# Study of $\bar{B}^{0}_{(s)}$ decaying into $D^{0}$ and $\pi^{+}\pi^{-}$,
$K^{+}K^{-}$, or $\pi^{0}\eta$ with final state interactions

Hiwa A. Ahmed<EMAIL_ADDRESS>Medical Physics Department,
College of Medicals and Applied Science, Charmo University, 46023, Chamchamal,
Sulaimania, Iraq  School of Nuclear Science and Technology, University of
Chinese Academy of Sciences, Beijing 100049, China  C. W. Xiao
<EMAIL_ADDRESS>School of Physics and Electronics, Hunan Key Laboratory of
Nanophotonics and Devices, Central South University, Changsha 410083, China

###### Abstract

We investigate the resonant contributions in the decays $\bar{B}^{0}_{(s)}\to
D^{0}\pi^{+}\pi^{-}$, $\bar{B}^{0}_{(s)}\to D^{0}K^{+}K^{-}$, and
$\bar{B}^{0}\rightarrow D^{0}\pi^{0}\eta$ by considering the final state
interactions within the chiral unitary approach. The scalar resonances
produced dynamically from the final state interactions of these decays are the
ones of $f_{0}(500)$, $f_{0}(980)$, and $a_{0}(980)$. In addition, the vector
mesons $\rho$ and $\phi$ produced in the $p$-wave are also taken into account.
From the invariant mass distributions of $B^{0}$ decays, one can see that the
contributions of the states $f_{0}(500)$ and $a_{0}(980)$ are remarkably
larger than the one of $f_{0}(980)$. However, for the case of $B^{0}_{s}$
decays only a clear structure closed to the $K^{+}K^{-}$ threshold appears,
which corresponds to the $f_{0}(980)$ state and has no signal for the
$f_{0}(500)$ resonance. The dominant contributions are from the vector meson
$\phi$ for the decay $\bar{B}^{0}_{s}\to D^{0}K^{+}K^{-}$, where the results
of the $K^{+}K^{-}$ invariant mass distributions are in agreement with the
experimental data up to $1.1$ GeV. Moreover, the branching fractions and the
ratios of branching fractions are also investigated, where some of them are
consistent with the experimental measurements, and the branching fraction
$\text{Br}(\bar{B}^{0}\rightarrow
D^{0}a_{0}(980)[a_{0}(980)\to\pi^{0}\eta])=(6.08\pm 0.72)\times 10^{-5}$ and
the ratio of branching fractions $\text{Br}(\bar{B}^{0}\rightarrow
D^{0}f_{0}(980))/\text{Br}(\bar{B}^{0}\rightarrow D^{0}a_{0}(980))=0.13\pm
0.03$ are predicted.

##  I Introduction

For investigating the $CP$-violation phenomena and understanding the physics
beyond the Standard Model (SM), many non-leptonic three-body $B$ meson decays
have been measured by the collaborations Belle, BaBar, CLEO, LHCb, and so on.
The charmless hadronic $B$ decays with three-pseudoscalar final states were
observed by the CLEO collaboration Eckhart:2002qr , where the branching
fractions of the decays $\bar{B}\rightarrow K^{0}\pi^{+}\pi^{-}$ and
$\bar{B}\rightarrow K^{*+}(892)\pi^{-}$ were measured. In Ref. Lees:2012kxa ,
the BABAR collaboration studied the decays $B^{0}\rightarrow
K^{+}K^{-}K_{s}^{0}$, $B^{+}\rightarrow K^{+}K^{-}K^{+}$, and
$B^{+}\rightarrow K_{s}^{0}K_{s}^{0}K^{+}$ with the amplitude analysis of
Dalitz plot approach, and measured the $CP$-violation parameters based on the
collected data sample of approximately $470\times 10^{6}$ $B\bar{B}$ decays.
Moreover, the Belle collaboration measured the $CP$-violation asymmetries in
the $B^{0}(\bar{B}^{0})\rightarrow K^{+}K^{-}K_{s}^{0}$ decays using the time-
dependent Dalitz approach Nakahama:2010nj , where the direct $CP$-violation
signal was not found. Recently, the LHCb collaboration has reported lots of
the results on the investigation of the $CP$-violation phenomena. The decays
$\bar{B}_{s}^{0}\rightarrow J/\psi f_{0}(980)$ and
$\bar{B}_{s}^{0}\rightarrow\phi\pi^{+}\pi^{-}$ were observed for the first
time by the LHCb collaboration in Ref. Aaij:2011fx  and Ref. Aaij:2016qnm ,
respectively, which could be used to detect the $CP$-violation in the
$B_{s}^{0}$ decays. The amplitude analysis was performed for the decays
$B_{s}^{0}\rightarrow K_{s}^{0}K^{\pm}\pi^{\mp}$,
$\bar{B}^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$, and $\bar{B}^{+}\rightarrow
D^{+}D^{-}K^{+}$ in Refs. Aaij:2019nmr ; Aaij:2019jaq ; Aaij:2020ypa , where
the $CP$-violation effect was discussed and the resonant contributions were
found. In Ref. Aaij:2017zgz , the resonance structures were analyzed in
details for the $\bar{B}_{s}^{0}\rightarrow J/\psi K^{+}K^{-}$ decay with the
$CP$-violation phase measured. In addition, to hint the $CP$-violation effect,
the relative branching ratios for the decays $B^{+}\rightarrow
K^{+}K^{+}K^{-}$, $\pi^{+}K^{+}K^{-}$, $K^{+}\pi^{+}\pi^{-}$, and
$\pi^{+}\pi^{+}\pi^{-}$, were measured precisely in Ref. Aaij:2020dsq  with a
model-independent approach. The $\bar{B}^{0}\rightarrow D^{0}\pi^{+}\pi^{-}$
decay was investigated by the Belle collaboration Kuzmin:2006mw , and the
total branching fraction of the three-body decay $Br(\bar{B}^{0}\rightarrow
D^{0}\pi^{+}\pi^{-})=(8.4\pm 0.4(stat.)\pm 0.8(syst.))\times 10^{-4}$ was
reported, where the intermediate resonance contributions were taken into
account. The LHCb collaboration measured the $\bar{B}^{0}\rightarrow
D^{0}\pi^{+}\pi^{-}$ decay more precisely with the Dalitz plot technique and
large data sample Aaij:2015sqa , of which the measured branching ratio was
$(8.46\pm 0.14\pm 0.29\pm 0.40)\times 10^{-4}$, and where many resonances
contributions were found. In Ref. Aaij:2015rqa , the
$\bar{B}_{s}^{0}\rightarrow D^{0}f_{0}(980)$ decay was searched by LHCb
collaboration using the collected data of $pp$ collisions during 2011 and
2012, where no significant signal was found and the upper limits on its
branching fraction were set. Moreover, the LHCb collaboration reported the
first observations of the decays $\bar{B}_{(s)}^{0}\rightarrow D^{0}\phi$
Aaij:2018jqv  and $\bar{B}_{(s)}^{0}\rightarrow D^{0}K^{+}K^{-}$ Aaij:2018rol
with the corresponding branching fractions determined.

Except for the hint of $CP$-violation, the two-body resonance information in
the three-body $B_{(s)}$ decays is also caught much theoretical attention.
Utilizing the light-cone sum rule approach, Ref. Shi:2021ste  investigated the
$B^{-}\to K^{+}K^{-}\pi^{-}$ decay with the contributions from the scalar,
vector and tensor resonances. Employing the factorization-assisted
topological-amplitude approach, the three-body charmed $B_{(s)}$ decays were
studied in Ref. Zhou:2021yys  by considering the resonant effects, where the
related branching fractions and the SU(3) symmetry breaking effect were
calculated. With the perturbative QCD (pQCD) approach, the charmless decays
$B_{(s)}\to V\pi\pi$ ($V$ presented the vector meson) were studied in Ref.
Yang:2021zcx , where the $f_{0}(980)$ resonance contribution was introduced in
the calculation of the corresponding branching fractions. The
$f_{0}(980)$-$f_{0}(500)$ mixing was discussed in Ref. Zhang:2016qvq  for the
charmed $B_{(s)}$ decays under the pQCD approach. The decays
$\bar{B}_{(s)}^{0}\rightarrow D^{0}(\bar{D}^{0})\pi^{+}\pi^{-}$ were studied
in Ref. Xing:2019xti  within the pQCD approach, where only the $s$-wave
contributions were taken into account and the Breit-Wigner model was used for
the scalar form factors of the resonances $f_{0}(500)$, $f_{0}(980)$,
$f_{0}(1500)$ and $f_{0}(1700)$. Also using the pQCD formalism, Ref.
Li:2020zng  investigated the three-body decays
$B_{(s)}\rightarrow[D^{(*)},\bar{D}^{(*)}]K^{+}K^{-}$ by considering the
resonance contributions in the $s$-, $p$-, and $d$-waves. Analogously,
applying the pQCD approach, the resonance contributions were taken into
account in the $B_{(s)}$ three-body decays in Refs. Zou:2020atb ; Ma:2020jsb ;
Li:2021cnd . Based on the QCD factorization approach and constrained by the
analyticity and unitarity, the parametrizations of three-body $B$ and $D$ weak
decay amplitudes were discussed in detail in Ref. Boito:2017jav  with the
isobar model for the quasi-two-body resonance contributions. In Ref.
Liang:2014ama , the decays of $\bar{B}_{(s)}^{0}$ into a $D^{0}$ meson and a
scalar meson or a vector meson were researched using the final state
interaction approach, where the scalar resonances $f_{0}(500)$, $f_{0}(980)$,
and $a_{0}(980)$, were dynamical generated in the subprocesses of quasi-two-
body interaction with the chiral unitary approach (ChUA) Oller:1997ti ;
Oset:1997it ; Kaiser:1998fi ; Oller:2000fj ; Oller:2000ma ; Hyodo:2011ur .
Furthermore, also combined with the final state interaction approach and the
ChUA, three-body hadronic $B_{(s)}$ decays were analyzed in Refs.
Liang:2014tia ; Liang:2015qva ; Bayar:2014qha  to study the dynamical
production properties of the scalar resonances $f_{0}(500)$, $f_{0}(980)$ and
$a_{0}(980)$. More discussions and applications about this approach can be
referred to Ref. Oset:2016lyh .

Recently, based on the BESIII measurements, Refs. Achasov:2020qfx ;
Achasov:2020aun ; Achasov:2021dvt  discussed the four-quark nature of the
states $f_{0}(500)$, $f_{0}(980)$ and $a_{0}(980)$ in the $D_{(s)}$
semileptonic decays and the $J/\psi$ radiative decays, where the four-quark
nature of these resonances had been early concerns in Refs. Achasov:1979xc ;
Achasov:1980gu ; Achasov:1980tb ; Achasov:1981kh ; Achasov:1987ts ;
Achasov:2003cn . Also based on the BESIII measurements and with the final
state interaction under the ChUA, Ref. Molina:2019udw  investigated the so-
called pure $W$-annihilation decay $D^{+}_{s}\to\pi^{+}\pi^{0}\eta$, and Ref.
Duan:2020vye  studied the single Cabibbo suppressed process
$D^{+}\rightarrow\pi^{+}\pi^{0}\eta$, where the main role of the $a_{0}(980)$
state generated in the subprocesses was found. In Ref. Ikeno:2021kzf  the
production of the $a_{0}(980)$ resonance was discussed in detail for the
single Cabibbo suppressed decays $D^{+}\to\pi^{+}\eta\eta$ and
$D^{+}\to\pi^{+}\pi^{0}\eta$, where both the internal and external
$W$-emission mechanisms were taken into account. Moreover, the reactions
$D^{+}\to K^{-}K^{+}K^{+}$ and $D_{s}^{+}\to\pi^{+}K_{s}^{0}K_{s}^{0}$ were
investigated in Refs. Roca:2020lyi  and Dai:2021owu , respectively, where some
resonances were generated in the two-body final state interactions. The decays
$B^{0}_{(s)}\to\phi\pi^{+}\pi^{-}$ were researched in our former work of Ref.
Ahmed:2020qkv , where the productions of the $f_{0}(980)$ resonance and some
vector mesons were examined. Thus, in the present work, inspired by the new
results of the LHCb collaboration for the decays $\bar{B}_{(s)}^{0}\rightarrow
D^{0}\phi$ Aaij:2018jqv  and $\bar{B}_{(s)}^{0}\rightarrow D^{0}K^{+}K^{-}$
Aaij:2018rol  as discussed before, we continue to investigate the
$\bar{B}^{0}_{(s)}$ decays with the processes of decaying into a $D^{0}$ meson
and a pair of pseudoscalar mesons $\pi^{+}\pi^{-}$, $\pi^{0}\eta$ and
$K^{+}K^{-}$ within the final state interaction approach, aiming to study the
productions of the resonances $f_{0}(500)$, $f_{0}(980)$ and $a_{0}(980)$ in
the three-body decay processes.

Our work is organized as following. In the next section, we will introduce the
formalism of the final state interaction in $s$-wave within the ChUA.
Furthermore, we also discuss the vector meson productions in $p$-wave under
the same weak decay mechanism. In the following section, we show the results
of the invariant mass distributions and the branching fractions of
corresponding decay channels. At the end, we make a short conclusion.

##  II Formalism

In the present work, we investigate the decays $\bar{B}^{0}_{(s)}\rightarrow
D^{0}\;\pi^{+}\pi^{-}(\pi^{0}\eta,K^{+}K^{-})$ within the final state
interaction approach, and take into account the $s$\- and $p$-waves resonance
contributions, which are from the resonances $f_{0}(500)$, $f_{0}(980)$, and
$a_{0}(980)$ in the $s$-wave productions, and the states $\rho$, $\omega$, and
$\phi$ in the $p$-wave case. In our formalism for the hadroniazation mechanism
of the three particles produced in the final states of the decay processes
$\bar{B}^{0}_{(s)}\rightarrow D^{0}\;\pi^{+}\pi^{-}(K^{+}K^{-},\pi^{0}\eta)$,
at the quark level the bottom quark goes through the weak transitions $b\to
c\bar{u}d$ and $b\to c\bar{u}s$ for the $\bar{B}^{0}$ and $\bar{B}^{0}_{s}$,
respectively, where the $c\bar{u}$ quarks produce the $D^{0}$ meson and the
remaining pair of quarks undergoes the hadronization procedure to generate the
meson pairs $\pi^{+}\pi^{-}$, $\pi^{0}\eta$, or $K^{+}K^{-}$ in the final
states; see more details below.

For the decays $\bar{B}^{0}_{(s)}\to D^{0}\pi^{+}\pi^{-}$,
$\bar{B}^{0}_{(s)}\to D^{0}K^{+}K^{-}$, and $\bar{B}^{0}\rightarrow
D^{0}\pi^{0}\eta$, the Feynman diagrams of these decay processes in the quark
level are shown in Fig. 1. These processes are Cabibbo suppressed and proceed
via the weak transitions of the quarks $b\to s$ and $b\to d$ at the end for
the decay of $\bar{B}^{0}_{s}$ and $\bar{B}^{0}$, respectively. Considering
the fact that the $\bar{d}$ quark in Fig. 1(a) and $\bar{s}$ quark in Fig.
1(b) are spectator, these two processes have similar weak decay mechanism; see
more details and discussions in Refs. Ahmed:2020qkv ; Liang:2014tia ;
Liang:2014ama .

(a)

(b)

Figure 1: Feynman diagrams for the decays of $\bar{B}^{0}_{(s)}$ into $D^{0}$
and a primary $q\bar{q}$ pair.

###  II.1 Scalar meson production in $s$-wave

In this subsection, we discuss the formalism of three pseudoscalar mesons
produced in the decays $\bar{B}^{0}_{(s)}\to D^{0}\pi^{+}\pi^{-}$,
$\bar{B}^{0}_{(s)}\to D^{0}K^{+}K^{-}$, and $\bar{B}^{0}\rightarrow
D^{0}\pi^{0}\eta$ in the quark level, where the final state interaction in the
hadron level is taken into account within the ChUA. The dominant weak decay
mechanism for the $\bar{B}^{0}_{(s)}$ decays is depicted in Fig. 1, which is
proceeded as,

$\displaystyle\bar{B}^{0}(b\bar{d})$ $\displaystyle\Rightarrow
V_{cb}c\mathit{W^{-}}\bar{d}\Rightarrow
V_{cb}V_{ud}^{*}(c\bar{u})(d\bar{d})\,,$ (1)
$\displaystyle\bar{B}^{0}_{s}(b\bar{s})$ $\displaystyle\Rightarrow
V_{cb}c\mathit{W^{-}}\bar{s}\Rightarrow
V_{cb}V_{us}^{*}(c\bar{u})(s\bar{s})\,,$ (2)

where $V_{q_{1}q_{2}}$ is the element of the CKM matrix for the transition of
the quarks $q_{1}\to q_{2}$. The $c\bar{u}$ quark pair produces the $D^{0}$
meson. The other primary $q\bar{q}$ pair creates a pair of mesons
$\pi^{+}\pi^{-}$, $\pi^{0}\eta$, or $K^{+}K^{-}$ in the final states. The
process of producing two pseudoscalar mesons is proceeded by adding an extra
$q\bar{q}$ pairs generated from the vacuum to the primary quark pair, written
as $u\bar{u}+d\bar{d}+s\bar{s}$, which is shown in Fig. 2. The hadronization
processes are mathematically formulated as

$\displaystyle\bar{B}^{0}$ $\displaystyle\Rightarrow
V_{cb}V_{ud}^{*}(c\bar{u}\to D^{0})[d\bar{d}\to
d\bar{d}\cdot(u\bar{u}+d\bar{d}+s\bar{s})]\,,$ (3)
$\displaystyle\bar{B}^{0}_{s}$ $\displaystyle\Rightarrow
V_{cb}V_{us}^{*}(c\bar{u}\to D^{0})[s\bar{s}\to
s\bar{s}\cdot(u\bar{u}+d\bar{d}+s\bar{s})]\,.$ (4)

The procedure of transferring the $q\bar{q}$ pairs to the physical mesons is
explicitly expressed as,

$\displaystyle d\bar{d}\cdot(u\bar{u}+d\bar{d}+s\bar{s})$
$\displaystyle\to(\Phi\cdot\Phi)_{22}=\pi^{+}\pi^{-}+\frac{1}{2}\pi^{0}\pi^{0}-\frac{2}{\sqrt{6}}\pi^{0}\eta+K^{0}\bar{K}^{0}+\frac{1}{3}\eta\eta,$
(5) $\displaystyle s\bar{s}\cdot(u\bar{u}+d\bar{d}+s\bar{s})$
$\displaystyle\to(\Phi\cdot\Phi)_{33}=K^{-}K^{+}+K^{0}\bar{K}^{0}+\frac{1}{3}\eta\eta\,,$

where $\Phi$ is the matrix of the meson fields. Taken the standard
$\eta-\eta\prime$ mixing into account, the $\Phi$ matrix is written as

$\Phi=\left(\begin{array}[]{ccc}{\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{3}}\eta+\frac{1}{\sqrt{6}}\eta\prime}&{\pi^{+}}&{K^{+}}\\\
{\pi^{-}}&{-\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{3}}\eta+\frac{1}{\sqrt{6}}\eta\prime}&{K^{0}}\\\
{K^{-}}&{\bar{K}^{0}}&{-\frac{1}{\sqrt{3}}\eta+\sqrt{\frac{2}{3}}\eta\prime}\end{array}\right).$
(6)

Figure 2: Procedure for the hadronization $q\bar{q}\rightarrow
q\bar{q}(u\bar{u}+d\bar{d}+s\bar{s})$. Figure 3: Diagrammatic representation
of the meson pair production in the final state interactions of the
$\bar{B}^{0}$ decay. Figure 4: Analogous to Fig. 3, but for the
$\bar{B}^{0}_{s}$ decay.

The final states of Eq. (5) are similar to the one obtained in Ref.
Ahmed:2020qkv , where only $K\bar{K}$ and $\eta\eta$ channels were produced in
the final states of the $\bar{B}^{0}_{s}$ decay. Even though, the final states
are hadronized after the weak decay process, they can go to further final
state interactions. In the case of the $B^{0}$ decay, we have three pairs of
the pseudoscalar mesons, $\pi^{+}\pi^{-}$, $\pi^{0}\eta$, and $K^{+}K^{-}$
alongside the $D^{0}$ meson as depicted in Fig. 3, where one can see that the
$\pi^{+}\pi^{-}$ and $\pi^{0}\eta$ can be produced from the direct and
rescattering mechanisms. However, the $K^{+}K^{-}$ are only produced from the
rescattering of the final states. For the $B_{s}^{0}$ decay, the situation is
different. The $\pi^{+}\pi^{-}$ are generated from the rescattering of the
final states, while the $K^{+}K^{-}$ can be produced from the direct
production and the interactions of the final states, as shown in Fig. 4.
Therefore, the scattering amplitudes of these decay precesses in Figs. 3 and 4
can be written as

$\begin{split}&t_{\bar{B}^{0}\rightarrow
D^{0}\pi^{+}\pi^{-}}=V_{P}V_{cb}V_{ud}^{*}\left(1+G_{\pi^{+}\pi^{-}}T_{\pi^{+}\pi^{-}\rightarrow\pi^{+}\pi^{-}}\right.\\\
&\quad+2\frac{1}{2}\frac{1}{2}G_{\pi^{0}\pi^{0}}T_{\pi^{0}\pi^{0}\rightarrow\pi^{+}\pi^{-}}\left.+G_{K^{0}\bar{K}^{0}}T_{K^{0}\bar{K}^{0}\rightarrow\pi^{+}\pi^{-}}+2\frac{1}{3}\frac{1}{2}G_{\eta\eta}T_{\eta\eta\rightarrow\pi^{+}\pi^{-}}\right),\end{split}$
(7)

$\begin{split}t_{\bar{B}^{0}\rightarrow
D^{0}\pi^{0}\eta}=V_{P}V_{cb}V_{ud}^{*}\left(-\frac{2}{\sqrt{6}}-\frac{2}{\sqrt{6}}G_{\pi^{0}\eta}T_{\pi^{0}\eta\rightarrow\pi^{0}\eta}+G_{K^{0}\bar{K}^{0}}T_{K^{0}\bar{K}^{0}\rightarrow\pi^{0}\eta}\right),\end{split}$
(8)

$\begin{split}&t_{\bar{B}^{0}\rightarrow
D^{0}K^{+}K^{-}}=V_{P}V_{cb}V_{ud}^{*}\left(G_{\pi^{+}\pi^{-}}T_{\pi^{+}\pi^{-}\rightarrow
K^{+}K^{-}}+2\frac{1}{2}\frac{1}{2}G_{\pi^{0}\pi^{0}}T_{\pi^{0}\pi^{0}\rightarrow
K^{+}K^{-}}\right.\\\
&\quad+2\frac{1}{3}\frac{1}{2}G_{\eta\eta}T_{\eta\eta\rightarrow
K^{+}K^{-}}-\frac{2}{\sqrt{6}}G_{\pi^{0}\eta}T_{\pi^{0}\eta\rightarrow
K^{+}K^{-}}\left.+G_{K^{0}\bar{K}^{0}}T_{K^{0}\bar{K}^{0}\rightarrow
K^{+}K^{-}}\right),\end{split}$ (9)

$\begin{split}&t_{\bar{B}_{s}^{0}\rightarrow
D^{0}\pi^{+}\pi^{-}}=V_{P}V_{cb}V_{us}^{*}\left(G_{K^{+}K^{-}}T_{K^{+}K^{-}\rightarrow\pi^{+}\pi^{-}}\right.\\\
&\left.\quad+G_{K^{0}\bar{K}^{0}}T_{K^{0}\bar{K}^{0}}\rightarrow\pi^{+}\pi^{-}+2\frac{1}{3}\frac{1}{2}G_{\eta\eta}T_{\eta\eta\rightarrow\pi^{+}\pi^{-}}\right),\end{split}$
(10)

$\begin{split}&t_{\bar{B}_{s}^{0}\rightarrow
D^{0}K^{+}K^{-}}=V_{P}V_{cb}V_{us}^{*}\left(1+G_{K^{+}K^{-}}T_{K^{+}K^{-}\rightarrow
K^{+}K^{-}}\right.\\\
&\left.\quad+G_{K^{0}\bar{K}^{0}}T_{K^{0}\bar{K}^{0}\rightarrow
K^{+}K^{-}}+2\frac{1}{3}\frac{1}{2}G_{\eta\eta}T_{\eta\eta\rightarrow
K^{+}K^{-}}\right),\end{split}$ (11)

where $V_{P}$ is the production vertex factor, which is universal for these
reactions due to the similar weak decay mechanism, and accounts for the direct
reproduction diagram. Note that, there is a factor of $2$ in the terms related
with the identical particles, such as the $\pi^{0}\pi^{0}$ and $\eta\eta$,
which has been cancelled with the factor of $\frac{1}{2}$ in their propagator
functions within our normalization scheme. In the case of the
$\bar{B}^{0}\rightarrow D^{0}K^{+}K^{-}$ decay, we have the contributions from
isospins $I=0$ and $I=1$, and thus, different isospin contributions in Eq. (9)
can be decomposited as,

$t_{\bar{B}^{0}\rightarrow D^{0}K^{+}K^{-}}=t^{I=0}+t^{I=1}\,,$ (12)

where $t^{I=0}$ and $t^{I=1}$ are given by

$\displaystyle t^{I=0}=$ $\displaystyle
V_{P}V_{cb}V_{ud}^{*}\left[G_{\pi\pi}T_{\pi^{+}\pi^{-}\rightarrow
K^{+}K^{-}}+\frac{1}{2}G_{\pi\pi}T_{\pi^{0}\pi^{0}\rightarrow
K^{+}K^{-}}\right.$ (13)
$\displaystyle\left.+\frac{1}{3}G_{\eta\eta}T_{\eta\eta\rightarrow
K^{+}K^{-}}+G_{K\bar{K}}\left(\frac{1}{2}T_{K^{0}\bar{K}^{0}\rightarrow
K^{+}K^{-}}+\frac{1}{2}T_{K^{+}K^{-}\rightarrow K^{+}K^{-}}\right)\right],$

$\displaystyle t^{I=1}=$ $\displaystyle
V_{P}V_{cb}V_{ud}^{*}\left[-\frac{2}{\sqrt{6}}G_{\pi^{0}\eta}T_{\pi^{0}\eta\rightarrow
K^{+}K^{-}}\right.$ (14)
$\displaystyle\left.+G_{K\bar{K}}\left(\frac{1}{2}T_{K^{0}\bar{K}^{0}\rightarrow
K^{+}K^{-}}-\frac{1}{2}T_{K^{+}K^{-}\rightarrow K^{+}K^{-}}\right)\right].$

The scattering amplitudes of the coupled channels $T_{ij}$ for the transition
of $i\to j$ channel are evaluated from the coupled channel Bethe-Salpeter
equation with the on-shell description,

$T=[1-VG]^{-1}V,$ (15)

where $V$ matrix is made of the transition potentials. For the coupled
channels $\pi^{+}\pi^{-}$, $\pi^{0}\pi^{0}$, $K^{+}K^{-}$, $K^{0}\bar{K}^{0}$,
$\eta\eta$, and $\pi^{0}\eta$ with the labels $1$ to $6$, respectively, we
take the potentials from Refs. Oller:1997ti ; Xiao:2019lrj ,

$\displaystyle V_{11}=-\frac{1}{2f^{2}}s,\quad
V_{12}=-\frac{1}{\sqrt{2}f^{2}}\left(s-m_{\pi}^{2}\right),\quad
V_{13}=-\frac{1}{4f^{2}}s,$ (16) $\displaystyle
V_{14}=-\frac{1}{4f^{2}}s,\quad
V_{15}=-\frac{1}{3\sqrt{2}f^{2}}m_{\pi}^{2},\quad V_{16}=0,$ $\displaystyle
V_{22}=-\frac{1}{2f^{2}}m_{\pi}^{2},V_{23}=-\frac{1}{4\sqrt{2}f^{2}}s,\quad
V_{24}=-\frac{1}{4\sqrt{2}f^{2}}s,$ $\displaystyle
V_{25}=-\frac{1}{6f^{2}}m_{\pi}^{2},\quad
V_{26}=0,V_{33}=-\frac{1}{2f^{2}}s,\quad V_{34}=-\frac{1}{4f^{2}}s,$
$\displaystyle
V_{35}=-\frac{1}{12\sqrt{2}f^{2}}\left(9s-6m_{\eta}^{2}-2m_{\pi}^{2}\right),$
$\displaystyle
V_{36}=-\frac{\sqrt{3}}{12f^{2}}\left(3s-\frac{8}{3}m_{K}^{2}-\frac{1}{3}m{\pi}^{2}-6m_{\eta}^{2}\right),$
$\displaystyle
V_{44}=-\frac{1}{2f^{2}}s,V_{45}=-\frac{1}{12\sqrt{2}f^{2}}\left(9s-6m_{\eta}^{2}-2m_{\pi}^{2}\right),$
$\displaystyle
V_{46}=\frac{\sqrt{3}}{12f^{2}}\left(3s-\frac{8}{3}m_{K}^{2}-\frac{1}{3}m{\pi}^{2}-6m_{\eta}^{2}\right),$
$\displaystyle
V_{55}=-\frac{1}{18f^{2}}\left(16m_{K}^{2}-7m_{\pi}^{2}\right),V_{56}=0,$
$\displaystyle V_{66}=-\frac{1}{3f^{2}}m_{\pi}^{2}\texttt{ .}$

And the diagonal matrix $G$ is constructed by the loop functions of two meson
propagators, given by

$G_{ii}(s)=\int_{0}^{q_{\max}}\frac{q^{2}dq}{(2\pi)^{2}}\frac{\omega_{1}+\omega_{2}}{\omega_{1}\omega_{2}\left[s-\left(\omega_{1}+\omega_{2}\right)^{2}+i\varepsilon\right]}\text{
,}$ (17)

with $q=|\vec{q}\,|$ and $\omega_{i}=(\vec{q}^{\>2}+m_{i}^{2})^{1/2}$, where
the cutoff $q_{max}$ is chosen as $600$ MeV for the case of including
$\eta\eta$ channel Liang:2014tia  and $931$ MeV for the one of excluding
$\eta\eta$ channel Xiao:2019lrj .

Finally, the invariant mass distributions for each decay channel can be
obtained from the evaluation of the differential decay width
$\frac{d\Gamma}{dM_{\text{inv}}}$. Thus, we have

$\frac{d\Gamma}{dM_{\text{inv}}}=\frac{1}{(2\pi)^{3}}\frac{1}{4M_{\bar{B}^{0}_{(s)}}^{2}}p_{D^{0}}\tilde{p}_{P}\bar{\sum}\sum\left|t_{B_{(s)}^{0}\rightarrow
D^{0}PP^{\prime}}\right|^{2},$ (18)

where $t_{B_{(s)}^{0}\rightarrow D^{0}PP^{\prime}}$ is the scattering
amplitude obtained from Eqs. (7-11) with $P^{(\prime)}$ stands for a
pseudoscalar meson, $p_{D^{0}}$ the $D^{0}$ momentum in the rest frame of the
decaying $\bar{B}^{0}_{(s)}$ meson, and $\tilde{p}_{P}$ the kaon (pion)
momentum in the rest frame of the $K^{+}K^{-}$ ($\pi^{+}\pi^{-}$,
$\pi^{0}\eta$) system, which are given by

$\displaystyle
p_{D^{0}}=\frac{\lambda^{1/2}\left(M_{\bar{B}^{0}_{(s)}}^{2},M_{D^{0}}^{2},M_{\text{inv}}^{2}\right)}{2M_{B_{(s)}}},$
(19)
$\displaystyle\tilde{p}_{P}=\frac{\lambda^{1/2}\left(M_{\text{inv}}^{2},m_{P}^{2},m_{P^{\prime}}^{2}\right)}{2M_{\text{inv
}}},$

with the usual Källen triangle function
$\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2(ab+ac+bc)$.

###  II.2 Vector meson production in $p$-wave

Despite producing the pseudoscalar meson pair alongside $D^{0}$, the vector
mesons in $p$-wave also can be generated in these decays, which finally decay
into the pseudoscalar meson pair. The $\rho^{0}$ and $\omega$ can be formed
from the primary $d\bar{d}$ quarks, and $\phi$ from the $s\bar{s}$ quark pair,
which can be decayed into $\pi^{+}\pi^{-}$ and $K^{+}K^{-}$, respectively. As
done in Ref. Liang:2014ama , we can write down the amplitudes for each
channel. It is worth mentioning that in the work of Ref. Liang:2014ama , the
decay channel of $\bar{B}^{0}\rightarrow D^{0}\phi$ had not been considered.
The allowed scattering amplitudes in $p$-wave are given by

$t_{\bar{B}^{0}\rightarrow
D^{0}\rho^{0}}=\frac{1}{\sqrt{2}}\tilde{V}_{P}V_{cb}V_{ud}^{*}\textbf{ },\quad
t_{\bar{B}^{0}\rightarrow
D^{0}\omega}=\frac{1}{\sqrt{2}}\tilde{V}_{P}V_{cb}V_{ud}^{*}\textbf{ },$ (20)

$t_{\bar{B}^{0}\rightarrow D^{0}\phi}=\tilde{V}_{P}V_{cb}V_{ud}^{*}\textbf{
},\quad t_{\bar{B}^{0}_{s}\rightarrow
D^{0}\phi}=\tilde{V}_{P}V_{cb}V_{us}^{*}\textbf{ },$ (21)

where the prefactor $\frac{-1}{\sqrt{2}}$ and $\frac{1}{\sqrt{2}}$ are the
$d\bar{d}$ component in $\rho^{0}$ and $\omega$, respectively, and
$\tilde{V}_{P}$ the $p$-wave production vertex factor. The widths for the
decays $\bar{B}^{0}_{(s)}\to D^{0}V$ with a vector meson ($V$) in the products
are given by

$\Gamma_{\bar{B}^{0}_{(s)}\rightarrow
D^{0}V}=\frac{1}{8\pi}\frac{1}{m_{B_{(s)}^{0}}^{2}}\left|t_{\bar{B}^{0}_{(s)}\rightarrow
D^{0}V}\right|^{2}p_{D}.$ (22)

Note that, we consider that the resonances $\rho^{0}$ and $\phi$ can only
decay into $\pi^{+}\pi^{-}$ and $K^{+}K^{-}$, respectively, of which the
contributions to the invariant mass distributions can be obtained by using the
spectral function Liang:2014ama ; Wang:2020pem ,

$\frac{d\Gamma_{\bar{B}^{0}\rightarrow
D^{0}\rho^{0}}}{dM_{\text{inv}}\left(\pi^{+}\pi^{-}\right)}=-\frac{1}{\pi}2m_{\rho}\operatorname{Im}\frac{1}{M_{\text{inv}}^{2}-m_{\rho}^{2}+im_{\rho}\tilde{\Gamma}_{\rho}\left(M_{\text{inv}}\right)}\Gamma_{\bar{B}^{0}\rightarrow
D^{0}\rho^{0}},$ (23)

$\frac{d\Gamma_{\bar{B}^{0}_{s}\rightarrow
D^{0}\phi}}{dM_{\text{inv}}\left(K^{+}K^{-}\right)}=-\frac{1}{\pi}m_{\phi}\operatorname{Im}\frac{1}{M_{\text{inv}}^{2}-m_{\phi}^{2}+im_{\phi}\tilde{\Gamma}_{\phi}\left(M_{\text{inv}}\right)}\Gamma_{\bar{B}^{0}_{s}\rightarrow
D^{0}\phi},$ (24)

where $\tilde{\Gamma}_{\rho}\left(M_{\text{inv}}\right)$ and
$\tilde{\Gamma}_{\phi}\left(M_{\text{inv}}\right)$ are the energy dependent
decay widths of the $\rho^{0}$ decaying into two pions and the $\phi$ decaying
into $K^{+}K^{-}$ with $\frac{1}{2}$ weight, respectively, given by

$\displaystyle\tilde{\Gamma}_{\rho}\left(M_{\text{inv}}\right)=\Gamma_{\rho}\textbf{
}\left(\frac{p_{\pi}^{\text{off }}}{p_{\pi}^{\text{on }}}\right)^{3},$ (25)
$\displaystyle
p_{\pi}^{\mathrm{off}}=\frac{\lambda^{1/2}\left(M_{\text{inv}}^{2},m_{\pi}^{2},m_{\pi}^{2}\right)}{2M_{\text{inv}}}\theta\left(M_{\text{inv}}-2m_{\pi}\right),$
$\displaystyle
p_{\pi}^{\mathrm{on}}=\frac{\lambda^{1/2}\left(m_{\rho}^{2},m_{\pi}^{2},m_{\pi}^{2}\right)}{2m_{\rho}},$

$\displaystyle\tilde{\Gamma}_{\phi}\left(M_{\text{inv}}\right)=\Gamma_{\phi}\textbf{
}\left(\frac{p_{K}^{\text{off }}}{p_{K}^{\text{on }}}\right)^{3},$ (26)
$\displaystyle
p_{K}^{\mathrm{off}}=\frac{\lambda^{1/2}\left(M_{\text{inv}}^{2},m_{K}^{2},m_{\bar{K}}^{2}\right)}{2M_{\text{inv}}}\theta\left(M_{\text{inv}}-2m_{K}\right),$
$\displaystyle
p_{K}^{\mathrm{on}}=\frac{\lambda^{1/2}\left(m_{\phi}^{2},m_{K}^{2},m_{\bar{K}}^{2}\right)}{2m_{\phi}},$

with $p^{\mathrm{on}}(p^{\mathrm{off}})$ the on-shell (off-shell) three
momentum in the rest frame.

##  III Results

In the present work, considering the final state interactions, we look for the
resonant contributions in the energy region lower than 1.2 GeV. The coupled
channels considered in our work are the ones $\pi^{+}\pi^{-}$,
$\pi^{0}\pi^{0}$, $K^{+}K^{-}$, $K^{0}\bar{K}^{0}$, $\eta\eta$, and
$\pi^{0}\eta$. As done in the work of Ahmed:2020qkv , in our results we also
estimate the theoretical uncertainties from including or excluding the
$\eta\eta$ channel contribution. In our formalism, the free parameters are the
cutoff and the production vertex factors. As discussed above, the value of the
cutoff is already taken as $q_{max}=600$ MeV for the case of including
$\eta\eta$ channel Liang:2014tia  and $931$ MeV for the one of excluding
$\eta\eta$ channel Xiao:2019lrj . Thus, the only unknown parameters left in
our formalism are two production vertex factors, one of which is $V_{p}$ in
the $s$-wave case and the other one $\tilde{V_{p}}$ in $p$-wave. More
discussions about the partial wave analysis and the final state interactions
can be found in the recent review of Ref. Yao:2020bxx . Next, we can determine
the $V_{p}$ using the experimental branching ratio of the
$\bar{B}^{0}\rightarrow D^{0}f_{0}(500)$ decay. For the cases of considering
the $\eta\eta$ channel and not including the $\eta\eta$ channel, the results
are given by Eq. (27) and Eq. (28), respectively, written

$\text{Br}(\bar{B}^{0}\rightarrow
D^{0}f_{0}(500))=\frac{\Gamma_{\bar{B}^{0}\rightarrow
D^{0}f_{0}(500)}}{\Gamma_{\bar{B}^{0}}}=\frac{\int_{2m_{\pi}}^{900}\frac{d\Gamma_{\bar{B}^{0}\rightarrow
D^{0}f_{0}(500)}}{dM_{inv}}dM_{inv}}{\Gamma_{\bar{B}^{0}}}\\\
=\frac{V_{p}^{2}}{\Gamma_{\bar{B}^{0}}}\times(1.92\times 10^{-5})\textbf{ },$
(27)

$\text{Br}(\bar{B}^{0}\rightarrow
D^{0}f_{0}(500))=\frac{\Gamma_{\bar{B}^{0}\rightarrow
D^{0}f_{0}(500)}}{\Gamma_{\bar{B}^{0}}}=\frac{\int_{2m_{\pi}}^{900}\frac{d\Gamma_{\bar{B}^{0}\rightarrow
D^{0}f_{0}(500)}}{dM_{inv}}dM_{inv}}{\Gamma_{\bar{B}^{0}}}\\\
=\frac{V_{p}^{2}}{\Gamma_{\bar{B}^{0}}}\times(2.58\times 10^{-5})\textbf{ }.$
(28)

Thus, using the measured branching fraction of
$\text{Br}(\bar{B}^{0}\rightarrow D^{0}f_{0}(500))=(0.68\pm 0.08)\times
10^{-4}$ in Particle Data Group ParticleDataGroup:2020ssz , we obtain
$\frac{V_{p}^{2}}{\Gamma_{\bar{B}^{0}}}=(3.54\pm 0.42)$ and
$\frac{V_{p}^{2}}{\Gamma_{\bar{B}^{0}}}=(2.63\pm 0.31)$ for the two cases,
respectively, where the uncertainties are estimated from the errors of the
experimental branching ratio. Furthermore, for the case of $\tilde{V_{p}}$ in
the $p$-wave formalism, using Eq. (22), we have

$\text{Br}(\bar{B}^{0}\rightarrow
D^{0}\rho^{0})=\frac{\Gamma_{\bar{B}^{0}\rightarrow
D^{0}\rho^{0}}}{\Gamma_{\bar{B}^{0}}}=\frac{\tilde{V}_{P}^{2}}{\Gamma_{\bar{B}^{0}}}\times
0.013.$ (29)

Then, taking the experimental results of $\text{Br}(\bar{B}^{0}\rightarrow
D^{0}\rho^{0})=(3.21\pm 0.21)\times 10^{-4}$ ParticleDataGroup:2020ssz , we
can get $\frac{\tilde{V}_{P}^{2}}{\Gamma_{\bar{B}^{0}}}=0.024\pm 0.002$, which
is a common value for the $p$-wave calculations.

In Fig. 5, we show the results of the invariant mass distributions of the
$\pi^{+}\pi^{-}$, $K^{+}K^{-}$, and $\pi^{0}\eta$ channels in the decay
processes $\bar{B}^{0}\rightarrow D^{0}PP^{\prime}$, where $P^{(\prime)}$
corresponds to a pseudoscalar meson, which are consistent with the results of
Ref. Liang:2014ama . Note that in Fig. 5 only the $s$-wave contribution is
taken into account and the one for $p$-wave discussed later. From the results
of Fig. 5(a) for the case of $\bar{B}^{0}\rightarrow D^{0}\pi^{+}\pi^{-}$, one
can see that the $f_{0}(500)$ state has a major contribution above the
$\pi\pi$ threshold, and the clear signal of the $f_{0}(980)$ resonance is seen
especially for the case of including $\eta\eta$ channels, see the solid (blue)
line, which shows up as a narrow and small peak near the $K\bar{K}$ threshold.
This behaviour of $\pi^{+}\pi^{-}$ mass distribution is similar to the results
found in the decays $\bar{B}^{0}\rightarrow J/\psi\pi^{+}\pi^{-}$
Liang:2014tia  and $\bar{B}^{0}\rightarrow\phi\pi^{+}\pi^{-}$ Ahmed:2020qkv .
As shown in Fig. 5(b), the $a_{0}(980)$ state appears in the $\pi^{0}\eta$
mass distributions and has a sizable strength, which is several times stronger
than the $f_{0}(980)$ state in Fig. 5(a), while the normalization factor is
the same for both cases. Another feature of Fig. 5(b) is the appearing of cusp
effect for the case of including the contribution of $\eta\eta$ channel, where
similar effect was found in the work of Ref. Ahmed:2020kmp . Note that, for
the invariant mass distribution of the $B^{0}$ decaying into $D^{0}$ and two
scalar mesons, there would be two contributions from isospins $I=0$ and 1 for
the two-body interactions. In the case of $\pi^{+}\pi^{-}$ components, the
contribution only comes from the parts of $I=0$, and the case of $\pi^{0}\eta$
channel only has $I=1$ parts contributed. However, the situation is different
for the case of $K^{+}K^{-}$ in the final states. In Fig. 5(c), the
$K^{+}K^{-}$ invariant mass distribution has a mixing of isospins $I=0$ and 1,
which means that this result has the contributions of both $f_{0}(980)$ and
$a_{0}(980)$ states. In order to show the strength of each individual state in
the $K^{+}K^{-}$ mass distribution, we use the isospin formalism and separate
each isospin part as discussed in the formalism, where the results are shown
in Fig. 6. From Fig. 6, one can see that there are interference effects for
the sum of two isospin scattering amplitudes for the case of including
$\eta\eta$ channel. In Fig. 7, we show the results of the $\pi^{+}\pi^{-}$ and
$K^{+}K^{-}$ invariant mass distributions in the decays
$\bar{B}_{s}^{0}\rightarrow D^{0}\pi^{+}\pi^{-}$ (Fig. 7(a)) and
$\bar{B}_{s}^{0}\rightarrow D^{0}K^{+}K^{-}$ (Fig. 7(b)) within the effective
energy range of the ChUA. Since each pair of pseudoscalar mesons is produced
from the primary $s\bar{s}$ quarks, both of the invariant mass distributions
are contributed from the isospin $I=0$ components. As found in the Refs.
Ahmed:2020qkv ; Liang:2014tia , there is no contribution from the $f_{0}(500)$
state for these $\bar{B}_{s}^{0}$ decay channels, and only a large peak in the
$\pi^{+}\pi^{-}$ mass distributions, which corresponds to the $f_{0}(980)$
resonance, see Fig. 7(a). For the case of $K^{+}K^{-}$, the distributions
reach the maximum strength close to the $K^{+}K^{-}$ threshold and then fall
down gradually, as shown in Fig. 7(b). As found in Ref. Ahmed:2020kmp , the
$f_{0}(500)$ state was came from the $\pi^{+}\pi^{-}\to\pi^{+}\pi^{-}$
interaction amplitude only, whereas, the $f_{0}(980)$ state was contributed
from the interactions $K^{+}K^{-}\to\pi^{+}\pi^{-}$ and $K^{+}K^{-}\to
K^{+}K^{-}$. One can see that there is no interaction part from
$\pi^{+}\pi^{-}\to\pi^{+}\pi^{-}$ in Eqs. (10) and (11). Therefore, it is not
surprising that there is no such signal of the $f_{0}(500)$ state in these
decays. Besides, one can also see that the results of Fig. 7(b) for the
$\bar{B}_{s}^{0}\rightarrow D^{0}K^{+}K^{-}$ decay are analogous to the
results of Fig. 5(c) for the case of $\bar{B}^{0}\rightarrow D^{0}K^{+}K^{-}$
decay, even though two precesses have different subprocesses, see Eqs. (9) and
(11) shown in Figs. 3 and 4, respectively.

(a) For the decay $\bar{B}^{0}\rightarrow D^{0}\pi^{+}\pi^{-}$.

(b) For the decay $\bar{B}^{0}\rightarrow D^{0}\pi^{0}\eta$.

(c) For the decay $\bar{B}^{0}\rightarrow D^{0}K^{+}K^{-}$.

Figure 5: $\pi^{+}\pi^{-}$, $\pi^{0}\eta$, and $K^{+}K^{-}$ invariant mass
distributions in the decays $\bar{B}^{0}\rightarrow D^{0}PP^{\prime}$, only
with $s$-wave contribution. The solid (blue) line is for the results including
$\eta\eta$ channel and dashed (red) line for not including it. Figure 6:
Contributions from different isospin components in the $K^{+}K^{-}$ invariant
mass distributions in the $\bar{B}^{0}\rightarrow D^{0}K^{+}K^{-}$ decay.

(a) For the $\bar{B}_{s}^{0}\rightarrow D^{0}\pi^{+}\pi^{-}$ decay.

(b) For the $\bar{B}_{s}^{0}\rightarrow D^{0}K^{+}K^{-}$ decay.

Figure 7: $\pi^{+}\pi^{-}$ and $K^{+}K^{-}$ invariant mass distributions in
the decays $\bar{B}_{s}^{0}\rightarrow D^{0}PP^{\prime}$, only with $s$-wave
contributions. The results for different lines are the same as Fig. 5.

The results of Figs. 5 and 7 are only taken into account the $s$-wave
contribution. However, as discussed in the subsection II.2 the $p$-wave also
has a significant contribution for the decays channels $\pi^{+}\pi^{-}$ and
$K^{+}K^{-}$, since the vector mesons $\rho$ and $\omega$ can decay into the
$\pi^{+}\pi^{-}$ channel and the $\phi$ into the $K^{+}K^{-}$. The results of
considering the $s$-wave and $p$-wave contributions are shown in Figs. 8 and 9
for the decays $\bar{B}^{0}\rightarrow D^{0}\pi^{+}\pi^{-}$ and
$\bar{B}_{s}^{0}\rightarrow D^{0}K^{+}K^{-}$, respectively. In Fig. 8, one can
see that the $p$-wave contribution of the $\rho$ meson is dominated the
invariant mass distributions, which are stronger than the contributions from
the $f_{0}(500)$ and $f_{0}(980)$ states in $s$-wave, and also consistent with
the results of Ref. Liang:2014ama . Indeed, this coincides with the fact that
the branching ratio for the decay $\bar{B}^{0}\rightarrow D^{0}\rho^{0}$ is
bigger than the result for the decay $\bar{B}^{0}\rightarrow D^{0}f_{0}(500)$
ParticleDataGroup:2020ssz ; see the discussions above for determining the
production vertex factors. Even though the contributions from the $f_{0}(500)$
and $f_{0}(980)$ states are small, their signals can be clear seen in the line
shape of the $\pi^{+}\pi^{-}$ invariant mass distributions. Likewise, the
contribution of the $\phi$ meson in $p$-wave surpasses the ones of the
$f_{0}(980)$ for the $K^{+}K^{-}$ invariant mass distributions in the case of
the $\bar{B}_{s}^{0}\rightarrow D^{0}K^{+}K^{-}$ decay; see the results of
Fig. 9. Note that, in Fig. 9 we have two arbitrary normalization factors to
match the experimental events, one for the $s$-wave part and the other one for
the $p$-wave contribution. These two factors are determined from the fitting
of the experimental data Aaij:2018jqv  with the sum of the contributions from
the $s$\- and $p$-waves, of which the values are $1.81$ and $2.47$ for the
$s$\- and $p$-waves, respectively, with $\chi^{2}/dof=1.63$. One can see that
our fitting results describe the experimental data well with the contributions
from both the $s$\- and $p$-waves. It looks like the signal of $f_{0}(980)$
resonance is not so visible. Once again, the results of Fig. 9 show that the
dominant contribution comes from the vector meson primarily produced in
$p$-waves, which indeed indicate the molecular nature of the resonances
$f_{0}(500)$, $f_{0}(980)$ and $a_{0}(980)$ generated from the subprocesses of
final state interactions and not produced in the primary $q\bar{q}$ quark
pair, as found in Ref. Wang:2021nxz . Analogous considerations can be found in
Ref. Hyodo:2008xr  for the origin nature of the resonances.

Figure 8: $\pi^{+}\pi^{-}$ invariant mass distributions in the
$\bar{B}^{0}\rightarrow D^{0}\pi^{+}\pi^{-}$ decay, where the contributions
from the $s$-wave and $p$-wave are shown as dashed-dot (blue) line and dashed
(green) line, respectively, and the sum of two of them is displayed as solid
(red) line. Figure 9: $K^{+}K^{-}$ invariant mass distributions in the
$\bar{B}_{s}^{0}\rightarrow D^{0}K^{+}K^{-}$ decay, where the contributions
from the $s$-wave and $p$-wave are shown as dashed-dot (blue) line and dashed
(green) line, respectively. The red line represents the sum of the $s$\- and
$p$-waves contributions (with $\chi^{2}/dof=1.63$), and the black dots are the
experimental data Aaij:2018jqv .

Next, we make a further calculation of the branching fractions for each decay
channel. The obtained branching fractions and the ratios of branching
fractions are given in Tables I and II. From the results in Table I, one can
see the consistency between our work and the experimental measurements within
the uncertainties. The uncertainties in our model come from the errors of the
branching fractions used to determine the vertex factors, as discussed above.
In Table I, for the case of including $\eta\eta$ channel, the branching
fraction of $\bar{B}^{0}\rightarrow D^{0}f_{0}(980)$ is in good agreement with
the experiment, whereas, for the one of eliminating $\eta\eta$ channel, the
result is affected to give a bit small value. This is analogous to the case of
the $\bar{B}^{0}\rightarrow\phi\pi^{+}\pi^{-}$ decay found in Ref.
Ahmed:2020qkv . Besides, for the case of $\bar{B}^{0}\rightarrow
D^{0}\pi^{+}\pi^{-}$ decay, if we consider full decay mode, including or
excluding $\eta\eta$ channel does not influence the outcomes; see the second
row of Table I. Moreover, the predicted branching ratio of
$\bar{B}^{0}\rightarrow D^{0}a_{0}(980)$ decay is one magnitude larger than
the result of $\bar{B}^{0}\rightarrow D^{0}f_{0}(980)$ decay, which is easily
expected from the results of Fig. 5. The results of
$\text{Br}(\bar{B}^{0}_{s}\rightarrow D^{0}f_{0}(980))$ and
$\text{Br}(\bar{B}^{0}_{s}\rightarrow D^{0}K^{+}K^{-})$ are also consistent
with the upper limit of the experimental measurements for the cases of both
including and excluding the $\eta\eta$ channel. Moreover, with the branching
fractions obtained, one can easily evaluate their ratios, as shown in Table
II, some of which are consistent with the experimental results within the
uncertainties. Note that, these values are not depended on the production
vertex factor, since it is universal for all the calculations with the
contributions from $s$-wave and cancelled in these fractions. Therefore, these
ratios are more reliable in our formalism.

Table I: Results of the branching fractions.

Branching ratios | Without $\eta\eta$ channel | With $\eta\eta$ channel | Exp. ParticleDataGroup:2020ssz
---|---|---|---
$\text{Br}(\bar{B}^{0}\rightarrow D^{0}f_{0}(980))$ | $(5.47\pm 0.64)\times 10^{-6}$ | $(8.02\pm 0.94)\times 10^{-6}$ | $8.0\pm 4.0\times 10^{-6}$
$\text{Br}(\bar{B}^{0}\rightarrow D^{0}\pi^{+}\pi^{-})$ | $(7.15\pm 0.51)\times 10^{-4}$ | $(7.17\pm 0.51)\times 10^{-4}$ | $(8.8\pm 0.5)\times 10^{-4}$
$\text{Br}(\bar{B}^{0}\rightarrow D^{0}a_{0}(980)$ | $(7.34\pm 0.86)\times 10^{-5}$ | $(6.08\pm 0.72)\times 10^{-5}$ | $\cdots$
$\text{Br}(\bar{B}^{0}\rightarrow D^{0}K^{+}K^{-})$ | $(6.96\pm 0.47)\times 10^{-5}$ | $(7.07\pm 0.49)\times 10^{-5}$ | $(6.1\pm 0.4\pm 0.3\pm 0.3)\times 10^{-5}$
$\text{Br}(\bar{B}^{0}_{s}\rightarrow D^{0}f_{0}(980))$ | $(3.10\pm 0.37)\times 10^{-6}$ | $(2.16\pm 0.25)\times 10^{-6}$ | $<3.1\times 10^{-6}$
$\text{Br}(\bar{B}^{0}_{s}\rightarrow D^{0}K^{+}K^{-})$ | $(3.41\pm 0.23)\times 10^{-5}$ | $(3.35\pm 0.22)\times 10^{-5}$ | $(5.7\pm 0.5\pm 0.4\pm 0.5)\times 10^{-5}$

Table II: Ratios of the branching fractions.

Ratios | Without $\eta\eta$ channel | With $\eta\eta$ channel | Exp. ParticleDataGroup:2020ssz
---|---|---|---
$\frac{\text{Br}(\bar{B}^{0}\rightarrow D^{0}K^{+}K^{-})}{\text{Br}(\bar{B}^{0}\rightarrow D^{0}\pi^{+}\pi^{-})}$ | $0.097\pm 0.014$ | $0.098\pm 0.014$ | $0.069\pm 0.004\pm 0.003$
$\frac{\text{Br}(\bar{B}^{0}_{s}\rightarrow D^{0}K^{+}K^{-})}{\text{Br}(\bar{B}^{0}\rightarrow D^{0}K^{+}K^{-})}$ | $0.49\pm 0.07$ | $0.47\pm 0.06$ | $0.93\pm 0.089\pm 0.069$
$\frac{\text{Br}(\bar{B}^{0}\rightarrow D^{0}f_{0}(980))}{\text{Br}(\bar{B}^{0}\rightarrow D^{0}f_{0}(500))}$ | $0.08\pm 0.02$ | $0.12\pm 0.03$ | $0.12\pm 0.06$
$\frac{\text{Br}(\bar{B}^{0}\rightarrow D^{0}f_{0}(980))}{\text{Br}(\bar{B}^{0}\rightarrow D^{0}a_{0}(980))}$ | $0.07\pm 0.02$ | $0.13\pm 0.03$ | $\cdots$

Furthermore, we can obtain the ratios of branching fractions and the branching
fractions for the other vector decay channels $D^{0}V$. For these cases, the
results are related to the CKM matrix elements for the intermediate weak decay
procedures, and thus, one can easily get the ratios as below,

$\begin{array}[]{l}R_{1}^{th}=\frac{\Gamma_{\bar{B}^{0}\rightarrow
D^{0}\rho^{0}}}{\Gamma_{\bar{B}^{0}_{s}\rightarrow
D^{0}\phi}}=\frac{1}{2}\left|\frac{V_{cb}V_{ud}^{*}}{V_{cb}V_{us}^{*}}\right|^{2}\frac{m_{\bar{B}^{0}_{s}}^{2}}{m_{\bar{B}^{0}}^{2}}(\frac{p_{\rho^{0}}}{p_{\phi}})^{3}=9.82,\\\
R_{2}^{th}=\frac{\Gamma_{\bar{B}^{0}\rightarrow
D^{0}\rho^{0}}}{\Gamma_{\bar{B}^{0}\rightarrow
D^{0}\omega}}=(\frac{p_{\rho^{0}}}{p_{\omega}})^{3}=1.002.\end{array}$ (30)

Besides, using the measured branching ratio of $\bar{B}^{0}\rightarrow
D^{0}\rho^{0}$ decay to determine the production vertex factor as discussed
above, we can also obtain the other two branching ratios,

$\begin{array}[]{l}\text{Br}(\bar{B}^{0}_{s}\rightarrow
D^{0}\phi)=\frac{\Gamma_{\bar{B}^{0}_{s}\rightarrow
D^{0}\phi}}{\Gamma_{B}}=(3.27\pm 0.21)\times 10^{-5},\\\
\text{Br}(\bar{B}^{0}\rightarrow
D^{0}\omega)=\frac{\Gamma_{\bar{B}^{0}\rightarrow
D^{0}\omega}}{\Gamma_{B}}=(3.2\pm 0.21)\times 10^{-4}.\end{array}$ (31)

which are consistent with the experimental results ParticleDataGroup:2020ssz
within the uncertainties,

$\begin{array}[]{l}\text{Br}(\bar{B}^{0}_{s}\rightarrow D^{0}\phi)=(3.0\pm
0.5)\times 10^{-5},\\\ \text{Br}(\bar{B}^{0}\rightarrow D^{0}\omega)=(2.54\pm
0.16)\times 10^{-4}.\end{array}$ (32)

##  IV Conclusions

In the present work, based on the chiral unitary approach, we utilize the
final state interaction formalism to investigate the decays
$\bar{B}^{0}_{(s)}\to D^{0}\pi^{+}\pi^{-}$, $\bar{B}^{0}_{(s)}\to
D^{0}K^{+}K^{-}$, and $\bar{B}^{0}\rightarrow D^{0}\pi^{0}\eta$. Moreover, we
take into account the resonant contributions from both the $s$\- and $p$-waves
in the relevant decay channels. The considered resonance contributions come
from the states $f_{0}(500)$, $f_{0}(980)$, $\rho$, and $\phi$ in the $s$\-
and $p$-wave calculations for the decays $\bar{B}^{0}\to D^{0}\pi^{+}\pi^{-}$
and $\bar{B}^{0}\to D^{0}K^{+}K^{-}$, the state $a_{0}(980)$ for the
$\bar{B}^{0}\rightarrow D^{0}\pi^{0}\eta$ decay, and the states $f_{0}(980)$
and $\phi$ in the $s$\- and $p$-wave contributions for the decays
$\bar{B}^{0}_{s}\to D^{0}\pi^{+}\pi^{-}$ and $\bar{B}^{0}_{s}\to
D^{0}K^{+}K^{-}$. From the invariant mass distribution of $B^{0}$ decays, it
is found that the $f_{0}(500)$ and $a_{0}(980)$ has sizeable structures
compared to the $f_{0}(980)$ state, which leads to the fact that the decaying
into $f_{0}(980)$ state has a small branching fraction. However, for the case
of $B^{0}_{s}$ decays, since the hadronization in these decay processes come
from the primary quark pair $s\bar{s}$, there is no contribution from isospin
$I=1$. Thus, there is only a peak closed to the $K^{+}K^{-}$ threshold, which
corresponds to the state $f_{0}(980)$, and no signal for the resonance
$f_{0}(500)$. Despite the $s$-wave contributions are considered, from Figs. 8
and 9 one can see that the vector mesons $\rho$ and $\phi$, produced in the
primary $q\bar{q}$ quark pair from the $p$-wave contributions, have the
dominant contributions in the decays $\bar{B}^{0}\to D^{0}\pi^{+}\pi^{-}$ and
$\bar{B}^{0}_{s}\to D^{0}K^{+}K^{-}$, respectively, which happen to indicate
the molecular nature of the resonances $f_{0}(500)$, $f_{0}(980)$ and
$a_{0}(980)$ generated from the subprocesses of final state interactions. In
addition to show the invariant mass distributions, one can also calculate the
branching fractions and the ratios of branching fractions. We investigate the
branching ratios for different decay processes with the scalar and vector
mesons produced in the final states, where some of our results are in
agreement with the experiments. We also predict the branching fraction of
$\bar{B}^{0}\rightarrow D^{0}a_{0}(980)[a_{0}(980)\to\pi^{0}\eta]$ decay,
obtained as $\text{Br}(\bar{B}^{0}\rightarrow D^{0}a_{0}(980)=(7.34\pm
0.86)\times 10^{-5}$ and $\text{Br}(\bar{B}^{0}\rightarrow
D^{0}a_{0}(980)=(6.08\pm 0.72)\times 10^{-5}$ for the cases of without and
with $\eta\eta$ channel contribution, respectively. Then, the relative weights
of the branching fractions are predicted with no free parameters due to the
cancellation of the production vertex factor. And thus, these predictions are
meaningful for future experiments, such as the ratio of
$\text{Br}(\bar{B}^{0}\rightarrow
D^{0}f_{0}(980))/\text{Br}(\bar{B}^{0}\rightarrow D^{0}a_{0}(980))$.

## Acknowledgments

We thank Prof. Mei Huang for careful reading the manuscript and useful
comments, and acknowledge Profs. Eulogio Oset and Nikolay Achasov for helpful
comments.

## References

  * (1) E. Eckhart et al. [CLEO], Phys. Rev. Lett. 89, 251801 (2002) [arXiv:hep-ex/0206024 [hep-ex]].
  * (2) J. P. Lees et al. [BaBar], Phys. Rev. D 85, 112010 (2012) [arXiv:1201.5897 [hep-ex]].
  * (3) Y. Nakahama et al. [Belle], Phys. Rev. D 82, 073011 (2010) [arXiv:1007.3848 [hep-ex]].
  * (4) R. Aaij et al. [LHCb], Phys. Lett. B 698, 115-122 (2011) [arXiv:1102.0206 [hep-ex]].
  * (5) R. Aaij et al. [LHCb], Phys. Rev. D 95, no.1, 012006 (2017) [arXiv:1610.05187 [hep-ex]].
  * (6) R. Aaij et al. [LHCb], JHEP 06, 114 (2019) [arXiv:1902.07955 [hep-ex]].
  * (7) R. Aaij et al. [LHCb], Phys. Rev. D 101, no.1, 012006 (2020) [arXiv:1909.05212 [hep-ex]].
  * (8) R. Aaij et al. [LHCb], Phys. Rev. D 102, 112003 (2020) [arXiv:2009.00026 [hep-ex]].
  * (9) R. Aaij et al. [LHCb], JHEP 08, 037 (2017) [arXiv:1704.08217 [hep-ex]].
  * (10) R. Aaij et al. [LHCb], Phys. Rev. D 102, 112010 (2020) [arXiv:2010.11802 [hep-ex]].
  * (11) A. Kuzmin et al. [Belle], Phys. Rev. D 76, 012006 (2007) [arXiv:hep-ex/0611054 [hep-ex]].
  * (12) R. Aaij et al. [LHCb], Phys. Rev. D 92, no.3, 032002 (2015) [arXiv:1505.01710 [hep-ex]].
  * (13) R. Aaij et al. [LHCb], JHEP 08, 005 (2015) [arXiv:1505.01654 [hep-ex]].
  * (14) R. Aaij et al. [LHCb], Phys. Rev. D 98, no.7, 071103 (2018) [arXiv:1807.01892 [hep-ex]].
  * (15) R. Aaij et al. [LHCb], Phys. Rev. D 98, no.7, 072006 (2018) [arXiv:1807.01891 [hep-ex]].
  * (16) Y. J. Shi, U. G. Meißner and Z. X. Zhao, [arXiv:2111.05647 [hep-ph]].
  * (17) S. H. Zhou, R. H. Li, Z. Y. Wei and C. D. Lu, [arXiv:2107.11079 [hep-ph]].
  * (18) L. Yang, Z. T. Zou, Y. Li, X. Liu and C. H. Li, Phys. Rev. D 103, no.11, 113005 (2021) [arXiv:2103.15031 [hep-ph]].
  * (19) Z. Q. Zhang, S. Y. Wang and X. K. Ma, Phys. Rev. D 93, no.5, 054034 (2016) [arXiv:1601.04137 [hep-ph]].
  * (20) Y. Xing and Z. P. Xing, Chin. Phys. C 43, no.7, 073103 (2019) [arXiv:1903.04255 [hep-ph]].
  * (21) Y. Li, D. C. Yan, Z. Rui, L. Liu, Y. T. Zhang and Z. J. Xiao, Phys. Rev. D 102, no.5, 056017 (2020) [arXiv:2007.13629 [hep-ph]].
  * (22) Z. T. Zou, Y. Li, Q. X. Li and X. Liu, Eur. Phys. J. C 80, no.5, 394 (2020) [arXiv:2003.03754 [hep-ph]].
  * (23) A. J. Ma and W. F. Wang, Phys. Rev. D 103, no.1, 016002 (2021) [arXiv:2010.12906 [hep-ph]].
  * (24) Y. Li, D. C. Yan, J. Hua, Z. Rui and H. n. Li, Phys. Rev. D 104, no.9, 096014 (2021) [arXiv:2105.03899 [hep-ph]].
  * (25) D. Boito, J. P. Dedonder, B. El-Bennich, R. Escribano, R. Kaminski, L. Lesniak and B. Loiseau, Phys. Rev. D 96, no.11, 113003 (2017) [arXiv:1709.09739 [hep-ph]].
  * (26) W. H. Liang, J. J. Xie and E. Oset, Phys. Rev. D 92, no.3, 034008 (2015) [arXiv:1501.00088 [hep-ph]].
  * (27) J. A. Oller and E. Oset, Nucl. Phys. A 620, 438 (1997) Erratum: [Nucl. Phys. A 652, 407 (1999)] [hep-ph/9702314].
  * (28) E. Oset and A. Ramos, Nucl. Phys. A 635, 99 (1998) [nucl-th/9711022].
  * (29) N. Kaiser, Eur. Phys. J. A 3, 307 (1998).
  * (30) J. A. Oller and U.-G. Meißner, Phys. Lett. B 500, 263 (2001) [hep-ph/0011146].
  * (31) J. A. Oller, E. Oset and A. Ramos, Prog. Part. Nucl. Phys. 45, 157 (2000) [hep-ph/0002193].
  * (32) T. Hyodo and D. Jido, Prog. Part. Nucl. Phys. 67, 55-98 (2012) [arXiv:1104.4474 [nucl-th]].
  * (33) W. H. Liang and E. Oset, Phys. Lett. B 737, 70-74 (2014) [arXiv:1406.7228 [hep-ph]].
  * (34) M. Bayar, W. H. Liang and E. Oset, Phys. Rev. D 90, no.11, 114004 (2014) [arXiv:1408.6920 [hep-ph]].
  * (35) W. H. Liang, J. J. Xie and E. Oset, Eur. Phys. J. C 75, no. 12, 609 (2015) [arXiv:1510.03175 [hep-ph]].
  * (36) E. Oset, W. H. Liang, M. Bayar, J. J. Xie, L. R. Dai, M. Albaladejo, M. Nielsen, T. Sekihara, F. Navarra and L. Roca, et al. Int. J. Mod. Phys. E 25, 1630001 (2016) [arXiv:1601.03972 [hep-ph]].
  * (37) N. N. Achasov, A. V. Kiselev and G. N. Shestakov, Phys. Rev. D 102, no.1, 016022 (2020) [arXiv:2005.06455 [hep-ph]].
  * (38) N. N. Achasov, J. V. Bennett, A. V. Kiselev, E. A. Kozyrev and G. N. Shestakov, Phys. Rev. D 103, no.1, 014010 (2021) [arXiv:2009.04191 [hep-ph]].
  * (39) N. N. Achasov, A. V. Kiselev and G. N. Shestakov, Phys. Rev. D 104, no.1, 016034 (2021) [arXiv:2106.10670 [hep-ph]].
  * (40) N. N. Achasov, S. A. Devyanin and G. N. Shestakov, Phys. Lett. B 88, 367-371 (1979)
  * (41) N. N. Achasov, S. A. Devyanin and G. N. Shestakov, Sov. J. Nucl. Phys. 32, 566 (1980) TF-109.
  * (42) N. N. Achasov, S. A. Devyanin and G. N. Shestakov, Phys. Lett. B 96, 168-170 (1980)
  * (43) N. N. Achasov, S. A. Devyanin and G. N. Shestakov, Phys. Lett. B 108, 134-139 (1982) [erratum: Phys. Lett. B 108, 435 (1982)]
  * (44) N. N. Achasov and V. N. Ivanchenko, Nucl. Phys. B 315, 465-476 (1989)
  * (45) N. N. Achasov, Nucl. Phys. A 728, 425-438 (2003) doi:10.1016/j.nuclphysa.2003.09.002 [arXiv:hep-ph/0309118 [hep-ph]].
  * (46) 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]].
  * (47) M. Y. Duan, J. Y. Wang, G. Y. Wang, E. Wang and D. M. Li, Eur. Phys. J. C 80, no.11, 1041 (2020) [arXiv:2008.10139 [hep-ph]].
  * (48) N. Ikeno, M. Bayar and E. Oset, Eur. Phys. J. C 81, no.4, 377 (2021) [arXiv:2102.01650 [hep-ph]].
  * (49) L. Roca and E. Oset, Phys. Rev. D 103, no.3, 034020 (2021) [arXiv:2011.05185 [hep-ph]].
  * (50) L. R. Dai, E. Oset and L. S. Geng, [arXiv:2111.10230 [hep-ph]].
  * (51) H. A. Ahmed, Z. Y. Wang, Z. F. Sun and C. W. Xiao, Eur. Phys. J. C 81, no.8, 695 (2021) [arXiv:2011.08758 [hep-ph]].
  * (52) C. W. Xiao, U.-G. Meißner and J. A. Oller, Eur. Phys. J. A 56, no. 1, 23 (2020) [arXiv:1907.09072 [hep-ph]].
  * (53) Z. Wang, Y. Y. Wang, E. Wang, D. M. Li and J. J. Xie, Eur. Phys. J. C 80, no.9, 842 (2020) [arXiv:2004.01438 [hep-ph]].
  * (54) D. L. Yao, L. Y. Dai, H. Q. Zheng and Z. Y. Zhou, Rept. Prog. Phys. 84, no.7, 076201 (2021) [arXiv:2009.13495 [hep-ph]].
  * (55) P. A. Zyla et al. [Particle Data Group], PTEP 2020, no.8, 083C01 (2020)
  * (56) H. A. Ahmed and C. Xiao, Phys. Rev. D 101, no.9, 094034 (2020) [arXiv:2001.08141 [hep-ph]].
  * (57) Z. Y. Wang, J. Y. Yi, Z. F. Sun and C. W. Xiao, [arXiv:2109.00153 [hep-ph]].
  * (58) T. Hyodo, D. Jido and A. Hosaka, Phys. Rev. C 78, 025203 (2008) [arXiv:0803.2550 [nucl-th]].