{\left(q_{1}^{2}-m_{D_{s}^{\ast+}}^{2}\right)\left(q_{2}^{2}-m_{\bar{D}^{0}}^{2}\right)\left(q_{3}^{2}-m_{{D}^{\ast
0}}^{2}\right)},$ (2) $\displaystyle\mathcal{A}_{c}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}q_{3}}{(2\pi)^{4}}\frac{{\rm
i}\mathcal{A}(B^{+}\to D_{s}^{\ast+}\bar{D}^{\ast
0})\mathcal{A}\left(D_{s}^{\ast+}\to{D}^{0}K^{+}\right)\mathcal{A}\left({D}^{0}\bar{D}^{\ast
0}\to
X(3872)\right)}{\left(q_{1}^{2}-m_{D_{s}^{\ast+}}^{2}\right)\left(q_{2}^{2}-m_{\bar{D}^{\ast
0}}^{2}\right)\left(q_{3}^{2}-m_{{D}^{0}}^{2}\right)},$ (3)
$\displaystyle\mathcal{A}_{d}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}q_{3}}{(2\pi)^{4}}\frac{{\rm
i}\mathcal{A}(B^{+}\to D_{s}^{+}\bar{D}^{\ast
0})\mathcal{A}\left(D_{s}^{+}\to{D}^{\ast
0}K^{+}\right)\mathcal{A}\left({D}^{\ast 0}\bar{D}^{\ast 0}\to
X_{2}(4013)\right)}{\left(q_{1}^{2}-m_{D_{s}^{+}}^{2}\right)\left(q_{2}^{2}-m_{\bar{D}^{\ast
0}}^{2}\right)\left(q_{3}^{2}-m_{{D}^{\ast 0}}^{2}\right)},$ (4)
$\displaystyle\mathcal{A}_{e}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}q_{3}}{(2\pi)^{4}}\frac{{\rm
i}\mathcal{A}(B^{+}\to D_{s}^{\ast+}\bar{D}^{\ast
0})\mathcal{A}\left(D_{s}^{\ast+}\to{D}^{\ast
0}K^{+}\right)\mathcal{A}\left({D}^{\ast 0}\bar{D}^{\ast 0}\to
X_{2}(4013)\right)}{\left(q_{1}^{2}-m_{D_{s}^{\ast+}}^{2}\right)\left(q_{2}^{2}-m_{\bar{D}^{\ast
0}}^{2}\right)\left(q_{3}^{2}-m_{{D}^{\ast 0}}^{2}\right)}.$ (5)

where $q_{1}$, $q_{2}$ and $q_{3}$ denote the momenta of $D_{s}^{(\ast)}$,
$\bar{D}^{(\ast)}$ and $D^{(\ast)}$, and the amplitudes for each vertex of the
triangle diagrams are listed in the Supplemental Material.

With the amplitudes for the decays of $B\to X(3872)/X_{2}(4013)K$ and $B\to
Z_{c}(3900)/Z_{c}(4020)K$ given above, one can compute the corresponding
partial decay widths

$\displaystyle\Gamma=\frac{1}{2J+1}\frac{1}{8\pi}\frac{|\vec{p}|}{m_{B}^{2}}{|\overline{M}|}^{2},$
(6)

where $J$ is the total angular momentum of the initial $B$ meson, the overline
indicates the sum over the polarization vectors of final states, and
$|\vec{p}|$ is the momentum of either final state in the rest frame of the $B$
meson.

Table 1: Ratios of the couplings in particle basis to the couplings in isospin basis.  Molecules |  ${D}^{\ast+}D^{-}$ |  ${D}^{+}D^{\ast-}$ |  ${D}^{\ast 0}\bar{D}^{0}$ |  ${D}^{0}\bar{D}^{\ast 0}$
---|---|---|---|---
$X(3872)$ |  $1/2$ |  $-1/2$ |  $1/2$ |  $-1/2$
$Z_{c}(3900)$ |  $1/2$ |  $1/2$ |  $-1/2$ |  $-1/2$
Molecules |  ${D}^{\ast+}D^{\ast-}$ |  ${D}^{\ast 0}\bar{D}^{\ast 0}$ |  |
$X_{2}(4013)$ |  $1/\sqrt{2}$ |  $1/\sqrt{2}$ |  |
$Z_{c}(4020)$ |  $1/\sqrt{2}$ |  $-1/\sqrt{2}$ |  |

Results and discussions.— As a $\bar{D}^{*}D$ bound state, $X(3872)$ contains
both a neutral component $\bar{D}^{*0}D^{0}/\bar{D}^{0}D^{\ast 0}$ and a
charged component $D^{*+}D^{-}/D^{+}D^{\ast-}$. The couplings to the neutral
and charged components are found to be, $g_{n}=3.86$ GeV and $g_{c}=3.39$ GeV,
which indicates that the neutral components play a more important role than
the charged component, consistent with the conclusions of Refs. Ortega  _et
al._ (2010); Gamermann and Oset (2009); Li and Zhu (2012); Guo  _et al._
(2014); Zhou and Xiao (2018); Yamaguchi  _et al._ (2020). Employing HQSS, we
obtain the potentials of the $\bar{D}^{*0}D^{\ast 0}/D^{*+}D^{\ast-}$ system
and predict the existence of a $J^{PC}=2^{++}$ bound state with a mass of
$m=4013.03$ MeV, corresponding to $X_{2}(4013)$. Similarly, the $X_{2}(4013)$
couplings to its neutral and charged components are estimated to be
$g_{n}^{\prime}=5.36$ GeV and $g_{c}^{\prime}=4.86$ GeV. Because the
$Z_{c}(3900)$ is located above the mass thresholds of the neutral and charged
components of $\bar{D}^{*}D$ by about 10 MeV, isospin breaking effects are
expected to be small. Therefore, we deal with the $Z_{c}$ states in the
isospin limit. By reproducing the mass and width of $Z_{c}(3900)$, we obtain
the coupling $g_{Z_{c}(3900)\bar{D}D^{*}}=7.10$ GeV. The HQSS dictates the
existence of a $\bar{D}^{*}D^{*}$ molecule with M$=4028$ MeV and $\Gamma=26$
MeV, whose coupling is estimated to be $g_{Z_{c}(4020)\bar{D}^{*}D^{*}}=1.77$.
In Table 1, we present the couplings in particle basis in terms of those in
isospin basis. We note that the couplings to the charged component and those
to the neutral component are of the same sign for the isoscalar states but are
of the opposite sign for the isovector states, which has important impact on
our understanding of the production of these molecules in $B$ decays as shown
below.

In our study, the dominant uncertainties originate from the couplings of the
three vertices contained in the triangle diagrams. For the weak interaction
vertices, the experimental uncertainties of the branching fractions of
$B\to\bar{D}^{(\ast)}D_{s}^{(\ast)}$ lead to about $10\%$ uncertainty for the
Wilson coefficient $a_{1}$ Xie  _et al._ (2022a). For the vertices describing
the dynamical generation of hadronic molecules, the uncertainties are mainly
from the cutoff $\Lambda$ of the form factor. If we increase the cutoff from 1
to $2$ GeV, the couplings decrease by about $10\%$. Therefore, we assign a
$10\%$ uncertainty for the couplings of the molecules to their constituents
Xie  _et al._ (2022a), a bit larger than the estimation for a cutoff variation
from $0.5$ GeV to $1$ GeV Guo  _et al._ (2014). As for the couplings
$g_{D_{s}^{(\ast)}D^{(\ast)}K}$ the large SU(4)-flavor symmetry breaking can
lead to an uncertainty of about $33\%$ 111The SU(3)-flavor symmetry breaking
can be characterized by the difference between the decay constants $f_{K}$ and
$f_{\pi}$, which is about $19\%$ Follana  _et al._ (2008); Carrasco  _et al._
(2015); Miller  _et al._ (2020). Along this line, the SU(4)-flavor symmetry
breaking is estimated to be about $33\%$ via the decay constants $f_{D}$ and
$f_{K}$ Follana  _et al._ (2008); Carrasco  _et al._ (2015), consistent with
Ref. Fontoura  _et al._ (2017). . Finally, we obtain the uncertainties of the
branching fractions induced by the uncertainties of these parameters via a
Monte Carlo sampling in their 1$\sigma$ intervals.

Figure 2: Top: branching fractions of $B^{+(0)}\to X(3872)K^{+(0)}$ (green
block), $B^{+(0)}\to X(4013)K^{+(0)}$ (blue block), $B^{+(0)}\to
Z_{c}(3900)K^{+(0)}$ (blue block) and $B^{+(0)}\to Z_{c}(4020)K^{+(0)}$
(yellow block). The left and right data points in each block are for the
$B^{+}$ an $B^{0}$ decays, respectively. Bottom: the corresponding ratios
between the branching fractions of $B^{+}$ and those of $B^{0}$ . The red
error bars and shadow parts are the corresponding experimental data.

In Fig. 2, we compare the predicted branching fractions of $\bar{D}^{*}D$ and
$\bar{D}^{*}D^{*}$ molecules in $B$ decays with the available experimental
data. The numbers are given in Table 4 of the Supplemental Material. One can
see that the branching fractions of the decays $B^{+}\to X(3872)K^{+}$ and
$B^{0}\to X(3872)K^{0}$ are in reasonable agreement with the experimental
data. We further compute the ratio $\mathcal{B}[B^{0}\to
X(3872)K^{0}]/\mathcal{B}[B^{+}\to X(3872)K^{+}]$ to be $0.62\pm 0.13$, in
agreement with the experimental value $0.52\pm 0.26$ within uncertainties. We
note that the uncertainty of the predicted ratio is much smaller than those of
the branching fractions. We stress that the fact that the branching fractions
of $X(3872)$ in $B$ decays can be reproduced in the $\bar{D}^{*}D$ molecular
picture provides a non-trivial support for the nature of $X(3872)$ as a
$\bar{D}^{*}D$ bound state.

The branching fractions of $\mathcal{B}[B^{+}\to Z_{c}(3900)K^{+}]$ and
$\mathcal{B}[B^{0}\to Z_{c}(3900)K^{0}]$ are about $(1.0\sim 3.3)\times
10^{-5}$ and $(0.6\sim 2.0)\times 10^{-5}$. The upper limit of the
experimental branching fraction $\mathcal{B}[B^{+}\to
Z_{c}(3900)(Z_{c}(3900)\to\eta_{c}\pi^{+}\pi^{-})K^{+}]$ is $4.7\times
10^{-5}$. Although due to the unknown branching fraction of
$\mathcal{B}[Z_{c}(3900)\to\eta_{c}\pi^{+}\pi^{-}]$, we can not determine
$\mathcal{B}[B^{+}\to Z_{c}(3900)K^{+}]$, our prediction is safely below the
experimental upper limit. We note that the ratio $\mathcal{B}[B^{+}\to
Z_{c}(3900)K^{+}]/\mathcal{B}[B^{0}\to Z_{c}(3900)K^{0}]=0.63\pm 0.29$ shows
large isospin breaking effects. However, unlike the case of $X(3872)$ and
$X_{2}(4013)$, this is not due to isospin breaking of the wave functions but
is mainly caused by the Wilson coefficient $a_{1}$ fitted to the $B^{+(0)}\to
D_{s}^{+}\bar{D}^{0}(D^{-})$ and $B^{+(0)}\to D_{s}^{\ast+}\bar{D}^{\ast
0}(D^{\ast-})$ decays (see the Supplemental Material for details). It is
interesting to compare the branching fractions of $\mathcal{B}[B\to
Z_{c}(3900)K]$ with those of $\mathcal{B}[B\to X(3872)K]$. The former are
smaller than the latter by one order of magnitude, which is consistent with
the fact that the $Z_{c}(3900)$ state has not been observed in $B$ decays. We
note that only the amplitude of Fig. 1 (a) and that of Fig. 1 (c) contribute
to the decays of the $B$ meson into the $\bar{D}^{*}D$ molecules, while the
contribution of Fig. (1b) is accidentally very small. The sign of the
amplitude of Fig. 1 (a) and that of Fig. 1 (c) depend on the sign of the
isospin wave functions of the $\bar{D}^{*}D$ molecules. From Table 1 we can
see that the sign is opposite for the isocalar molecules but the same for the
isovector molecules. As the two amplitudes for the isoscalar molecules add
constructively, but those for the isovector molecules add destructively, the
production rates of $Z_{c}(3900)$ in $B$ decays are lower than those of
$X(3872)$ in $B$ decays.

We now turn to the branching fractions of $X_{2}(4013)$ and $Z_{c}(4020)$. The
predicted branching fractions of $\mathcal{B}[B^{+}\to X_{2}(4013)K^{+}]$ and
$\mathcal{B}[B^{0}\to X_{2}(4013)K^{0}]$ are $(1.5\sim 3.1)\times 10^{-5}$ and
$(1.1\sim 2.3)\times 10^{-5}$, and the ratio $\mathcal{B}[B^{0}\to
X_{2}(4013)K^{0}]/\mathcal{B}[B^{+}\to X_{2}(4013)K^{+}]$ is estimated to be
$0.75\pm 0.16$. We note that the isospin breaking of the ratio is mainly
caused by the isospin breaking of the $\bar{D}^{*}D^{*}$ wave function.
Similarly, we predict the branching fractions $\mathcal{B}[B\to Z_{c}(4020)K]$
to be around $1\times 10^{-6}$, which are lower than those of
$\mathcal{B}[B\to Z_{c}(3900)K]$ as well as $\mathcal{B}[B\to X_{2}(4013)K]$
by one order of magnitude. This implies that it will be more difficult to
observe them in $B$ decays.

A few remarks are in order. In this study, we only considered the dominant
$\bar{D}^{*}D^{(\ast)}$ contribution to the $XZ$ states. However, other
channels, such as $\bar{D}_{s}D_{s}$, $\bar{D}^{*}D^{*}$ and
$\bar{D}_{s}^{*}D_{s}^{*}$, can also play a role in forming the $X(3872)$
Gamermann and Oset (2009); Ji  _et al._ (2022a). In addition, the $X(3872)$
may contain a $c\bar{c}$ component Yamaguchi  _et al._ (2020). The fact that
the $\bar{D}^{*}D$ contribution alone can describe the branching fractions of
$X(3872)$ in $B$ decays indicates that $X(3872)$ contains a sizable or
dominant $\bar{D}^{*}D$ component. As for $Z_{c}(3900)$, purely based on the
$\bar{D}^{*}D$ invariant mass distributions, it can also be explained either
as a cusp effect or as a virtual state Wang  _et al._ (2013); Chen  _et al._
(2013); Liu and Li (2013); Swanson (2015); Albaladejo  _et al._ (2016);
Pilloni  _et al._ (2017); Ortega  _et al._ (2019), which would affect its
couplings to $\bar{D}^{*}D$ and therefore modify $\mathcal{B}[B\to
Z_{c}(3900)K]$. As a result, future experimental measurements of $Z_{c}(3900)$
in $B$ decays will help either confirm or refute its nature as a
$\bar{D}^{*}D$ resonant state.

Summary.— In this letter, we proposed a unified framework to compute the
branching fractions of $\bar{D}^{*}D$ and $\bar{D}^{*}D^{*}$ molecules in $B$
decays, where the former molecules refer to $X(3872)$ and $Z_{c}(3900)$, and
the latter to $X_{2}(4013)$ and $Z_{c}(4020)$. Our framework, with no free
parameters, predicted the branching fractions of $\mathcal{B}[B^{+}\to
X(3872)K^{+}]$ and $\mathcal{B}[B^{0}\to X(3872)K^{0}]$, $(0.87\sim
2.11)\times 10^{-4}$ and $(0.54\sim 1.32)\times 10^{-4}$, consistent with the
experimental data. The branching fractions of $\mathcal{B}[B^{+}\to
Z_{c}(3900)K^{+}]$ and $\mathcal{B}[B^{0}\to Z_{c}(3900)K^{0}]$ are found to
be about the order of $10^{-5}$, smaller than the experimental upper limit.
Moreover, we predicted the branching fractions of $\mathcal{B}[B\to
X_{2}(4013)K]$ and $\mathcal{B}[B\to Z_{c}(4020)K]$ to be of the order of
$10^{-5}$ and $10^{-6}$.

We emphasize that the ratios of branching fractions are more precise than the
absolute branching fractions in our framework and can provide more insights
into the molecular nature of the states studied. The ratios of
$\mathcal{B}[B^{+}\to X(3872)K^{+}]/\mathcal{B}[B^{0}\to X(3872)K^{0}]$ and
$\mathcal{B}[B^{+}\to X_{2}(4013)K^{+}]/\mathcal{B}[B^{0}\to
X_{2}(4013)K^{0}]$ were found about $0.62$ and $0.75$, the former consistent
with the experimental data. The large isospin breaking effects are attributed
to the isospin breaking of the $\bar{D}^{*}D$ and $\bar{D}^{*}D^{*}$ neutral
and charged components. On the other hand, the isospin breaking ratio
$\mathcal{B}[B^{+}\to Z_{c}(3900)K^{+}]/\mathcal{B}[B^{0}\to
Z_{c}(3900)K^{0}]=0.63$ mainly originates from the Wilson coefficient $a_{1}$
determined by fitting to the weak decay processes of $B^{+(0)}\to
D_{s}^{+}\bar{D}^{0}(D^{-})$ and $B^{+(0)}\to D_{s}^{\ast+}\bar{D}^{\ast
0}(D^{\ast-})$. In addition, our results show that the branching fractions of
$\mathcal{B}[B\to Z_{c}(3900)K]$ are smaller than those of $\mathcal{B}[B\to
X(3872)K]$ by one order of magnitude, which is consistent with the fact
$Z_{c}(3900)$ has not been observed in $B$ decays. The predicted hierarchy in
the branching fractions of $\mathcal{B}[B\to Z_{c}(3900)K]$ and
$\mathcal{B}[B\to X(3872)K]$ serve as a highly nontrivial test on the
molecular nature of $X(3872)$ and $Z_{c}(3900)$ and should be checked by
future experiments.

Acknowledgement.— This work is supported in part by the National Natural
Science Foundation of China under Grants No.11975041 and No.11961141004. Ming-
Zhu Liu acknowledges support from the National Natural Science Foundation of
China under Grant No.12105007 and China Postdoctoral Science Foundation under
Grants No. 2022M710317, and No. 2022T150036.

## References

  * Choi  _et al._ (2003) S. K. Choi _et al._ (Belle), Phys. Rev. Lett. 91, 262001 (2003), arXiv:hep-ex/0309032 .
  * Aubert  _et al._ (2004) B. Aubert _et al._ (BaBar), Phys. Rev. Lett. 93, 041801 (2004), arXiv:hep-ex/0402025 .
  * Acosta  _et al._ (2004) D. Acosta _et al._ (CDF), Phys. Rev. Lett. 93, 072001 (2004), arXiv:hep-ex/0312021 .
  * Abazov  _et al._ (2004) V. M. Abazov _et al._ (D0), Phys. Rev. Lett. 93, 162002 (2004), arXiv:hep-ex/0405004 .
  * Chatrchyan  _et al._ (2013) S. Chatrchyan _et al._ (CMS), JHEP 04, 154 (2013), arXiv:1302.3968 [hep-ex] .
  * Aaij  _et al._ (2012) R. Aaij _et al._ (LHCb), Eur. Phys. J. C 72, 1972 (2012), arXiv:1112.5310 [hep-ex] .
  * Ablikim  _et al._ (2014) M. Ablikim _et al._ (BESIII), Phys. Rev. Lett. 112, 092001 (2014), arXiv:1310.4101 [hep-ex] .
  * Aaij  _et al._ (2013) R. Aaij _et al._ (LHCb), Phys. Rev. Lett. 110, 222001 (2013), arXiv:1302.6269 [hep-ex] .
  * Godfrey and Isgur (1985) S. Godfrey and N. Isgur, Phys. Rev. D 32, 189 (1985).
  * Abe  _et al._ (2005) K. Abe _et al._ (Belle), (2005), arXiv:hep-ex/0505037 .
  * del Amo Sanchez  _et al._ (2010) P. del Amo Sanchez _et al._ (BaBar), Phys. Rev. D 82, 011101 (2010), arXiv:1005.5190 [hep-ex] .
  * Ablikim  _et al._ (2019) M. Ablikim _et al._ (BESIII), Phys. Rev. Lett. 122, 232002 (2019), arXiv:1903.04695 [hep-ex] .
  * Swanson (2004) E. S. Swanson, Phys. Lett. B 588, 189 (2004), arXiv:hep-ph/0311229 .
  * Voloshin (2004) M. B. Voloshin, Phys. Lett. B 579, 316 (2004), arXiv:hep-ph/0309307 .
  * AlFiky  _et al._ (2006) M. T. AlFiky, F. Gabbiani, and A. A. Petrov, Phys. Lett. B 640, 238 (2006), arXiv:hep-ph/0506141 .
  * Liu  _et al._ (2008) Y.-R. Liu, X. Liu, W.-Z. Deng, and S.-L. Zhu, Eur. Phys. J. C56, 63 (2008), arXiv:0801.3540 [hep-ph] .
  * Sun  _et al._ (2011) Z.-F. Sun, J. He, X. Liu, Z.-G. Luo, and S.-L. Zhu, Phys. Rev. D84, 054002 (2011), arXiv:1106.2968 [hep-ph] .
  * Nieves and Valderrama (2012) J. Nieves and M. P. Valderrama, Phys. Rev. D86, 056004 (2012), arXiv:1204.2790 [hep-ph] .
  * Guo  _et al._ (2013) F.-K. Guo, C. Hidalgo-Duque, J. Nieves, and M. P. Valderrama, Phys. Rev. D88, 054007 (2013), arXiv:1303.6608 [hep-ph] .
  * Karliner and Rosner (2015) M. Karliner and J. L. Rosner, Phys. Rev. Lett. 115, 122001 (2015), arXiv:1506.06386 [hep-ph] .
  * Braaten and Kusunoki (2005a) E. Braaten and M. Kusunoki, Phys. Rev. D 72, 054022 (2005a), arXiv:hep-ph/0507163 .
  * Gamermann and Oset (2009) D. Gamermann and E. Oset, Phys. Rev. D80, 014003 (2009), arXiv:0905.0402 [hep-ph] .
  * Ortega  _et al._ (2010) P. G. Ortega, J. Segovia, D. R. Entem, and F. Fernandez, Phys. Rev. D 81, 054023 (2010), arXiv:0907.3997 [hep-ph] .
  * Hanhart  _et al._ (2012) C. Hanhart, Y. S. Kalashnikova, A. E. Kudryavtsev, and A. V. Nefediev, Phys. Rev. D 85, 011501 (2012), arXiv:1111.6241 [hep-ph] .
  * Li and Zhu (2012) N. Li and S.-L. Zhu, Phys. Rev. D 86, 074022 (2012), arXiv:1207.3954 [hep-ph] .
  * Takeuchi  _et al._ (2014) S. Takeuchi, K. Shimizu, and M. Takizawa, PTEP 2014, 123D01 (2014), [Erratum: PTEP 2015, 079203 (2015)], arXiv:1408.0973 [hep-ph] .
  * Zhou and Xiao (2018) Z.-Y. Zhou and Z. Xiao, Phys. Rev. D 97, 034011 (2018), arXiv:1711.01930 [hep-ph] .
  * Wu  _et al._ (2021) Q. Wu, D.-Y. Chen, and T. Matsuki, Eur. Phys. J. C 81, 193 (2021), arXiv:2102.08637 [hep-ph] .
  * Ablikim  _et al._ (2013a) M. Ablikim _et al._ (BESIII), Phys. Rev. Lett. 110, 252001 (2013a), arXiv:1303.5949 [hep-ex] .
  * Liu  _et al._ (2013) Z. Q. Liu _et al._ (Belle), Phys. Rev. Lett. 110, 252002 (2013), [Erratum: Phys.Rev.Lett. 111, 019901 (2013)], arXiv:1304.0121 [hep-ex] .
  * Albaladejo  _et al._ (2016) M. Albaladejo, F.-K. Guo, C. Hidalgo-Duque, and J. Nieves, Phys. Lett. B 755, 337 (2016), arXiv:1512.03638 [hep-ph] .
  * Yang  _et al._ (2021) Z. Yang, X. Cao, F.-K. Guo, J. Nieves, and M. P. Valderrama, Phys. Rev. D 103, 074029 (2021), arXiv:2011.08725 [hep-ph] .
  * Meng  _et al._ (2020) L. Meng, B. Wang, and S.-L. Zhu, Phys. Rev. D 102, 111502 (2020), arXiv:2011.08656 [hep-ph] .
  * Baru  _et al._ (2022) V. Baru, E. Epelbaum, A. A. Filin, C. Hanhart, and A. V. Nefediev, Phys. Rev. D 105, 034014 (2022), arXiv:2110.00398 [hep-ph] .
  * Yan  _et al._ (2021) M.-J. Yan, F.-Z. Peng, M. Sánchez Sánchez, and M. Pavon Valderrama, Phys. Rev. D 104, 114025 (2021), arXiv:2102.13058 [hep-ph] .
  * Du  _et al._ (2022) M.-L. Du, M. Albaladejo, F.-K. Guo, and J. Nieves, Phys. Rev. D 105, 074018 (2022), arXiv:2201.08253 [hep-ph] .
  * Ablikim  _et al._ (2021) M. Ablikim _et al._ (BESIII), Phys. Rev. Lett. 126, 102001 (2021), arXiv:2011.07855 [hep-ex] .
  * Baru  _et al._ (2016) V. Baru, E. Epelbaum, A. A. Filin, C. Hanhart, U.-G. Meißner, and A. V. Nefediev, Phys. Lett. B 763, 20 (2016), arXiv:1605.09649 [hep-ph] .
  * Wang  _et al._ (2022) X. L. Wang _et al._ (Belle), Phys. Rev. D 105, 112011 (2022), arXiv:2105.06605 [hep-ex] .
  * Ablikim  _et al._ (2013b) M. Ablikim _et al._ (BESIII), Phys. Rev. Lett. 111, 242001 (2013b), arXiv:1309.1896 [hep-ex] .
  * Braaten  _et al._ (2004) E. Braaten, M. Kusunoki, and S. Nussinov, Phys. Rev. Lett. 93, 162001 (2004), arXiv:hep-ph/0404161 .
  * Braaten and Kusunoki (2005b) E. Braaten and M. Kusunoki, Phys. Rev. D 71, 074005 (2005b), arXiv:hep-ph/0412268 .
  * Wang  _et al._ (2023) H.-N. Wang, L.-S. Geng, Q. Wang, and J.-J. Xie, Chin. Phys. Lett. 40, 021301 (2023), arXiv:2211.14994 [hep-ph] .
  * Chau (1983) L.-L. Chau, Phys. Rept. 95, 1 (1983).
  * Chau and Cheng (1987) L.-L. Chau and H.-Y. Cheng, Phys. Rev. D 36, 137 (1987), [Addendum: Phys.Rev.D 39, 2788–2791 (1989)].
  * Molina  _et al._ (2020) R. Molina, J.-J. Xie, W.-H. Liang, L.-S. Geng, and E. Oset, Phys. Lett. B 803, 135279 (2020), arXiv:1908.11557 [hep-ph] .
  * Liu  _et al._ (2022) M.-Z. Liu, X.-Z. Ling, L.-S. Geng, En-Wang, and J.-J. Xie, Phys. Rev. D 106, 114011 (2022), arXiv:2209.01103 [hep-ph] .
  * Xie  _et al._ (2022a) J.-M. Xie, M.-Z. Liu, and L.-S. Geng, (2022a), arXiv:2207.12178 [hep-ph] .
  * Guo  _et al._ (2014) F. K. Guo, C. Hidalgo-Duque, J. Nieves, A. Ozpineci, and M. P. Valderrama, Eur. Phys. J. C 74, 2885 (2014), arXiv:1404.1776 [hep-ph] .
  * Yamaguchi  _et al._ (2020) Y. Yamaguchi, A. Hosaka, S. Takeuchi, and M. Takizawa, J. Phys. G 47, 053001 (2020), arXiv:1908.08790 [hep-ph] .
  * Follana  _et al._ (2008) E. Follana, C. Davies, G. Lepage, and J. Shigemitsu (HPQCD, UKQCD), Phys. Rev. Lett. 100, 062002 (2008), arXiv:0706.1726 [hep-lat] .
  * Carrasco  _et al._ (2015) N. Carrasco _et al._ , Phys. Rev. D 91, 054507 (2015), arXiv:1411.7908 [hep-lat] .
  * Miller  _et al._ (2020) N. Miller _et al._ , Phys. Rev. D 102, 034507 (2020), arXiv:2005.04795 [hep-lat] .
  * Fontoura  _et al._ (2017) C. E. Fontoura, J. Haidenbauer, and G. Krein, Eur. Phys. J. A 53, 92 (2017), arXiv:1705.09408 [nucl-th] .
  * Ji  _et al._ (2022a) T. Ji, X.-K. Dong, M. Albaladejo, M.-L. Du, F.-K. Guo, J. Nieves, and B.-S. Zou,  (2022a), 10.1016/j.scib.2023.02.034, arXiv:2212.00631 [hep-ph] .
  * Wang  _et al._ (2013) Q. Wang, C. Hanhart, and Q. Zhao, Phys. Rev. Lett. 111, 132003 (2013), arXiv:1303.6355 [hep-ph] .
  * Chen  _et al._ (2013) D.-Y. Chen, X. Liu, and T. Matsuki, Phys. Rev. D88, 036008 (2013), arXiv:1304.5845 [hep-ph] .
  * Liu and Li (2013) X.-H. Liu and G. Li, Phys. Rev. D 88, 014013 (2013), arXiv:1306.1384 [hep-ph] .
  * Swanson (2015) E. S. Swanson, Phys. Rev. D 91, 034009 (2015), arXiv:1409.3291 [hep-ph] .
  * Pilloni  _et al._ (2017) A. Pilloni, C. Fernandez-Ramirez, A. Jackura, V. Mathieu, M. Mikhasenko, J. Nys, and A. P. Szczepaniak (JPAC), Phys. Lett. B 772, 200 (2017), arXiv:1612.06490 [hep-ph] .
  * Ortega  _et al._ (2019) P. G. Ortega, J. Segovia, D. R. Entem, and F. Fernández, Eur. Phys. J. C 79, 78 (2019), arXiv:1808.00914 [hep-ph] .
  * Workman  _et al._ (2022) R. L. Workman _et al._ (Particle Data Group), PTEP 2022, 083C01 (2022).
  * Ali  _et al._ (1998) A. Ali, G. Kramer, and C.-D. Lu, Phys. Rev. D 58, 094009 (1998), arXiv:hep-ph/9804363 .
  * Qin  _et al._ (2014) Q. Qin, H.-n. Li, C.-D. Lü, and F.-S. Yu, Phys. Rev. D 89, 054006 (2014), arXiv:1305.7021 [hep-ph] .
  * Bauer  _et al._ (1987) M. Bauer, B. Stech, and M. Wirbel, Z. Phys. C 34, 103 (1987).
  * Verma (2012) R. C. Verma, J. Phys. G 39, 025005 (2012), arXiv:1103.2973 [hep-ph] .
  * Aoki  _et al._ (2020) S. Aoki _et al._ (Flavour Lattice Averaging Group), Eur. Phys. J. C 80, 113 (2020), arXiv:1902.08191 [hep-lat] .
  * Li  _et al._ (2017) Y. Li, P. Maris, and J. P. Vary, Phys. Rev. D 96, 016022 (2017), arXiv:1704.06968 [hep-ph] .
  * Azevedo and Nielsen (2004) R. S. Azevedo and M. Nielsen, Phys. Rev. C 69, 035201 (2004), arXiv:nucl-th/0310061 .
  * Bracco  _et al._ (2006) M. E. Bracco, A. Cerqueira, Jr., M. Chiapparini, A. Lozea, and M. Nielsen, Phys. Lett. B 641, 286 (2006), arXiv:hep-ph/0604167 .
  * Wang and Wan (2006) Z. G. Wang and S. L. Wan, Phys. Rev. D 74, 014017 (2006), arXiv:hep-ph/0606002 .
  * Liu  _et al._ (2021) M.-Z. Liu, T.-W. Wu, M. Sánchez Sánchez, M. P. Valderrama, L.-S. Geng, and J.-J. Xie, Phys. Rev. D 103, 054004 (2021), arXiv:1907.06093 [hep-ph] .
  * Peng  _et al._ (2020) F.-Z. Peng, M.-Z. Liu, M. Sánchez Sánchez, and M. Pavon Valderrama, Phys. Rev. D 102, 114020 (2020), arXiv:2004.05658 [hep-ph] .
  * Hidalgo-Duque  _et al._ (2013) C. Hidalgo-Duque, J. Nieves, and M. P. Valderrama, Phys. Rev. D 87, 076006 (2013), arXiv:1210.5431 [hep-ph] .
  * Albaladejo  _et al._ (2015) M. Albaladejo, F. K. Guo, C. Hidalgo-Duque, J. Nieves, and M. P. Valderrama, Eur. Phys. J. C75, 547 (2015), arXiv:1504.00861 [hep-ph] .
  * Ji  _et al._ (2022b) T. Ji, X.-K. Dong, M. Albaladejo, M.-L. Du, F.-K. Guo, and J. Nieves, Phys. Rev. D 106, 094002 (2022b), arXiv:2207.08563 [hep-ph] .
  * Liu  _et al._ (2019) M.-Z. Liu, Y.-W. Pan, F.-Z. Peng, M. Sánchez Sánchez, L.-S. Geng, A. Hosaka, and M. Pavon Valderrama, Phys. Rev. Lett. 122, 242001 (2019), arXiv:1903.11560 [hep-ph] .
  * Xie  _et al._ (2022b) J.-M. Xie, X.-Z. Ling, M.-Z. Liu, and L.-S. Geng, Eur. Phys. J. C 82, 1061 (2022b), arXiv:2204.12356 [hep-ph] .

##  I Supplemental Material

In this Supplemental Material, we provide some details about how the Feynman
diagrams of Fig.1 of the main manuscript are calculated in the effective
Lagrangian approach. We first explain how the amplitudes corresponding to the
three vertices in the triangle diagrams are described, and then present the
contact range effective field theory in which all the $\bar{D}^{*}D$ and
$\bar{D}^{*}D^{*}$ molecules are dynamically generated. In the end, we show
the numerical branching fractions and the isospin breaking ratios obtained in
the present work.

###  I.1 Amplitudes for weak decays $B^{+(0)}\to
D_{s}^{(\ast)+}\bar{D}^{(\ast)0}(D^{(\ast)-})$

Figure 3: External $W$-emission mechanism for $B^{+(0)}\to
D_{s}^{(\ast)+}\bar{D}^{(\ast)0}(D^{(\ast)-})$.

At the quark level, the decays of $B^{+(0)}\to
D_{s}^{(\ast)+}\bar{D}^{(\ast)0}(D^{(\ast)-})$ mainly proceed via the external
$W$-emission mechanism shown in Fig. 3, because according to the topological
classification of weak decays, the external $W$-emission mechanism often
provides the dominant contributions Chau (1983); Chau and Cheng (1987); Molina
_et al._ (2020). We tabulate the experimental branching fractions
$\mathcal{B}[B\to D_{s}^{(\ast)}\bar{D}^{(\ast)}]$ in Table 2. In Refs.
Braaten  _et al._ (2004); Braaten and Kusunoki (2005b), Braaten et al.
proposed that the weak decays $B\to\bar{D}^{(*)}D^{(*)}K$ are responsible for
dynamically generating the $\bar{D}^{*}D$ molecules. However, as shown in
Table 2, the branching fractions of
$\mathcal{B}[B\to\bar{D}^{(*)}D_{s}^{(*)}]$ are always larger those of
$\mathcal{B}[B\to\bar{D}^{(*)}D^{(*)}K]$ by at least two times and even up to
one order of magnitude, which indicates that the weak vertices $B\to
D_{s}^{(\ast)}\bar{D}^{(\ast)}$ are more favorable to produce the
$\bar{D}^{*}D^{(*)}$ molecules via the triangle mechanism proposed in this
work.

Table 2: Branching fractions ($10^{-3}$) of $B\to D_{s}^{(\ast)}\bar{D}^{(\ast)}$ and $B\to\bar{D}^{(*)}D^{(*)}K$.  | Decay mode |  RPP. Workman  _et al._ (2022) |  Decay mode |  RPP. Workman  _et al._ (2022) |  Decay mode |  RPP. Workman  _et al._ (2022)
---|---|---|---|---|---|---
$a_{1+}^{\prime}$ |  $B^{+}\to\bar{D}^{0}D_{s}^{+}$ |  $9.0\pm 0.9$ |  $B^{+}\to\bar{D}^{0}D^{+}K^{0}$ |  $1.55\pm 0.21$ |  $B^{+}\to\bar{D}^{0}D^{0}K^{+}$ |  $1.45\pm 0.33$
$a_{10}^{\prime}$ |  ${B}^{0}\to{D}^{-}D_{s}^{+}$ |  $7.2\pm 0.8$ |  ${B}^{0}\to{D}^{-}D^{+}K^{0}$ |  $0.75\pm 0.17$ |  ${B}^{0}\to{D}^{-}D^{0}K^{+}$ |  $1.07\pm 0.11$
$a_{1+}^{\ast}$ |  $B^{+}\to\bar{D}^{0}D_{s}^{\ast+}$ |  $7.6\pm 1.6$ |  $B^{+}\to\bar{D}^{0}D^{\ast+}K^{0}$ |  $3.8\pm 0.4$ |  $B^{+}\to\bar{D}^{0}D^{\ast 0}K^{+}$ |  $6.3\pm 0.5$
$a_{10}^{\ast}$ |  ${B}^{0}\to{D}^{-}D_{s}^{\ast+}$ |  $7.4\pm 1.6$ |  ${B}^{0}\to{D}^{-}D^{\ast+}K^{0}$ |  $3.2\pm 0.25$ |  ${B}^{0}\to{D}^{-}D^{\ast 0}K^{+}$ |  $3.5\pm 0.4$
$a_{1+}$ |  $B^{+}\to\bar{D}^{\ast 0}D_{s}^{+}$ |  $8.2\pm 1.7$ |  $B^{+}\to\bar{D}^{\ast 0}D^{+}K^{0}$ |  $2.1\pm 0.5$ |  $B^{+}\to\bar{D}^{\ast 0}D^{0}K^{+}$ |  $2.26\pm 0.23$
$a_{10}$ |  ${B}^{0}\to{D}^{\ast-}D_{s}^{+}$ |  $8.0\pm 1.1$ |  ${B}^{0}\to{D}^{\ast-}D^{+}K^{0}$ |  $3.2\pm 0.25$ |  ${B}^{0}\to{D}^{\ast-}D^{0}K^{+}$ |  $2.47\pm 0.21$
$a_{1+}^{\prime\ast}$ |  $B^{+}\to\bar{D}^{\ast 0}D_{s}^{\ast+}$ |  $17.1\pm 2.4$ |  $B^{+}\to\bar{D}^{\ast 0}D^{\ast+}K^{0}$ |  $9.2\pm 1.2$ |  $B^{+}\to\bar{D}^{\ast 0}D^{\ast 0}K^{+}$ |  $11.2\pm 1.3$
$a_{10}^{\prime\ast}$ |  ${B}^{0}\to{D}^{\ast-}D_{s}^{\ast+}$ |  $17.7\pm 1.4$ |  ${B}^{0}\to{D}^{\ast-}D^{\ast+}K^{0}$ |  $8.1\pm 0.7$ |  ${B}^{0}\to{D}^{\ast-}D^{\ast 0}K^{+}$ |  $10.6\pm 0.9$

For the weak interaction vertices, the decay amplitudes of $B^{+(0)}\to
D_{s}^{(\ast)+}\bar{D}^{(\ast)0}(D^{(\ast)-})$ can be expressed as the
products of two hadronic matrix elements Ali  _et al._ (1998); Qin  _et al._
(2014)

$\displaystyle\mathcal{A}\left(B^{+}\to D_{s}^{+}\bar{D}^{\ast 0}\right)$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cb}V_{cs}a_{1}\left\langle
D_{s}^{+}|(s\bar{c})|0\right\rangle\left\langle\bar{D}^{\ast
0}|(c\bar{b})|B^{+}\right\rangle,$ (7) $\displaystyle\mathcal{A}\left(B^{+}\to
D_{s}^{+}\bar{D}^{0}\right)$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cb}V_{cs}a_{1}^{\prime}\left\langle
D_{s}^{+}|(s\bar{c})|0\right\rangle\left\langle\bar{D}^{0}|(c\bar{b})|B^{+}\right\rangle,$
(8) $\displaystyle\mathcal{A}\left(B^{+}\to D_{s}^{\ast+}\bar{D}^{0}\right)$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cb}V_{cs}a_{1}^{*}\left\langle
D_{s}^{\ast+}|(s\bar{c})|0\right\rangle\left\langle\bar{D}^{0}|(c\bar{b})|B^{+}\right\rangle,$
(9) $\displaystyle\mathcal{A}\left(B^{+}\to D_{s}^{\ast+}\bar{D}^{\ast
0}\right)$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cb}V_{cs}a_{1}^{\prime\ast}\left\langle
D_{s}^{\ast+}|(s\bar{c})|0\right\rangle\left\langle\bar{D}^{\ast
0}|(c\bar{b})|B^{+}\right\rangle,$ (10)

where $a_{1}=c_{1}^{eff}+c_{2}^{eff}/N_{c}$ with $N_{c}$ the number of colors
and $a_{1}$ can be obtained in the factorization approach Bauer  _et al._
(1987). In the present work, we determine $a_{1}$ by fitting to the
experimental branching fractions.

The current matrix elements between a pseudoscalar meson or vector meson and
the vacuum have the following form:

$\displaystyle\left\langle D_{s}^{+}|(s\bar{c})|0\right\rangle$
$\displaystyle=$ $\displaystyle
f_{D_{s}^{+}}p^{\mu}_{D_{s}^{+}},~{}~{}\left\langle
D_{s}^{\ast+}|(s\bar{c})|0\right\rangle=m_{D_{s}^{\ast+}}f_{D_{s}^{\ast+}}\epsilon_{\mu}^{*},$
(11)

where $f_{D_{s}^{+}}$ and $f_{D_{s}^{\ast+}}$ are the decay constants for
$D_{s}^{+}$ and $D_{s}^{\ast+}$, respectively, and $\epsilon_{\mu}^{*}$
denotes the polarization vector of a vector particle. In this work, we take
$G_{F}=1.166\times 10^{-5}~{}{\rm GeV}^{-2}$, $V_{cb}=0.041$, $V_{cs}=0.987$,
$f_{D_{s}}=250$ MeV, and $f_{D_{s}^{\ast+}}=272$ MeV of Refs. Workman  _et
al._ (2022); Verma (2012); Aoki  _et al._ (2020); Li  _et al._ (2017).

The hadronic matrix elements are parameterised in terms of six form factors
Verma (2012)

$\displaystyle\left\langle\bar{D}^{\ast
0}|(c\bar{b})|B^{+}\right\rangle=\epsilon_{\alpha}^{*}\left\\{-g^{\mu\alpha}(m_{\bar{D}^{\ast
0}}+m_{B^{+}})A_{1}\left(q^{2}\right)+P^{\mu}P^{\alpha}\frac{A_{2}\left(q^{2}\right)}{m_{\bar{D}^{\ast
0}}+m_{B^{+}}}\right.$ (12)
$\displaystyle+i\varepsilon^{\mu\alpha\beta\gamma}P_{\beta}q_{\gamma}\left.\frac{V\left(q^{2}\right)}{m_{\bar{D}^{\ast
0}}+m_{B^{+}}}+q^{\mu}P^{\alpha}\left[\frac{m_{\bar{D}^{\ast
0}}+m_{B^{+}}}{q^{2}}A_{1}\left(q^{2}\right)-\frac{m_{B^{+}}-m_{\bar{D}^{\ast
0}}}{q^{2}}A_{2}\left(q^{2}\right)-\frac{2m_{\bar{D}^{\ast
0}}}{q^{2}}A_{0}\left(q^{2}\right)\right]\right\\},$
$\displaystyle\left\langle\bar{D}^{0}|(c\bar{b})|B^{+}\right\rangle=\left[(p_{B^{+}}+p_{\bar{D}^{0}})^{\mu}-\frac{m_{B^{+}}^{2}-m_{\bar{D}^{0}}^{2}}{q^{\prime
2}}q^{\prime}_{\mu}\right]F_{1D}(q^{\prime
2})+\frac{m_{B^{+}}^{2}-m_{\bar{D}^{0}}^{2}}{q^{\prime
2}}q^{\prime}_{\mu}F_{0D}(q^{\prime 2}),$ (13)

where $q$ and $q^{\prime}$ represent the momentum transfer of
$p_{B^{+}}-p_{\bar{D}^{\ast 0}}$ and $p_{B^{+}}-p_{\bar{D}^{0}}$,
respectively, and $P=p_{B^{+}}+p_{\bar{D}^{\ast 0}}$.

The form factors of $F_{1,0D}(t)$, $A_{0}(t)$, $A_{1}(t)$, $A_{2}(t)$, and
$V(t)$ with $t\equiv q^{(\prime)2}$ can be parameterized in the following form
Verma (2012)

$X(t)=\frac{X(0)}{1-a\left(t/m_{B}^{2}\right)+b\left(t^{2}/m_{B}^{4}\right)},$
(14)

which could reproduce the transition form factors of $B\to\bar{D}^{(\ast)}$ as
shown in Ref. Verma (2012).

The values of the parameters in these form factors are taken from the
covariant light-front quark model, i.e.,
$(F_{1}(0),a,b)^{B\to\bar{D}}=(0.67,1.22,0.36)$,
$(F_{0}(0),a,b)^{B\to\bar{D}}=(0.67,0.63,0.01)$,
$(A_{0}(0),a,b)^{B\to\bar{D}^{\ast}}=(0.68,1.21,0.36)$,
$(A_{1}(0),a,b)^{B\to\bar{D}^{\ast}}=(0.65,0.60,0.00)$,
$(A_{2}(0),a,b)^{B\to\bar{D}^{\ast}}=(0.61,1.12,0.31)$, and
$(V_{0}(0),a,b)^{B\to\bar{D}^{\ast}}=(0.77,1.25,0.38)$ Verma (2012).

The weak decay amplitudes of $B\to D_{s}^{(\ast)}\bar{D}^{(\ast)}$ have the
following form

$\displaystyle\mathcal{A}(B\to D_{s}\bar{D}^{\ast})$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}V_{cb}V_{cs}a_{1}f_{D_{s}}\\{-q_{1}\cdot\varepsilon(q_{2})(m_{\bar{D}^{\ast
0}}+m_{B^{+}})A_{1}\left(q_{1}^{2}\right)$ (15)
$\displaystyle+(k_{0}+q_{2})\cdot\varepsilon(q_{2})q_{1}\cdot(k_{0}+q_{2})\frac{A_{2}\left(q_{1}^{2}\right)}{m_{\bar{D}^{\ast
0}}+m_{B^{+}}}+(k_{0}+q_{2})\cdot\varepsilon(q_{2})$
$\displaystyle[(m_{\bar{D}^{\ast
0}}+m_{B^{+}})A_{1}(q_{1}^{2})-(m_{B^{+}}-m_{\bar{D}^{\ast
0}})A_{2}(q_{1}^{2})-2m_{\bar{D}^{\ast 0}}A_{0}(q_{1}^{2})]\\},$
$\displaystyle\mathcal{A}(B\to D_{s}\bar{D})$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}V_{cb}V_{cs}a_{1}^{\prime}f_{D_{s}}(m_{B}^{2}-m_{D}^{2})F_{0}(q_{1}^{2}),$
$\displaystyle\mathcal{A}(B^{+}\to D_{s}^{\ast+}\bar{D}^{0})$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}V_{cb}V_{cs}a_{1}^{*}m_{D_{s}^{\ast+}}f_{D_{s}^{\ast+}}(k_{0}+q_{2})\cdot\varepsilon(q_{1})F_{1}(q_{1}^{2}),$
$\displaystyle\mathcal{A}(B^{+}\to D_{s}^{\ast+}\bar{D}^{\ast 0})$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}V_{cb}V_{cs}a_{1}^{*\prime}m_{D_{s}^{\ast+}}f_{D_{s}^{\ast+}}\varepsilon_{\mu}(q_{1})\left[(-g^{\mu\alpha}(m_{\bar{D}^{\ast
0}}+m_{B^{+}})A_{1}\left(q_{1}^{2}\right)\right.$
$\displaystyle+\left.(k_{0}+q_{2})^{\mu}(k_{0}+q_{2})^{\alpha}\frac{A_{2}\left(q_{1}^{2}\right)}{m_{\bar{D}^{\ast
0}}+m_{B^{+}}}+i\varepsilon^{\mu\alpha\beta\gamma}(k_{0}+q_{2})_{\beta}q_{1\gamma}\frac{V\left(q_{1}^{2}\right)}{m_{\bar{D}^{\ast
0}}+m_{B^{+}}}\right]\varepsilon_{\alpha}(q_{2}).$

By fitting to the eight branching fractions of $B^{+(0)}\to
D_{s}^{(\ast)+}\bar{D}^{(\ast)0}(D^{(\ast)-})$ tabulated in Table 2, we obtain
$a_{1+}=0.929$, $a_{10}=0.955$, $a_{1+}^{\prime}=0.791$,
$a_{10}^{\prime}=0.736$, $a_{1+}^{\ast}=0.812$, $a_{10}^{\ast}=0.834$,
$a_{1+}^{\prime\ast}=0.833$, and $a_{10}^{\prime\ast}=0.880$. The subscript
with $+$ and $0$ denote the $B^{+}$ and $B^{0}$ decay modes, the values of
which show small isospin breaking effects of less than 10%.

###  I.2 Amplitudes for $D_{s}^{(*)}\to D^{(*)}K$

The Lagrangians describing the interactions between charmed mesons and the
kaon read

$\displaystyle\mathcal{L}_{D_{s}^{\ast}DK}$ $\displaystyle=$ $\displaystyle-
ig_{D_{s}^{\ast}DK}(D\partial^{\mu}KD_{s\mu}^{\ast{\dagger}}-D_{s\mu}^{\ast}\partial^{\mu}KD^{{\dagger}}),$
(16) $\displaystyle\mathcal{L}_{D_{s}D^{\ast}K}$ $\displaystyle=$
$\displaystyle-
ig_{D_{s}D^{\ast}K}(D_{s}\partial^{\mu}KD^{\ast{\dagger}}_{\mu}-D_{\mu}^{*}\partial^{\mu}KD_{s}^{{\dagger}}),$
$\displaystyle\mathcal{L}_{D_{s}^{\ast}D^{\ast}K}$ $\displaystyle=$
$\displaystyle-
g_{D_{s}^{\ast}D^{\ast}K}\varepsilon_{\mu\nu\alpha\beta}\partial^{\mu}D_{s}^{\ast\nu}{\partial}^{\alpha}{D}^{\ast\beta{\dagger}}K,$

where $g_{D_{s}D^{\ast}K}$, $g_{D_{s}^{\ast}DK}$ and
$g_{D_{s}^{\ast}D^{\ast}K}$ are the couplings to be determined. From these
Lagrangains, one can obtain the amplitudes for the decays of the
${D}_{s}^{(\ast)}\to{D}^{(\ast)}K$

$\displaystyle\mathcal{A}\left({D}_{s}^{\ast}\to{D}K\right)$ $\displaystyle=$
$\displaystyle g_{{D}_{s}^{\ast}DK}p_{1}\cdot\varepsilon(q_{1}),$ (17)
$\displaystyle\mathcal{A}\left(D_{s}\to{D}^{\ast}K\right)$ $\displaystyle=$
$\displaystyle-g_{D_{s}{D}^{\ast}K}p_{1}\cdot\varepsilon(q_{3}),$ (18)
$\displaystyle\mathcal{A}\left({D}_{s}^{*}\to{D}^{\ast}K\right)$
$\displaystyle=$ $\displaystyle
g_{{D}_{s}^{*}{D}^{\ast}K}\varepsilon_{\mu\nu\alpha\beta}q_{1}^{\mu}\varepsilon^{\nu}(q_{1})q_{3}^{\alpha}\varepsilon^{\beta}(q_{3}).$
(19)

Assuming SU(3)-flavor symmetry and SU(4)-flavor symmetry the coupling
$g_{D_{s}D^{\ast}K}$ is estimated to be $16.6$ Xie  _et al._ (2022a) and $10$
Azevedo and Nielsen (2004), while the QCD sum rule yields $5$ Bracco  _et al._
(2006); Wang and Wan (2006). In view of this large variance, we adopt the
couplings estimated by SU(4) symmetry, which are in between those estimated
utilyzing SU(3) symmetry and by the QCD sum rule, i.e.,
$g_{D_{s}D^{\ast}K}=g_{D_{s}^{\ast}DK}=10$ and $g_{D_{s}^{\ast}D^{\ast}K}=7.0$
GeV-1 Azevedo and Nielsen (2004). It is well known that the SU(4) symmetry is
heavily broken. Therefore, following Refs. Follana  _et al._ (2008); Carrasco
_et al._ (2015), we estimate the breaking of SU(4) symmetry using the $D$ and
$K$ decay constants as references, which is $33\%$, consistent with Ref.
Fontoura  _et al._ (2017).

###  I.3  Amplitudes for the dynamical generation of $\bar{D}^{*}D$ and
$\bar{D}^{*}D^{*}$ molecules

The Lagrangian describing the interactions between $X(3872)$/$X_{2}(4013)$ and
their constituents are

$\displaystyle\mathcal{L}_{X\bar{D}D^{\ast}}$ $\displaystyle=$ $\displaystyle
g_{X\bar{D}D^{\ast}}X^{\mu}{D}^{\ast}_{\mu}\bar{D},$ (20)
$\displaystyle\mathcal{L}_{X_{2}\bar{D}^{*}D^{\ast}}$ $\displaystyle=$
$\displaystyle
g_{X_{2}\bar{D}^{*}D^{\ast}}X_{2\mu\nu}{D}^{\ast\mu}\bar{D}^{\ast\nu},$

where $g_{X\bar{D}D^{\ast}}$ and $g_{X_{2}\bar{D}^{*}D^{\ast}}$ are the
couplings to be determined.

In the following, we denote $Z_{c}(3900)$ and $Z_{c}(4020)$ as $Z_{c}$ and
$Z_{c}^{\prime}$, respectively. The interactions between the
$Z_{c}(3900)/Z_{c}(4020)$ resonant states and their constituents can be
expressed by the following effective Lagrangians:

$\displaystyle\mathcal{L}_{Z_{c}D\bar{D}^{*}}$ $\displaystyle=$ $\displaystyle
g_{Z_{c}D\bar{D}^{*}}Z_{c}^{\mu}D\bar{D}^{*}_{\mu},$ (21)
$\displaystyle\mathcal{L}_{Z_{c}^{\prime}D^{*}\bar{D}^{*}}$ $\displaystyle=$
$\displaystyle
ig_{Z_{c}^{\prime}D^{*}\bar{D}^{*}}\varepsilon_{\mu\nu\alpha\beta}\partial^{\mu}Z_{c}^{\prime\nu}D^{\ast\alpha}\bar{D}^{\ast\beta},$

where $g_{Z_{c}D\bar{D}^{*}}$ and $g_{Z_{c}^{\prime}D^{*}\bar{D}^{*}}$ are the
corresponding couplings.

From the above Lagrangians, one can obtain the amplitudes for the coupling of
$X(3872)$, $X(4013)$, $Z_{c}(3900)$, $Z_{c}(4020)$ to the $\bar{D}D^{*}$ and
$\bar{D}D^{*}$ channels

$\displaystyle\mathcal{A}(\bar{D}D^{*}\to X(3872))$ $\displaystyle=$
$\displaystyle
g_{\bar{D}D^{*}X}\varepsilon^{\mu}(p_{2})\varepsilon_{\mu}(q_{3}),$ (22)
$\displaystyle\mathcal{A}(\bar{D}^{*}D\to X(3872))$ $\displaystyle=$
$\displaystyle
g_{\bar{D}D^{*}X}\varepsilon^{\mu}(p_{2})\varepsilon_{\mu}(q_{2}),$ (23)
$\displaystyle\mathcal{A}(\bar{D}^{*}D^{*}\to X(4013))$ $\displaystyle=$
$\displaystyle
g_{\bar{D}^{*}D^{*}X}\varepsilon^{\mu\nu}(p_{2})\varepsilon_{\mu}(q_{3})\varepsilon_{\nu}(q_{2}),$
(24) $\displaystyle\mathcal{A}(\bar{D}D^{*}\to Z_{c}(3900))$ $\displaystyle=$
$\displaystyle
g_{\bar{D}D^{*}Z_{c}}\varepsilon^{\mu}(p_{2})\varepsilon_{\mu}(q_{3}),$ (25)
$\displaystyle\mathcal{A}(\bar{D}^{*}D\to Z_{c}(3900))$ $\displaystyle=$
$\displaystyle
g_{\bar{D}D^{*}Z_{c}}\varepsilon^{\mu}(p_{2})\varepsilon_{\mu}(q_{2}),$ (26)
$\displaystyle\mathcal{A}(\bar{D}^{*}D^{*}\to Z_{c}(4020))$ $\displaystyle=$
$\displaystyle
g_{\bar{D}^{*}D^{*}Z_{c}^{\prime}}\varepsilon_{\mu\nu\alpha\beta}p_{2}^{\mu}\varepsilon^{\nu}(p_{2})\varepsilon^{\alpha}(q_{3})\varepsilon^{\beta}(q_{2}),$
(27)

where $k_{0}$, $p_{1}$ and $p_{2}$ refer to the momentum of $B$, $K$ and
$X/Z$, $q_{1}$, $q_{2}$ and $q_{3}$ to the momentum of $D_{s}^{(\ast)}$,
$\bar{D}^{(\ast)}$ and $D^{(\ast)}$, $\varepsilon^{\mu}(q)$ denotes the
polarization vector of a particle with spin $S=1$, and
$\varepsilon^{\mu\nu}(q)$ denotes the polarization vector of a particle with
spin $S=2$.

In the isospin limit, one can decompose the isospin wave functions of the
isovector molecules as

$\displaystyle|Z_{c}(3900)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[(|{D}^{\ast+}D^{-}\rangle-|{D}^{\ast
0}\bar{D}^{0}\rangle)+(|{D}^{+}D^{\ast-}\rangle-|D^{0}\bar{D}^{\ast
0}\rangle)\right],$ (28) $\displaystyle|Z_{c}(4020)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(|{D}^{\ast+}D^{\ast-}\rangle-|{D}^{\ast
0}\bar{D}^{\ast 0}\rangle\right),$

which allows one to obtain the couplings of $Z_{c}$ to each component. To
compare with the isovector molecules, we decompose the isospin wave functions
of the isoscalar molecules as

$\displaystyle|X(3872)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[(|{D}^{\ast+}D^{-}\rangle+|{D}^{\ast
0}\bar{D}^{0}\rangle)-(|{D}^{+}D^{\ast-}\rangle+|D^{0}\bar{D}^{\ast
0}\rangle)\right],$ (29) $\displaystyle|X(4013)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(|{D}^{\ast+}D^{\ast-}\rangle+|{D}^{\ast
0}\bar{D}^{\ast 0}\rangle\right).$

Although the couplings of the isoscalar molecules to their constituents are
derived in the isospin limit, one immediately see that the couplings of the
isoscalar and isovector $\bar{D}^{*}D$ molecule to the charged components or
neutral components are of opposite sign. This can have large impact on the
isospin breaking of the branching fractions of our interests as shown in the
main text.

###  I.4 Contact-range effective field theory for the $\bar{D}^{*}D$ and
$\bar{D}^{*}D^{*}$ interactions

In this subsection, we briefly describe the contact-range effective field
theory in which the $\bar{D}^{*}D$ and $\bar{D}^{*}D^{*}$ interactions can
dynamically generate the $X(3872)$, $X_{2}(4013)$, $Z_{c}(3900)$, and
$Z_{c}(4020)$. More details can be found in Ref. Nieves and Valderrama (2012).

In the heavy quark limit, the contact potential of the $J^{PC}=1^{++}$
$\bar{D}^{*}D$ channel is expressed as a sum of two low-energy constants,
$C_{a}+C_{b}$, where $C_{a}$ characterizes the spin independent interaction
and $C_{b}$ accounts for the spin-spin interaction Nieves and Valderrama
(2012). Because the $X(3872)$ is quite close to the mass threshold of
$\bar{D}^{\ast 0}D^{0}$ but below the mass threshold of $\bar{D}^{\ast+}D^{-}$
by $8$ MeV, it is important to take into account the isospin breaking effects
of the $\bar{D}^{*}D$ wave function. It contains both a neutral component and
a charged component, which are defined as
$C_{n}=\frac{1}{\sqrt{2}}(\bar{D}^{\ast 0}D^{0}-\bar{D}^{0}D^{\ast 0})$ and
$C_{c}=\frac{1}{\sqrt{2}}(D^{\ast+}D^{-}-{D}^{+}D^{\ast-})$. The contact
potential for $C_{n}$ and $C_{c}$ read Nieves and Valderrama (2012):

$V_{C_{n}-C_{c}}=\begin{pmatrix}C_{a}^{\prime}+C_{b}^{\prime}&C_{a}^{\prime\prime}+C_{b}^{\prime\prime}\\\
C_{a}^{\prime\prime}+C_{b}^{\prime\prime}&C_{a}^{\prime}+C_{b}^{\prime}\end{pmatrix}.$
(30)

In principle, the contact potentials should be determined by reproducing the
experimental data. Due to the scarcity of experimental data one needs to turn
to other approaches such as the light meson saturation mechanism, which
dictates that the couplings are saturated by the light meson ($\sigma$,
$\rho$, and $\omega$) exchanges in the one boson exchange model. $C_{a}$
receives contributions from both the scalar and vector meson exchanges, but
$C_{b}$ only receives contributions from the vector meson excanges, i.e.,

$\displaystyle C_{a}^{S}+C_{a}^{V}$ $\displaystyle\propto$ $\displaystyle
C_{a}^{sat}(\Lambda\sim m_{\sigma},m_{V}),$ (31) $\displaystyle C_{b}^{V}$
$\displaystyle\propto$ $\displaystyle C_{b}^{sat}(\Lambda\sim
m_{\sigma},m_{V}),$

where

$\displaystyle C_{a}^{sat(\sigma)}(\Lambda\sim m_{\sigma})$
$\displaystyle\propto$ $\displaystyle-\frac{g_{\sigma}^{2}}{m_{\sigma}^{2}},$
(32) $\displaystyle C_{a}^{sat(V)}(\Lambda\sim m_{V})$ $\displaystyle\propto$
$\displaystyle-\frac{g_{v}^{2}}{m_{v}^{2}}(1-\vec{\tau}_{1}\cdot\vec{\tau}_{2}),$
$\displaystyle C_{b}^{sat(V)}(\Lambda\sim m_{V})$ $\displaystyle\propto$
$\displaystyle-\frac{f_{v}^{2}}{4M^{2}}(1-\vec{\tau}_{1}\cdot\vec{\tau}_{2}),$

where $g_{\sigma_{1}}$ denotes the coupling of the charm meson to the sigma
meson, $g_{v1}$ and $f_{v1}$ denote the electric-type and magnetic-type
couplings between the charm meson and a light vector meson, and $M$ is a mass
scale to render $f_{v1}$ dimensionless. The proportionality constant is
unknown and depends on the details of the renormalization procedure. However,
assuming that the constant is the same for $C_{a}^{sat}$ and $C_{b}^{sat}$, we
can calculate their ratio. Such an approach has been verified in the studies
of the $\bar{D}^{(*)}\Sigma_{c}^{(*)}$ system and it was found that the ratio
estimated by the light meson saturation is consistent with that obtained by
reproducing the masses of the $P_{c}$ states Liu  _et al._ (2021).

As for the $C_{n}-C_{c}$ system, the diagonal potential receives contributions
from the $\sigma$, $\omega$, and $\rho^{0}$ mesons but the off diagonal
potentials only receive contributions from the $\rho^{+}$ meson, which can
determine the relative strength between the off diagonal potential and the
diagonal potential. Following Ref. Peng  _et al._ (2020), the couplings of
$C_{a}^{\prime}$, $C_{b}^{\prime}$, $C_{a}^{\prime\prime}$ and
$C_{b}^{\prime\prime}$ in the light meson saturation approach read

$\displaystyle C_{a}^{\prime}$ $\displaystyle\propto$
$\displaystyle-\frac{g_{\sigma}^{2}}{m_{\sigma}^{2}}-\frac{g_{v}^{2}}{m_{v}^{2}}-\frac{g_{v}^{2}}{m_{v}^{2}},$
(33) $\displaystyle C_{b}^{\prime}$ $\displaystyle\propto$
$\displaystyle-\frac{f_{v}^{2}}{m_{v}^{2}}-\frac{f_{v}^{2}}{m_{v}^{2}},$
$\displaystyle C_{a}^{\prime\prime}$ $\displaystyle\propto$ $\displaystyle
2\frac{g_{v}^{2}}{m_{v}^{2}},$ $\displaystyle C_{b}^{\prime\prime}$
$\displaystyle\propto$ $\displaystyle 2\frac{f_{v}^{2}}{m_{v}^{2}},$

from which we obtain the ratio of $C_{a}^{\prime\prime}+C_{b}^{\prime\prime}$
to $C_{a}^{\prime}+C_{b}^{\prime}$:

$\displaystyle\frac{C_{a}^{\prime\prime}+C_{b}^{\prime\prime}}{C_{a}^{\prime}+C_{b}^{\prime}}\approx
0.5,$ (34)

consistent with the estimations of Refs. Hidalgo-Duque  _et al._ (2013);
Albaladejo  _et al._ (2015); Ji  _et al._ (2022b). As a result, the unknown
couplings of the contact-range potentials are reduced to one. By reproducing
the mass of $X(3872)$, we obtain $C_{a}^{\prime}+C_{b}^{\prime}=12.551$ GeV-2
and the corresponding couplings to the neutral and charged components of its
wave function, i.e., $g_{n}=3.86$ GeV and $g_{c}=3.39$ GeV. Taking into
account HQSS, one can obtain the potentials of the $\bar{D}^{*0}D^{\ast
0}/D^{*+}D^{\ast-}$ system and predict the existence of a $J^{PC}=2^{++}$
bound state with a mass of $m=4013.03$ MeV, corresponding to $X_{2}(4013)$.
Finally, the $X_{2}(4013)$ couplings to its neutral and charged components are
determined as $g_{n}^{\prime}=5.36$ GeV and $g_{c}^{\prime}=4.86$ GeV.

Table 3:  Values of the couplings of the $\bar{D}^{*}D^{(*)}$ molecules to their neutral and charged components.  Molecules |  $g_{n}$ |  $g_{c}$
---|---|---
$X(3872)$ |  $3.86$ GeV |  $3.39$ GeV
$X_{2}(4013)$ |  $5.36$ GeV |  $4.86$ GeV
$Z_{c}(3900)$ |  $5.02$ GeV |  $5.02$ GeV
$Z_{c}(4020)$ |  $1.25$ |  $1.25$

To generate resonant states, the contact potential has to supplemented with a
$q^{2}$ dependent term Yang  _et al._ (2021), where $q$ is the relative three
momentum. Identifying $Z_{c}(3900)$ as a $\bar{D}D^{\ast}$ resonant state with
the form of $C_{s}+C_{d}~{}q^{2}$, we obtain $C_{s}=-7.7$ GeV-2 and
$C_{d}=-211$ GeV-4 for a cutoff of $\Lambda=1$ GeV, and then the coupling
$g_{Z_{c}(3900)\bar{D}D^{*}}=7.10$ GeV. Taking into account HQSS, we predict
the existence of a $\bar{D}^{*}D^{*}$ molecule with $M=4028$ MeV and
$\Gamma=26$ MeV, in perfect agreement with the experimental measurements, and
then obtain the coupling $g_{Z_{c}(4020)\bar{D}^{*}D^{*}}=1.77$. In Table 3,
we collect the values of the molecular couplings.

In the following, for the sake of completeness, we explain how these couplings
are calculated from the residue of poles on the complex plane by solving the
Lippmann-Schwinger equation Liu  _et al._ (2019)

$\displaystyle T=(1-VG)^{-1}V,$ (35)

where $V$ is the hadron-hadron potential determined by the contact EFT
approach described above, and $G$ is the two-body propagator. In evaluating
the loop function $G$, we introduce a regulator of Gaussian form
$e^{-2q^{2}/\Lambda^{2}}$ in the integral as

$\displaystyle
G(s)=\int\frac{d^{3}q}{(2\pi)^{3}}\frac{e^{-2q^{2}/\Lambda^{2}}}{{\sqrt{s}}-m_{1}-m_{2}-q^{2}/(2\mu_{12})+i\varepsilon},$
(36)

where $\sqrt{s}$ is the total energy in the c.m.frame of $m_{1}$ and $m_{2}$,
$\mu_{12}=\frac{m_{1}m_{2}}{m_{1}+m_{2}}$ is the reduced mass, and $\Lambda$
is the momentum cutoff. Following our previous works Liu  _et al._ (2019); Xie
_et al._ (2022b), we take $\Lambda=1$ GeV in the present work, which reflects
the internal structure of the involved hadrons and provides an estimate of the
uncertainties of the numerical results. The dynamically generated states
correspond to poles in the unphysical sheet. In this sheet, the loop function
of Eq. (36) becomes

$\displaystyle
G^{II}(s,m_{1},m_{2})=G^{I}(s,m_{1},m_{2})+i\mu_{12}\frac{p}{2\pi}{e^{-2p^{2}/\Lambda^{2}}},$
(37)

where the c.m. momentum $p$ is

$\displaystyle p=\sqrt{2\mu_{12}\left(\sqrt{s}-m_{1}-m_{2}\right)}.$ (38)

Using the contact potentials we can search for poles generated by the obtained
interactions, and determine the couplings between the molecular states and
their constituents from the residues of the corresponding poles,

$\displaystyle
g_{i}g_{j}=\lim_{{\sqrt{s}}\to{\sqrt{s_{0}}}}\left({\sqrt{s}}-{\sqrt{s_{0}}}\right)T_{ij}(\sqrt{s}),$
(39)

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

###  I.5 Results

Table 4: Branching fractions ($10^{-4}$) of $B^{+(0)}\to X(3872)/X_{2}(4012)K^{+(0)}$ and $B^{+(0)}\to Z_{c}(3900)/Z_{c}(4020)K^{+(0)}$ and ratios $\mathcal{B}(B^{0}\to)$/$\mathcal{B}(B^{+}\to)$.  Decay modes |  Our predictions |  Exp. Workman  _et al._ (2022) |  Ratio |  Exp. data Workman  _et al._ (2022)
---|---|---|---|---
$B^{+}\to X(3872)K^{+}$ |  $1.49\pm 0.62$ |  $2.1\pm 0.7$ |  $0.62\pm 0.13$ |  $0.52\pm 0.26$
$B^{0}\to X(3872)K^{0}$ |  $0.93\pm 0.39$ |  $1.1\pm 0.4$
$B^{+}\to X(4013)K^{+}$ |  $0.23\pm 0.08$ |  $-$ |  $0.75\pm 0.16$ |  $-$
$B^{0}\to X(4013)K^{0}$ |  $0.17\pm 0.06$ |  $-$
$B^{+}\to Z_{c}(3900)K^{+}$ |  $0.21\pm 0.11$ |  $<4.7\times 10^{-5}$ |  $0.63\pm 0.29$ |  $-$
$B^{0}\to Z_{c}(3900)K^{0}$ |  $0.13\pm 0.07$ |  $-$
$B^{+}\to Z_{c}(4020)K^{+}$ |  $0.0095\pm 0.0033$ |  $<1.6\times 10^{-5}$ |  $1.05\pm 0.14$ |  $-$
$B^{0}\to Z_{c}(4020)K^{0}$ |  $0.0100\pm 0.0034$ |  $-$

In Table 4, we present the branching fractions of the decays of $B\to
X(3872)/X_{2}(4013)K$ and $B\to Z_{c}(3900)/Z_{c}(4020)K$. In the following,
we analyse the origin of the isospin breaking in the ratios
$\mathcal{B}[B^{0}\to X(3872)K^{0}]/\mathcal{B}[B^{+}\to X(3872)K^{+}]$ and
$\mathcal{B}[B^{+}\to Z_{c}(3900)K^{+}]/\mathcal{B}[B^{0}\to
Z_{c}(3900)K^{0}]$.

In the particle basis, the Wilson coefficients
$a_{1}^{\prime}/a_{1}^{\prime*}$ and the couplings $g_{n}/g_{c}$ for the
decays of $B^{0}\to X(3872)K^{0}$ and $B^{+}\to X(3872)K^{+}$ are different,
resulting in a ratio $\mathcal{B}[B^{0}\to X(3872)K^{0}]/\mathcal{B}[B^{+}\to
X(3872)K^{+}]=0.62\pm 0.13$. With different couplings $g_{n}/g_{c}$ but the
same Wilson coefficients $a_{1}^{\prime}/a_{1}^{\prime*}$, the ratio becomes
$\mathcal{B}[B^{0}\to X(3872)K^{0}]/\mathcal{B}[B^{+}\to X(3872)K^{+}]=0.66\pm
0.14$. On the other hand, with the same couplings $g_{n}/g_{c}$ but different
Wilson coefficients $a_{1}^{\prime}/a_{1}^{\prime*}$, the ratio becomes
$\mathcal{B}[B^{0}\to X(3872)K^{0}]/\mathcal{B}[B^{+}\to X(3872)K^{+}]=0.81\pm
0.17$. Clearly, the isospin breaking of the ratio $\mathcal{B}[B^{0}\to
X(3872)K^{0}]/\mathcal{B}[B^{+}\to X(3872)K^{+}]$ is mainly caused by the
isospin breaking of the $\bar{D}^{*}D$ wave function.

For the $Z_{c}(3900)$, the different Wilson coefficients
$a_{1}^{\prime}/a_{1}^{\prime*}$ lead to the ratio $\mathcal{B}[B^{+}\to
Z_{c}(3900)K^{+}]/\mathcal{B}[B^{0}\to Z_{c}(3900)K^{0}]=0.63\pm 0.29$. With
the same Wilson coefficients $a_{1}^{\prime}/a_{1}^{\prime*}$, the ratio
$\mathcal{B}[B^{+}\to Z_{c}(3900)K^{+}]/\mathcal{B}[B^{0}\to
Z_{c}(3900)K^{0}]$ becomes $0.98\pm 0.39$, which shows no isospin breaking. As
a result, the isospin breaking effect of the ratio $\mathcal{B}[B^{+}\to
Z_{c}(3900)K^{+}]/\mathcal{B}[B^{0}\to Z_{c}(3900)K^{0}]$ originates from the
Wilson coefficients fitted to the experimental data.

For $X_{2}(4013)$, in the particle basis, the ratio $\mathcal{B}[B^{0}\to
X_{2}(4013)K^{0}]/\mathcal{B}[B^{+}\to X_{2}(4013)K^{+}]$ is estimated to be
$0.75\pm 0.16$. With the same couplings $g_{n}^{\prime}/g_{c}^{\prime}$, the
ratio becomes $0.92\pm 0.20$. With the same Wilson coefficients, the ratio
becomes $0.70\pm 0.15$. Clearly, the isospin breaking of the neutral and
charged components in its wave function is responsible for the large isospin
breaking of this ratio.