# Nonleptonic two-body weak decays of charmed baryons

Chia-Wei Liu  Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai
200240, China

###### Abstract

We systematically analyze the two-body nonleptonic weak decays of charmed
baryons, employing the pole approximation in tandem with the $SU(3)_{F}$
symmetry. Leveraging the Körner-Pati-Woo theorem, we demonstrate a significant
reduction in the number of free parameters. Within the general pole scenario,
we successfully explain most of the experimental data of ${\bf
B}_{c}^{A}\to{\bf B}_{n}P$, though some inconsistencies are detected and
recommended for reexamination in future research. Assuming the dominance of
low-lying intermediate baryons, we are able to make novel predictions for
decay channels such as $\Omega_{c}^{0}\to{\bf B}_{n}P$ and ${\bf
B}_{cc}\to{\bf B}_{c}^{A,S}P$. Here, ${\bf B}_{n}$, ${\bf B}_{c}^{A}$, ${\bf
B}_{c}^{S}$ and ${\bf B}_{cc}$ are the low-lying octet, antitriplet charmed,
sextet charmed and doubly charmed baryons, respectively, and $P$ is the
pseudoscalar meson. Our findings also reveal that the fitted effective Wilson
coefficient ${\cal C}_{+}=0.469$ is notably smaller than the naïve
expectation, and the low-lying pole scenario fails to account for ${\cal
B}(\Lambda_{c}^{+}\to n\pi^{+},\Xi^{0}K^{+})$ , despite consistencies with the
soft meson limit. We further recommend the decay channel
$\Xi_{cc}^{+}\to\Xi_{c}^{0}\pi^{+}\to\Xi^{-}\pi^{+}\pi^{+}\pi^{+}\pi^{-}$ for
exploring evidence of $\Xi_{cc}^{+}$, estimating the branching fraction at
$(1.1\pm 0.6)\times 10^{-3}$. The predictions for nonleptonic weak decay
channels are compiled in the appendices, providing valuable references for
future experimental validation.

##  I Introductions

The investigation into charmed baryon decays has attracted significant
theoretical interest, driven further by the progress in experiments
ParticleDataGroup:2022pth . For a review, readers are referred to Refs.
Cheng:2021qpd ; Groote:2021pxt . At the BESIII facility, the lightest charmed
baryon, $\Lambda_{c}^{+}$, has been rigorously examined through $e^{+}e^{-}$
interactions at a central energy of $\sqrt{s}=4.6$ GeV BESIII:2015qfd . These
investigations have yielded remarkably precise measurements of branching
fractions and decay asymmetries BESIII:2015bjk ; BESIII:SigmapK ; BESIII:new .
The resonance structure of $e^{+}e^{-}\to\Lambda_{c}^{+}\Lambda_{c}^{-}$,
providing a clean background, has facilitated the BESIII collaboration’s
ability to measure $\Lambda_{c}\to n\pi^{+}$ in spite of the challenges posed
by neutrons BESIII:2022bkj . However, the comprehensive study of the entire
charmed baryon family necessitates the synthesis of results from multiple
experimental facilities, as only $\Lambda_{c}^{+}$ is currently accessible at
BESIII.

Through the $B$ meson decay chain, the Belle collaboration has access to all
the low-lying antitriplet charmed baryons $({\bf
B}_{c}^{A}=\Lambda_{c}^{+},\Xi_{c}^{+},\Xi_{c}^{0})$ Belle:2013ntc ;
Belle:2015wxn ; Belle:2021avh ; Belle:new . A significant recent breakthrough
includes the measurement of absolute branching fractions for
$\Xi_{c}^{0}\to\Xi^{-}\pi^{+}$ Belle:2018kzz  and
$\Xi_{c}^{0}\to\Xi^{-}e^{+}\nu_{e}$ Belle:2021crz , revealing substantial
$SU(3)_{F}$ symmetry breaking He:2021qnc . Conversely, the LHCb collaboration
has obtained the largest charmed hadron samples from $pp$ collisions at
$\sqrt{s}=7,8,12$ GeV. Despite more complex backgrounds compared to those at
BESIII and Belle, the majority of new charmed baryon discoveries LHCb:2021ptx
, including the famed doubly charmed baryon LHCb:2017iph , have been made at
LHCb. Additionally, Belle and LHCb have revisited the lifetimes of certain
baryons Belle-II:2022ggx ; LHCb:lifetime , with notable deviations found in
the measured lifetimes of $\Xi_{c}^{0}$ and $\Omega_{c}^{0}$ compared to
previous experiments ParticleDataGroup:2004fcd . These measurements, however,
are consistent with the heavy quark expansion (HQE) Cheng:2018rkz .

Thanks to the optical theorem, the inclusive decay widths of charmed hadrons
can be at least qualitatively studied Lenz:2013aua . It is understood that the
contributions of the dimension-6 operators in the HQE, suppressed by
$(\Lambda_{\text{QCD}}/M_{c})^{3}$, may exceed those of the dimension-3
operators due to phase space enhancement Guberina:1979xw . This emphasizes the
leading role of the $W$-exchange diagrams in decays. However, as of now, there
is no reliable method derived from first principles to address the
$W$-exchange diagrams in exclusive decays, leading to the need for several
approximations Cheng:1991sn ; Xu:1992vc ; Korner:1992wi ; Cheng:1993gf ;
Uppal:1994pt ; Ivanov:1997ra ; thSharmaVerma ; Cheng:2018hwl ; Hu:2020nkg ;
Meng:2020euv ; Niu:2020gjw ; Charmed-Cheng ; Xu:2021mkg ; Zeng:2022egh . One
less model-dependent approach is to perform a global fit using the $SU(3)$
flavor $(SU(3)_{F})$ symmetry, which has become popular Geng:2017mxn ;
Geng:2018plk ; Geng:2018bow ; Geng:2018upx ; Geng:single ; Asymmetries ;
Wang:2019dls ; Geng:2019awr ; Jia:2019zxi ; Geng:updow ; Geng:2020zgr ;
Huang:2021aqu ; Zhong:2022exp ; Wang:2022kwe ; Xing:2023bzh ; Hsiao:2021nsc ;
Chau:1995gk ; He:2018joe ; Savage:1989qr ; Sharma:1996sc ; Lu:2016ogy .
Nevertheless, even in the simplest case of ${\bf B}_{c}^{A}\to{\bf B}_{n}P$,
where ${\bf B}_{n}$ and $P$ represent the octet baryon and pseudoscalar meson
respectively, this method requires dozen one-time parameters. While the
results of the global fit often align with the experimental data used for
fitting, the predictive accuracy is disputable. The predicted branching
fractions significantly diverge across various theoretical studies relying on
the $SU(3)_{F}$ symmetry, illustrating that the free parameters are not
tightly constrained by the existing experimental data.

In an effort to reduce the number of free parameters in the $SU(3)_{F}$ global
fit, Geng, Tsai, and the author of this work considered the pole approximation
in 2019 Asymmetries ; Geng:single . This approach, grounded in the Körner-
Pati-Woo (KPW) theorem Korner:1970xq , enables the exclusion of six parameters
from $O_{+}^{qq^{\prime}}$ . Here $O_{+}^{qq^{\prime}}$ is the four-quark
operator in the effective Hamiltonian Buchalla:1995vs

$\mathcal{H}_{eff}=\sum_{q,q^{\prime}=d,s}\frac{G_{F}}{\sqrt{2}}V_{cq}^{*}V_{uq^{\prime}}\left(c_{+}O_{+}^{qq^{\prime}}+c_{-}O_{-}^{qq^{\prime}}\right)\,,$
(1)

with

$O_{\pm}^{qq^{\prime}}=\frac{1}{2}\left[\left(\bar{u}q^{\prime}\right)_{V-A}\left(\bar{q}c\right)_{V-A}\pm\left(\bar{q}q^{\prime}\right)_{V-A}(\bar{u}c)_{V-A}\right]\,,$
(2)

where $G_{F}$ is the Fermi constant and $V_{qq^{\prime}}$ is the Cabibbo-
Kobayashi-Maskawa matrix element. After considering the factorizable
contributions of $O_{+}^{qq^{\prime}}$, the smallness of ${\cal
B}(\Lambda_{c}^{+}\to p\pi^{0})$ is explained Geng:single . More importantly,
Ref. Asymmetries  predicted that

$\displaystyle{\cal B}(\Lambda_{c}^{+}\to\Sigma^{+}K_{S}^{0})={\cal
B}(\Lambda_{c}^{+}\to\Sigma^{0}K^{+})$ (3) $\displaystyle\frac{{\cal
B}(\Xi_{c}^{0}\to\Sigma^{0}K_{S}^{0})}{{\cal
B}(\Xi_{c}^{0}\to\Xi^{0}\pi^{-})}=(2.3\pm 1.8)\%$ (4)

which were not measured at that time. In particular, Eq. (3) is an critical
prediction stemming from the KPW theorem and the modest ratio in Eq. (4) is
quite surprising as both of them are Cabibbo favored (CF). These theoretical
benchmarks have since been found consistent with recent experimental results
BESIII:SigmapK ; Belle:2021avh .

In the present study, we build upon the framework established in Ref.
Asymmetries , extending it to include the decays of $\Omega_{c}^{0}$ and
doubly charmed baryons. To accomplish this, we make two critical
approximations:

  1. 1.

We assume that the intermediate states are principally dominated by the low-
lying baryons with spin-parity $\frac{1}{2}^{+}$ and $\frac{1}{2}^{-}$.

  2. 2.

We posit that the flavor of the spectator quark exerts only a minimal
influence on the baryon matrix elements.

It is worth noting that these approximations have been found to hold in the
majority of model-dependent studies, and a detailed discussion on them will be
provided later in this work.

This paper is structured in the following manner. In Section II, we delineate
the $SU(3)_{F}$ representations of the charmed baryons. Section III is devoted
to the evaluation of the factorizable contributions, utilizing the form
factors derived from Lattice QCD (LQCD). In Section IV, we thoroughly analyze
the pole amplitudes and explore the dependencies on mass. Section V presents
the numerical results, and finally, we conclude our findings in Section VI.

##  II $SU(3)_{F}$ representation and Körner-Pati-Woo theorem

In general, the amplitudes of ${\bf B}_{i}\to{\bf B}_{f}P$ read

${\cal M}=\langle{\bf B}_{f}P;t\to\infty|{\cal H}_{eff}|{\bf
B}_{i}\rangle=i\overline{u}_{f}\left(A+B\gamma_{5}\right)u_{i}\,,$ (5)

where $u_{i(f)}$ is the Dirac spinor of the initial(final) baryon and $A(B)$
is the parity-violating(conserving) amplitude, corresponding to the
$S(P)$-partial wave. If the final state interaction is absent, one can freely
interchange $t\to\pm\infty$ and $A$ and $B$ must be real. The decay width
$\Gamma$ and up-down asymmetry $\alpha$ are calculated by

$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\frac{p_{f}}{8\pi}\left(\frac{(M_{i}+M_{f})^{2}-M_{P}^{2}}{M_{i}^{2}}|A|^{2}+\frac{(M_{i}-M_{f})^{2}-M_{P}^{2}}{M_{i}^{2}}|B|^{2}\right)\,,$
(6) $\displaystyle\alpha=-\frac{2\kappa\text{
Re}(A^{*}B)}{|A|^{2}+\kappa^{2}|B|^{2}}\,,\hskip
56.9055pt\kappa=\frac{p_{f}}{E_{f}+M_{f}}\,,$

where $M_{i,f}$ and $M_{P}$ are the masses of ${\bf B}_{i,f}$ and $P$,
respectively and $p_{f}$ and $E_{f}$ are the magnitudes of the $3$-momentum
and energy of ${\bf B}_{f}$ at the rest frame of ${\bf B}_{i}$.

Figure 1:  The $SU(4)_{F}$ 20 multiplet represented by
---
(taken from Particle Data Group ParticleDataGroup:2022pth ).

To relate the decays with the $SU(3)_{F}$ symmetry, one has to write down the
hadron representations in the $SU(3)_{F}$ group. We start with the low-lying
pseudoscalar mesons. The responsible $SU(3)_{F}$ tensor is given by

$\displaystyle
P^{i}_{j}=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{6}}\eta_{8}+\frac{1}{\sqrt{2}}\pi^{0}&\pi^{+}&K^{+}\\\
\pi^{-}&\frac{1}{\sqrt{6}}\eta_{8}-\frac{1}{\sqrt{2}}\pi^{0}&K^{0}\\\
K^{-}&\overline{K}^{0}&-\frac{2}{\sqrt{6}}\eta_{8}\end{array}\right)_{ij}\,,$
(10)

which is related to the flavor part of wave functions according to

$|P\rangle=P^{i}_{j}|q_{i}\overline{q}^{j}\rangle\,.$ (11)

Here, the superscript and subscript of $P^{i}_{j}$ describe the quark and
antiquark flavors with $i,j\in\\{1,2,3\\}$ and $(q_{1},q_{2},q_{3})=(u,d,s)$.

We exclusively consider the $SU(4)_{F}$ ${\bf 20}$ multiplets, where the low-
lying $\frac{1}{2}^{+}$ baryons are depicted in FIG. 1. We start with the
idempotent of  in the Young tableau, which projects out a subspace of the
$SU(4)_{F}$ group, given by

$e_{23}=(1-(2,3))(1+(1,2))\,,$ (12)

where $(1,2)$ interchange the first and second elements and $(2,3)$ the second
and third. For instance, we have

$e_{23}|q_{a}q_{b}q_{c}\rangle=|q_{a}q_{b}q_{c}\rangle+|q_{b}q_{a}q_{c}\rangle-|q_{a}q_{c}q_{b}\rangle-|q_{b}q_{c}q_{a}\rangle\,.$
(13)

It is clear that after operating $e_{23}$, states are antisymmetric in regard
to the second and third quarks. The idempotent in Eq. (12) generates a
subspace in the sense that $e_{23}e_{A}=e_{23}e_{S}=0\,,$ where $e_{S(A)}$ are
the totally (anti)symmetric idempotent, given by

$\displaystyle e_{S}=1+(1,2)+(2,3)+(1,3)+(2,3)(1,2)+(1,2)(2,3)\,,$
$\displaystyle e_{A}=1-(1,2)-(2,3)-(1,3)+(2,3)(1,2)+(1,2)(2,3)\,,$ (14)

We stress that throughout this work the $SU(4)_{F}$ representations are merely
book-keeping tools to unify the expressions and we do not take advantage of
the $SU(4)_{F}$ symmetry.

If a light quark $(u,d,s)$ pair is in antisymmetric, we utilize that the
totally antisymmetric tensor $\epsilon^{ijk}$ is invariant under the
$SU(3)_{F}$ transformation to simplify the indices, i.e. two antisymmetric
quarks transform as an antiquark. As a result, the light quarks of ${\bf
B}_{c}^{A}$ are presented by one lower index as

$({\bf B}^{A}_{c})_{i}=(\Xi_{c}^{0},\Xi_{c}^{+},\Lambda_{c}^{+})_{i}\,.$ (15)

Eq. (15) can be translated back to a tensor with three quarks by

$({\bf B}^{A}_{c})^{i[jk]}=\frac{1}{\sqrt{12}}({\bf
B}_{c})_{l}\left(2\delta^{i}_{4}\epsilon^{ljk}+\delta^{j}_{4}\epsilon^{lik}-\delta^{k}_{4}\epsilon^{lij}\right)\,,$
(16)

with $q_{4}=c$. Here, Eq. (16) is derived by

$\small
e_{23}\frac{1}{\sqrt{12}}\left(|cud\rangle-|cdu\rangle\right)=\frac{1}{\sqrt{12}}\left(2|cud\rangle-2|cdu\rangle+|ucd\rangle-|dcu\rangle-|udc\rangle+|duc\rangle\right)\,,$
(17)

where we have used $\Lambda_{c}^{+}$ as an instance. We start with
$|cud\rangle-|cdu\rangle$ to make sure its isospin vanishes. One arrives at
$|\Sigma_{c}^{+}\rangle$ if $|cud\rangle+|cdu\rangle$ is used instead.

On the other hand, the other low-lying baryons with spin-parity
$\frac{1}{2}^{+}$ are

$\displaystyle({\bf B}_{n})^{i}_{j}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\frac{1}{\sqrt{6}}\Lambda+\frac{1}{\sqrt{2}}\Sigma^{0}&\Sigma^{+}&p\\\
\Sigma^{-}&\frac{1}{\sqrt{6}}\Lambda-\frac{1}{\sqrt{2}}\Sigma^{0}&n\\\
\Xi^{-}&\Xi^{0}&-\sqrt{\frac{2}{3}}\Lambda\end{array}\right)_{ij}\,,$ (21)
$\displaystyle({\bf B}^{S}_{c})^{ij}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\Sigma^{++}_{c}&\frac{1}{\sqrt{2}}\Sigma^{+}_{c}&\frac{1}{\sqrt{2}}\Xi^{\prime+}_{c}\\\
\frac{1}{\sqrt{2}}\Sigma^{+}_{c}&\Sigma^{0}_{c}&\frac{1}{\sqrt{2}}\Xi^{\prime
0}_{c}\\\ \frac{1}{\sqrt{2}}\Xi^{\prime+}_{c}&\frac{1}{\sqrt{2}}\Xi^{\prime
0}_{c}&\Omega^{0}_{c}\end{array}\right)_{ij}\,.$ (25) $\displaystyle({\bf
B}_{cc})^{i}$ $\displaystyle=$
$\displaystyle(\Xi_{cc}^{++},\Xi_{cc}^{+},\Omega_{cc}^{+})_{i}\,,$ (26)

where ${\bf B}_{c}^{S}$ and ${\bf B}_{cc}$ are the singly charmed sextet and
doubly charmed baryons, respectively. Similarly, they are translated to
tensors with three quark indices by

$\displaystyle({\bf B}_{n})^{i[jk]}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}({\bf B}_{n})^{i}_{l}\epsilon^{ljk}\,,$
$\displaystyle({\bf B}^{S}_{c})^{i[jk]}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(({\bf B}_{c})^{ij}\delta^{k}_{4}-({\bf
B}_{c})^{ik}\delta^{j}_{4}\right)\,,$ $\displaystyle({\bf B}_{cc})^{i[jk]}$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}\left(({\bf
B}_{cc})^{j}\delta^{i}_{4}\delta^{k}_{4}-({\bf
B}_{cc})^{k}\delta^{i}_{4}\delta^{j}_{4}\right)\,,$ (27)

which would lead us to the convention in Ref. Groote:2021pxt  up to some
unphysical overall phase factors. In the quark model, the spin-flavor wave
functions are obtained by

$|{\bf B}\rangle=(1+(1,2)+(1,3))\frac{\sqrt{2}}{3}\left[{\bf
B}^{i[jk]}|q_{i}q_{j}q_{k}\rangle\otimes\left(|\uparrow\uparrow\downarrow\rangle-|\uparrow\downarrow\uparrow\rangle\right)\right]\,,$
(28)

with ${\bf B}\in\\{{\bf B}_{n},{\bf B}^{A,S}_{c},{\bf B}_{cc}\\}$.

The effective Hamiltonian can be written in a compact way of

${\cal H}_{eff}=\frac{G_{F}}{\sqrt{2}}V_{ud}V_{cs}^{*}\left({\cal
H}^{ij}_{kl}(\overline{q}_{i}q^{k})_{V-A}(\overline{q}_{j}q^{l})_{V-A}\right)\,,$
(29)

where the nonzero elements are

$\displaystyle{\cal H}^{13}_{24}=c_{1}\,,~{}~{}~{}~{}~{}~{}~{}~{}{\cal
H}^{31}_{24}=c_{2}\,,~{}~{}~{}~{}~{}~{}~{}~{}{\cal
H}^{13}_{34}=c_{1}s_{c}\,,~{}~{}~{}~{}~{}~{}{\cal H}^{31}_{34}=c_{2}s_{c}\,,$
$\displaystyle{\cal H}^{12}_{24}=-c_{1}s_{c}\,,~{}~{}~{}{\cal
H}^{21}_{24}=-c_{2}s_{c}\,,~{}~{}~{}{\cal
H}^{12}_{34}=-c_{1}s_{c}^{2}\,,~{}~{}~{}{\cal H}^{21}_{34}=-c_{2}s_{c}^{2}\,,$
(30)

$s_{c}=V_{us}/V_{ud}=0.23$, $c_{1}=c_{+}+c_{-}$ and $c_{2}=c_{+}-c_{-}$ .
Similar to the baryon states, one decomposes the effective Hamiltonian
according to the permutation symmetry by

${\cal H}(\overline{{\bf 6}})_{kl}\epsilon^{lij}=\frac{1}{c_{-}}\left({\cal
H}^{ij}_{k4}-{\cal H}^{ji}_{k4}\right)\,,~{}~{}~{}{\cal H}({{\bf
15}})^{ij}_{k}=\frac{1}{2c_{+}}\left({\cal H}^{ij}_{k4}+{\cal
H}^{ji}_{k4}\right)\,.$ (31)

The factors of $1/c_{-}$ and $1/2c_{+}$ are included to match the convention.
Comparing to Eq. (1), it is clear that ${\cal H}(\overline{{\bf 6}})$ and
${\cal H}({\bf 15})$ take account $O_{-}$ and $O_{+}$ in the effective
Hamiltonian.

Figure 2: The $s$\- (left) and $u$-channels (right) of the pole contributions
in ${\bf B}_{i}\to{\bf B}_{f}P$, where $\times$ denotes the insertion of the
effective Hamiltonian and ${\bf B}_{I}$ is the intermediate state.

By far we have only considered the quark flavors and here is an appropriate
place to further consider their colors also. With the Fierz transformation, it
is straightforward to show that the color structure of $\overline{q}$ and
$\overline{u}$ in $O_{+}^{qq^{\prime}}$ is symmetric, and the same also
applies to $c$ and $q^{\prime}$. Recall that baryons are antisymmetric in
color, we arrive at

$\langle q_{a}q_{b}q_{c}|O_{+}^{qq^{\prime}}|{\bf B}_{i}\rangle=0\,,$ (32)

where the initial and final states are an arbitrary baryon and three quark
state, respectively. The same also applies to $\langle{\bf
B}_{f}|O_{+}^{qq^{\prime}}|q_{a}q_{b}q_{c}\rangle=0$ with ${\bf B}_{f}$ the
final state baryon.

In the decays ${\bf B}_{i}\to{\bf B}_{f}P$, the nonfactorizable contributions
can be approximated by the pole diagrams shown in FIG. 2, where the symbol
${\bf\times}$ marks the insertion of the effective Hamiltonian. This
approximation results in the well-known KPW theorem, which states that $O_{+}$
contributes solely to the factorizable amplitudes. Notably, Eq. (32) is scale-
independent, as $O_{\pm}$ do not undergo mixing in the renormalization group
evolution Buchalla:1995vs . While a hard gluon exchange could challenge the
KPW theorem, any breaking effect is likely below 10%. For a deeper dive into
this topic, readers can consult Ref. On_the_smallness . There, the small
branching fraction of ${\cal B}(B^{0}\to p\overline{p})$ is attributed to a
violation against the KPW theorem 111 To be explicit, Ref. On_the_smallness
shows that the amplitude of $B^{0}\to p\overline{p}$ is proportional to
$c_{+}$ instead of $c_{-}$. . Since this deviation is even less significant
than that of the $SU(3)_{F}$ breaking, we uphold the KPW theorem in this
study.

To identify the factorizable contributions of $O_{+}$, we observe the direct
product of ${\cal H}({\bf 15})^{ij}_{k}$ and $(P^{\dagger})^{l}_{m}$ has the
representation of

${\bf 15}\otimes{\bf 8}={\bf 42}\oplus\overline{{\bf 24}}\oplus{\bf
15}_{1}\oplus{\bf 15}_{1}\oplus{\bf 15}_{2}\oplus\overline{{\bf 6}}\oplus{\bf
3}\,.$ (33)

Hermitian conjugate is taken in $P$ as it appears in the final states. The
factorizable condition demands that the quark lines of $P$ originate from
$O_{+}$ exclusively. In other word, all the indices of $(P^{\dagger})^{l}_{m}$
shall contract to the ones of ${\cal H}({\bf 15})^{ij}_{k}$. Symbolically it
means that

$\includegraphics[width=195.12767pt]{eq20.eps}\,,$ (34)

where $\delta$ is the Kronecker delta, $F^{i}:={\cal H}({\bf
15})^{ij}_{k}(P^{\dagger})^{k}_{j}$ and the other linear combinations do not
contribute to ${\bf B}_{i}\to{\bf B}_{f}P$. It shows that only the ${\bf 3}$
representation in Eq. (33) contributes, reducing numbers of free parameters.

By identifying the factorizable contribution, we reduce the number of free
parameters from $14$ to $8$ for ${\bf B}_{c}^{A}\to{\bf B}_{n}P$ and arrive at
Asymmetries

$\displaystyle A$ $\displaystyle=$ $\displaystyle a_{1}{\cal H}(\overline{{\bf
6}})_{ij}T^{ik}({\bf B}_{n}^{\dagger})_{k}^{l}(P^{\dagger})_{l}^{j}+a_{2}{\cal
H}(\overline{{\bf 6}})_{ij}T^{ik}(P^{\dagger})_{k}^{l}({\bf
B}_{n}^{\dagger})_{l}^{j}+a_{3}{\cal H}(\overline{{\bf 6}})_{ij}({\bf
B}_{n}^{\dagger})_{k}^{i}(P^{\dagger})_{l}^{j}T^{kl}$ (35)
$\displaystyle+a_{6}({\bf B}_{n})^{j}_{i}F^{i}({\bf B}^{A}_{c})_{j}\,,$

where $T^{ij}\equiv({\bf B}_{c}^{A})_{k}\epsilon^{kij}$ and $a_{1,2,3,6}$ are
free parameters in general. We note that we do not consider $\eta^{\prime}$ as
its mass differs largely from the other pseudoscalar mesons. On the other
hand, the ${\bf B}_{c}^{S}$ and ${\bf B}_{cc}$ decays are parameterized by

$\displaystyle A$ $\displaystyle=$ $\displaystyle a_{1}{\cal H}(\overline{{\bf
6}})_{ij}({\bf B}_{c}^{S})^{ik}({\bf
B}_{n}^{\dagger})_{k}^{l}(P^{\dagger})_{l}^{j}+a_{2}{\cal H}(\overline{{\bf
6}})_{ij}({\bf B}_{c}^{S})^{ik}(P^{\dagger})_{k}^{l}({\bf
B}_{n}^{\dagger})_{l}^{j}+a_{3}{\cal H}(\overline{{\bf 6}})_{ij}({\bf
B}_{n}^{\dagger})_{k}^{i}(P^{\dagger})_{l}^{j}({\bf B}_{c}^{S})^{kl}$ (36)
$\displaystyle+a_{4}{\cal H}(\overline{{\bf 6}})_{ij}({\bf
B}_{n}^{\dagger})_{k}^{l}(P^{\dagger})_{l}^{k}({\bf
B}_{c}^{S})^{ij}+a_{6}({\bf B}_{n}^{\dagger})^{j}_{k}F^{i}({\bf
B}^{S}_{c})^{kl}\epsilon_{ijl}\,,$

for ${\bf B}_{c}^{S}\to{\bf B}_{n}P$,

$\displaystyle A$ $\displaystyle=$ $\displaystyle
a_{1}(P^{\dagger})_{i}^{j}({\bf B}_{c}^{A\dagger})^{i}{\cal H}(\overline{{\bf
6}})_{jk}({\bf B}_{cc})^{k}+a_{2}(P^{\dagger})_{i}^{j}({\bf
B}_{c}^{A\dagger})^{k}{\cal H}(\overline{{\bf 6}})_{jk}({\bf B}_{cc})^{i}$
(37) $\displaystyle+a_{6}F^{i}({\bf B}_{cc})^{j}({\bf
B}_{c}^{A\dagger})^{k}\epsilon_{ijk}\,,$

for ${\bf B}_{cc}\to{\bf B}_{c}^{A}P$, and

$\displaystyle A$ $\displaystyle=$ $\displaystyle
a_{1}(P^{\dagger})_{i}^{j}({\bf B}_{c}^{S\dagger})_{jk}{\cal H}(\overline{{\bf
6}})_{lm}({\bf B}_{cc})^{l}\epsilon^{ikm}+a_{2}(P^{\dagger})_{i}^{l}({\bf
B}_{c}^{S\dagger})_{jk}{\cal H}(\overline{{\bf 6}})_{lm}({\bf
B}_{cc})^{j}\epsilon^{ikm}$ (38) $\displaystyle+a_{6}F^{i}({\bf
B}_{cc})^{j}({\bf B}_{c}^{S\dagger})_{ij}\,,$

for ${\bf B}_{cc}\to{\bf B}_{c}^{S}P$. The $P$-wave amplitudes share the same
flavor structures with the $S$-wave ones and are obtained by

$B=A(a_{1,2,3,4,6}\to b_{1,2,3,4,6})\,.$ (39)

Please note that the same symbols are used to denote the parameters in Eqs.
(35), (36), (37), and (38). Although the symbols are the same, it is important
to recognize that they do not represent identical values in each equation. We
have used the same symbols in these different contexts due to a limitation in
the available symbols. The above parameterizaions with (35), (36), (37), and
(38) would be referred to as the general pole (GP) scenario.

One of the shortcoming of the GP scenario is that there are too many
parameters. As there are few available input for ${\bf B}_{c}^{S}$ and ${\bf
B}_{cc}$ decays, the GP scenario does not have concrete predictions except for
several direct relations. To overcome this problem, we assume that the
intermediate baryons ${\bf B}_{I}$ depicted in FIG. 2 are dominated by the
low-lying ones, which would be referred to as the low-lying pole (LP)
scenario. It allows us to infer the baryon matrix elements exhibited in ${\bf
B}_{c}^{S}\to{\bf B}_{n}P$ and ${\bf B}_{cc}\to{\bf B}_{c}^{A,S}P$ from ${\bf
B}_{c}^{A}\to{\bf B}_{n}P$. To this end, the next section is devoted to
calculating the factorizable contributions, and in the section following the
next one, we relate the four-quark operator matrix element in the decays of
$\mathbf{B}_{cc}$ and $\mathbf{B}_{c}^{A}$ for evaluating the pole diagrams.

##  III Factorization contributions

The amplitude is decomposed into the factorizable and nonfactorizable parts as

${\cal M}={\cal M}^{\text{fac}}+{\cal M}^{\text{pole}}\,,$ (40)

followed by $A=A^{\text{fac}}+A^{\text{pole}}$ and
$B=B^{\text{fac}}+B^{\text{pole}}$. The factorizable amplitude reads

${\cal M}^{\text{fac}}=\frac{G_{F}}{\sqrt{2}}\tilde{\cal H}^{ij}_{kl}\langle
P|\overline{q}_{i}\gamma_{\mu}(1-\gamma_{5})\overline{q}^{k}|0\rangle\langle{\bf
B}_{f}|\overline{q}_{j}\gamma^{\mu}(1-\gamma_{5})q^{l}|{\bf B}_{i}\rangle\,.$
(41)

Expressing the baryon matrix element with the $SU(3)_{F}$ symmetry, we find

$\displaystyle A^{\text{fac}}$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cq}^{*}V_{uq^{\prime}}{\cal
C}_{+,0}f_{P}\left(M_{i}-M_{f}\right)F_{V}$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}f_{P}\left(M_{i}-M_{f}\right)\tilde{\cal
H}^{ij}_{kl}(P^{\dagger})^{k}_{i}\left(A_{1}^{\text{fac}}{\bf B}^{m[nl]}{\bf
B}_{m[nj]}^{\dagger}+A_{2}^{fac}{\bf B}^{m[nl]}{\bf
B}_{n[jm]}^{\dagger}\right)\,,$ $\displaystyle B^{\text{fac}}$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cq}^{*}V_{uq^{\prime}}{\cal
C}_{+,0}f_{P}\left(M_{i}+M_{f}\right)G_{V}\,,$ (42) $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}f_{P}\left(M_{i}+M_{f}\right)\tilde{\cal
H}^{ij}_{kl}(P^{\dagger})_{i}^{k}\left(B_{1}^{\text{fac}}{\bf B}^{m[nl]}{\bf
B}_{m[nj]}^{\dagger}+B_{2}^{fac}{\bf B}^{m[nl]}{\bf
B}_{n[jm]}^{\dagger}\right)\,,$

where ${\cal C}_{+,0}$ are the effective Wilson coefficients with the
subscript denoting the charge of $P$, $f_{P}$ is the meson decay constant,
$F_{V}$ and $G_{V}$ are the leading vector and axial-vector form factors,
respectively, $\tilde{{\cal H}}$ is obtained by substituting ${\cal C}_{+,0}$
for $c_{1,2}$ in Eq. (II), and ${\bf B}$ and ${\bf B}^{\dagger}$ are the
tensors of ${\bf B}_{i}$ and ${\bf B}_{f}$, respectively, given in Eqs. (16)
and (II). From Eq. (28), we have $A_{2}^{\text{fac}}/A_{1}^{\text{fac}}=1/2$
and $B_{2}^{\text{fac}}/B_{1}^{\text{fac}}=5/4$ .

For $c\to s$ and $c\to u/d$ transitions in ${\bf B}_{c}^{A,s}$ decays, we fix
$A_{1}^{\text{fac}}$ and $B_{1}^{\text{fac}}$ with $\Lambda_{c}\to\Lambda$ and
$\Lambda_{c}\to n$ from LQCD at $q^{2}=0$ latL ; latn ; LcForm

$(F_{V},G_{V})_{c\to s}=(0.643,0.572)\,,~{}~{}~{}(F_{V},G_{V})_{c\to
u/d}=(0.672,0.602)\,,$ (43)

and arrive at

$(A_{1}^{\text{fac}},B_{1}^{\text{fac}})_{c\to
s}=(-2.572,-1.525)\,,~{}~{}~{}~{}(A_{1}^{\text{fac}},B_{1}^{\text{fac}})_{c\to
u/d}=(-2.195,-1.311)\,.$ (44)

At the limit of the $SU(3)_{F}$ symmetry, the form factors of $c\to s$ and
$c\to u/d$ would be numerically the same. Here we see that they deviate
roughly 15%, which is a common size of the $SU(3)_{F}$ breaking.

The form factors of ${\bf B}_{cc}\to{\bf B}_{c}^{A,S}$ from LQCD are not
available yet. Nonetheless, we utilize the approximation that the form factors
are independent of the spectator quark flavors, which allows us to infer them
from $\Lambda_{c}^{+}\to\Lambda/n$. We match the form factors of ${\bf
B}_{cc}\to{\bf B}_{c}^{A,S}$ to those of ${\bf B}_{c}^{A}\to{\bf B}_{n}$ at
the same $\omega=v_{i}\cdot v_{f}$ with $v_{i(f)}$ representing the 4-velocity
of ${\bf B}_{i(f)}$. By using the form factors provided in Refs. latL ; latn ,
we arrive at

$(A_{1}^{\text{fac}},B_{1}^{\text{fac}})_{c\to
s}=(-3.615-1.939)\,,~{}~{}~{}~{}(A_{1}^{\text{fac}},B_{1}^{\text{fac}})_{c\to
d}=(-3.518,-1.813)\,.$ (45)

for ${\bf B}_{cc}\to{\bf B}_{c}^{A,S}$. The main difference between Eqs. (44)
and (45) arises from the $\omega$ dependencies of the form factors.
Specifically, the values of $(\omega-1)$ for the transitions
$\Xi_{cc}^{++}\to\Xi^{+}_{c}\pi^{+}$ and $\Lambda_{c}^{+}\to\Lambda\pi^{+}$
are $0.074$ and $0.269$, respectively, which deviate significantly from each
other.

In this work, we fix ${\cal C}_{0}=-0.36\pm 0.04$ by ${\cal
B}_{exp}(\Lambda_{c}^{+}\to p\phi)$ from the experiment as shown in Appendix A
while ${\cal C}_{+}$ is treated as a free parameter in general.

##  IV Pole contributions

The amplitude of the $s$-channel can be illustratively represented in the
form:

$\propto g_{{\bf B}_{f}{\bf B}_{I}P}\frac{1}{M_{i}-M_{I}}\langle{\bf
B}_{I}|{\cal H}_{\text{eff}}|{\bf B}_{i}\rangle\,,$ (46)

where ${\bf B}_{I}$ and $M_{I}$ denote the intermediate baryon and its
corresponding mass, respectively. The coupling of ${\bf B}_{I}-{\bf B}_{f}-P$
is represented by $g_{{\bf B}_{I}{\bf B}_{f}P}$. The $u$-channel amplitude can
be parameterized in a manner akin to the above expression.

In this work, the baryon-meson couplings of $g_{{\bf B}{\bf B}^{(*)}P}$ are
extracted by the generalized Goldberg-Treiman relations

$g_{{\bf B}^{\prime}{\bf B}P}=\frac{\sqrt{2}}{f_{P}}(M^{\prime}+M)g^{P}_{{\bf
B}^{\prime}{\bf B}}\,,~{}~{}~{}g_{{\bf B}^{*}{\bf
B}P}=\frac{\sqrt{2}}{f_{P}}(M^{*}-M)g^{P}_{{\bf B}^{*}{\bf B}}\,,$ (47)

where

$\displaystyle(P^{\dagger})_{j}^{i}\langle{\bf
B}^{\prime}|\overline{q}_{i}\gamma_{\mu}\gamma_{5}q^{j}|{\bf B}\rangle$
$\displaystyle=$ $\displaystyle\overline{u}_{{\bf
B}^{\prime}}\left(g^{P}_{{\bf B}^{\prime}{\bf
B}}\gamma_{\mu}-ig_{2}\sigma_{\mu\nu}q^{\nu}+g_{3}q_{\mu}\right)\gamma_{5}u_{{\bf
B}}\,,$ $\displaystyle(P^{\dagger})_{j}^{i}\langle{\bf
B}^{*}|\overline{q}_{i}\gamma_{\mu}\gamma_{5}q^{j}|{\bf B}\rangle$
$\displaystyle=$ $\displaystyle\overline{u}_{{\bf B}^{*}}\left(g^{P}_{{\bf
B}^{*}{\bf
B}}\gamma_{\mu}-ig_{2}\sigma_{\mu\nu}q^{\nu}+g_{3}q_{\mu}\right)u_{{\bf
B}}\,.$ (48)

The symbols ${\bf B}^{\prime}$ and ${\bf B}^{*}$ denote the intermediate
baryons with spin-parity $\frac{1}{2}^{+}$ and $\frac{1}{2}^{-}$,
respectively. The corresponding masses of ${\bf B}^{(\prime)}$ and ${\bf
B}^{*}$ are represented by $M^{(\prime)}$ and $M^{*}$. The Goldberg-Treiman
relations are derived by operating $q^{\mu}$ on both sides of Eq. (IV) and
impose the equation of motion. The actual values of $g_{2}$ would be
irrelevant to this work and $g_{3}$ is mainly contributed by the baryon-meson
couplings.

The baryon matrix elements of the effective Hamiltonian with $\Delta c=-1$ are
decomposed as

$\displaystyle\langle{\bf B}^{\prime}|{\cal H}_{eff}|{\bf
B}\rangle=\overline{u}_{{\bf B}^{\prime}}\left(a_{{\bf B}^{\prime}{\bf
B}}-b_{{\bf B}^{\prime}{\bf B}}\gamma_{5}\right)u_{{\bf B}}\,,$
$\displaystyle\langle{\bf B}^{*}|{\cal H}_{eff}|{\bf
B}\rangle=\overline{u}_{{\bf B}^{*}}b_{{\bf B}^{*}{\bf B}}u_{{\bf
B}}\,,~{}~{}~{}\langle{\bf B}|{\cal H}_{eff}|{\bf
B}^{*}\rangle=\overline{u}_{{\bf B}}b_{{\bf B}{\bf B}^{*}}u_{{\bf B}^{*}}\,.$
(49)

In the following, $b_{{\bf B}^{\prime}{\bf B}}$ will be dropped as it is tiny
Cheng:1985dw . Collecting Eqs. (46), (47) and (IV), we are led to

$\displaystyle A^{\text{pole }}({\bf B}_{c}\to{\bf
B}_{n}P)=\frac{\sqrt{2}}{f_{P}}\sum_{{\bf
B}_{n,c}^{*}}\left(R_{c}^{A_{s}}g_{{\bf B}_{n}{\bf B}_{n}^{*}}^{P}b_{{\bf
B}_{n}^{*}{\bf B}_{c}}+R_{c}^{A_{u}}b_{{\bf B}_{n}{\bf B}_{c}^{*}}g_{{\bf
B}_{c}^{*}{\bf B}_{c}}^{P}\right)\,,$ (50) $\displaystyle B^{\text{pole
}}({\bf B}_{c}\to{\bf B}_{n}P)=\frac{\sqrt{2}}{f_{P}}\sum_{{\bf
B}_{n,c}^{\prime}}\left(R_{c}^{B_{s}}g^{P}_{{\bf B}_{n}{\bf
B}_{n}^{\prime}}a_{{\bf B}_{n}^{\prime}{\bf B}_{c}}+R_{c}^{B_{u}}a_{{\bf
B}_{n}{\bf B}_{c}^{\prime}}g^{P}_{{\bf B}_{c}^{\prime}{\bf B}_{c}}\right)\,,$

and

$\displaystyle A^{\text{pole }}({\bf B}_{cc}\to{\bf
B}_{c}P)=\frac{\sqrt{2}}{f_{P}}\sum_{{\bf
B}_{c,cc}^{*}}\left(R_{cc}^{A_{s}}g_{{\bf B}_{c}{\bf B}_{c}^{*}}^{P}b_{{\bf
B}_{c}^{*}{\bf B}_{cc}}+R_{cc}^{A_{u}}b_{{\bf B}_{c}{\bf B}_{cc}^{*}}g_{{\bf
B}_{cc}^{*}{\bf B}_{cc}}^{P}\right)\,,$ (51) $\displaystyle B^{\text{pole
}}({\bf B}_{cc}\to{\bf B}_{c}P)=\frac{\sqrt{2}}{f_{P}}\sum_{{\bf
B}_{c,cc}^{\prime}}\left(R_{cc}^{B_{s}}g^{P}_{{\bf B}_{c}{\bf
B}_{c}^{\prime}}a_{{\bf B}_{c}^{\prime}{\bf B}_{cc}}+R_{cc}^{B_{u}}a_{{\bf
B}_{c}{\bf B}_{cc}^{\prime}}g^{P}_{{\bf B}_{cc}^{\prime}{\bf
B}_{cc}}\right)\,,$

where the mass ratios are defined by

$\displaystyle
R_{c}^{A_{s}}=\frac{M_{n}-M_{n^{*}}}{M_{c}-M_{n^{*}}}\,,~{}~{}~{}R_{c}^{A_{u}}=\frac{M_{c}-M_{c^{*}}}{M_{{c}^{*}}-M_{n}}\,,$
$\displaystyle
R_{c}^{B_{s}}=\frac{M_{n}+M_{n^{\prime}}}{M_{c}-M_{n^{\prime}}}\,,~{}~{}~{}R_{c}^{B_{u}}=\frac{M_{c}+M_{c^{\prime}}}{M_{n}-M_{c^{\prime}}}\,,$
(52)

and

$\displaystyle
R_{cc}^{A_{s}}=\frac{M_{c}-M_{c^{*}}}{M_{cc}-M_{{c}^{*}}}\,,~{}~{}~{}R_{cc}^{A_{u}}=\frac{M_{d}-M_{cc^{*}}}{M_{cc^{*}}-M_{c}}\,,$
$\displaystyle
R_{cc}^{B_{s}}=\frac{M_{c}+M_{c^{\prime}}}{M_{cc}-M_{c^{\prime}}}\,,~{}~{}~{}R_{cc}^{B_{u}}=\frac{M_{cc}+M_{cc^{\prime}}}{M_{c}-M_{cc^{\prime}}}\,.$
(53)

Here, $M_{n,c,d^{(\prime,*)}}$ represent the masses of ${\bf
B}_{n,c,d}^{(\prime,*)}$, respectively.

Up to the present, there is no ample data to accurately fit the unknown
hadronic parameters for $\Omega_{c}^{0}$ and ${\bf B}_{cc}$ decays. In the
subsequent analysis, we will utilize two essential approximations, as
delineated in the Introduction:

  * •

The intermediate states ${\bf B}_{I}$ are exclusively confined to the low-
lying ${\bf 20}$ multiplets of the $SU(4)_{F}$ group. Here, ${\bf 20}={\bf
8}\oplus\overline{{\bf 3}}\oplus{\bf 6}\oplus{\bf 3}$ in the $SU(3)_{F}$
group.

  * •

The baryon matrix elements are independent of the spectator quarks, implying
that the amplitudes shown in Fig. 3 do not depend on $q^{(\prime)}$.

The reliability of our predictions hinges on the validity of these two
approximations. The first approximation emphasizes that ${\bf
B}^{\prime}\in\\{{\bf B}_{c}^{A,S},{\bf B}_{n},{\bf B}_{cc}\\}$ and ${\bf
B}^{*}$ belong to the representation of ${\bf 20}$ also.

On the other hand, we have already used the second approximation to extract
the form factors of ${\bf B}_{cc}\to{\bf B}_{c}^{A,S}$ in Eq. (45), which are
essentially two-quark operator baryon matrix elements. For the four-quark
operators, it facilitates the parameterization expressed in

$a_{{\bf B}^{\prime}{\bf B}}=\tilde{a}{\bf B}^{i[jk]}{\cal H}_{jk}^{lm}{\bf
B}^{\prime\dagger}_{ilm}\,,~{}~{}~{}~{}b_{{\bf B}^{*}{\bf B}}=\tilde{b}{\bf
B}^{i[jk]}{\cal H}_{jk}^{lm}{\bf B}^{*\dagger}_{ilm}\,,~{}~{}b_{{\bf B}{\bf
B}^{*}}=\tilde{b}^{\prime}{\bf B}^{*i[jk]}{\cal H}_{jk}^{lm}{\bf
B}^{\dagger}_{ilm}\,,$ (54)

and

$\displaystyle g^{P}_{{\bf B}^{\prime}{\bf B}}$ $\displaystyle=g_{1}{\bf
B}^{i[jk]}{\bf B}_{i[jl]}^{\prime\dagger}(P^{\dagger})^{l}_{k}+g_{2}{\bf
B}^{i[jk]}{\bf B}_{j[li]}^{\prime\dagger}(P^{\dagger})^{l}_{k}\,,$
$\displaystyle g^{P}_{{\bf B}^{*}{\bf B}}$
$\displaystyle=g^{\overline{P}}_{{\bf B}^{*}{\bf B}}=g^{\prime}_{1}{\bf
B}^{i[jk]}{\bf B}^{*\dagger}_{i[jl]}(P^{\dagger})^{l}_{k}+g_{2}^{\prime}{\bf
B}^{i[jk]}{\bf B}_{j[li]}^{*\dagger}(P^{\dagger})^{l}_{k}\,.$ (55)

Furthermore, by implementing Eq. (28), we obtain the ratio $g_{2}/g_{1}=5/4$,
leading to the vanishing of $g^{P}_{{\bf B}_{c}^{A}{\bf B}_{c}^{A}}$
Yan:1992gz . Incorporating Eqs. (54) and (IV) into Eqs. (50) and (51) and
summing over ${\bf B}_{I}$, we eliminate the tensors of the intermediate
states by employing the completeness relation Groote:2021pxt

$\displaystyle\sum_{{\bf B}_{c}^{A}}({\bf B}_{c}^{A})_{i}({\bf
B}_{c}^{A\dagger})^{j}=\delta^{j}_{i}\,,~{}~{}~{}~{}~{}~{}\sum_{{\bf
B}_{n}}({\bf B}_{n})^{i}_{j}({\bf
B}_{n}^{\dagger})^{k}_{l}=\delta^{i}_{l}\delta^{k}_{j}-\frac{1}{3}\delta^{i}_{j}\delta^{k}_{l}\,,$
$\displaystyle\sum_{{\bf B}_{cc}}({\bf B}_{cc})^{i}({\bf
B}_{cc}^{\dagger})_{j}=\delta^{i}_{j}\,,~{}~{}~{}~{}~{}~{}~{}\sum_{{\bf
B}_{c}^{S}}({\bf B}_{c}^{S})^{ij}({\bf
B}_{c}^{S\dagger})_{kl}=\frac{1}{2}\left(\delta^{i}_{k}\delta^{j}_{l}+\delta^{j}_{k}\delta^{i}_{l}\right)\,,$
(56)

where we have taken the baryons with spin-parity $\frac{1}{2}^{+}$ as
examples. The same relation would hold for ${\bf B}^{*}_{n,c,cc}$ as they
belong to the same $SU(3)_{F}$ group, which allows us to consider the
contributions of negative baryons without specifying them.

Figure 3: The topological diagrams for the baryon matrix elements of the two-
quark and four-quark operators. We use the approximation of that their
magnitudes do not depend on $q^{(\prime)}$.

To calculate $R_{c,cc}^{B_{s,u}}$, the masses of ${\bf B}^{(\prime)}$ are
readily available from experimental measurements ParticleDataGroup:2022pth .
However, the masses of ${\bf B}^{*}$ are not fully available yet. For the
charmless octet baryons, we consider the states $N(1535)$ and $\Sigma(1750)$,
taking the average mass value of $M_{n^{*}}=1643\,\text{MeV}$. For the charmed
baryons with negative parity ${\bf B}_{c}^{*}$, we identify the candidates as
$\Lambda_{c}^{+}(2595)$, $\Xi_{c}^{+}(2790)$, and $\Sigma_{c}^{+}(2792)$, from
which we calculate the average masses $M_{c^{*}}=2700\,\text{MeV}$ and
$M_{c^{*}}=2900\,\text{MeV}$ for the ${\overline{{\bf 3}}}$ and ${\bf 6}$
representations, respectively. In the case of the doubly charmed baryons with
$J=\frac{1}{2}^{-}$, we adopt the value $M_{cc^{*}}=3932\,\text{MeV}$
Yu:2022lel . Summarizing, the mass ratios related to the $J=\frac{1}{2}^{-}$
baryons utilized in this work are expressed as:

$\displaystyle(R_{c}^{A_{u}},R_{c}^{A_{s}}(\overline{\bf
3}),R_{c}^{A_{s}}({\bf 6}))$ $\displaystyle=(-0.671,-0.207,-0.298),$
$\displaystyle(R_{cc}^{A_{u}}(\overline{\bf 3}),R_{cc}^{A_{u}}({\bf
6}),R_{cc}^{A_{s}})$ $\displaystyle=(-0.350,-0.725,-0.201)\,,$ (57)

where the parenthesis denotes the representation of $M_{c}^{*}$. We note that
focusing solely on ${\bf B}^{A}_{c}$ decays, the uncertainties in
$R_{c,cc}^{A_{s,u}}$ would be incorporated into the baryon matrix elements of
$g_{1,2}^{\prime}$ and $\tilde{b}^{(\prime)}$. Consequently, the uncertainties
in Eq. (IV) would only influence the predictions for the $\Omega_{c}^{0}$ and
${\bf B}_{cc}$ decays.

##  V numerical results

The numerical results of this study are organized into several subsections. In
Subsec. A, we recall the experimental data of the ${\bf B}_{c}^{A}$ decays,
and the free parameters in both the LP and GP scenarios are extracted
accordingly. Although the GP scenario provides more reliable predictions for
${\bf B}_{c}^{A}$ decays, the LP scenario has broader applications, i.e., its
parameters can be applied to both $\Omega_{c}^{0}$ and ${\bf B}_{cc}$ decays.
Subsec. B and C are devoted to the study of $\Omega_{c}^{0}$ and ${\bf
B}_{cc}$ decays in the LP scenario, respectively.

###  V.1 Results of ${\bf B}_{c}^{A}$ decays

Since $g_{{\bf B}^{\prime}{\bf B}}^{P}$($g_{{\bf B}^{*}{\bf B}}^{P}$) is
always followed by $a_{{\bf B}^{\prime}{\bf B}}(b_{{\bf B}^{*}{\bf B}})$ (see
Eqs. (50) and (51)), we absorb $g_{1}(g_{1}^{\prime})$ into
$\tilde{a}(\tilde{b}^{(\prime)})$ so that $g_{1}^{(\prime)}=1$. For the
nonfactorizable amplitudes, there remain $(\tilde{a})$ and
$(\tilde{b},\tilde{b}^{\prime},g_{2}^{\prime})$ to be fitted in the $P$\- and
$S$-waves, respectively. Comparing to the GP scenario, the parameters of the
nonfactorizable amplitudes in the $P$-waves have been reduced from $3$ to $1$.
It is due to that we have related $g_{{\bf B}_{n}{\bf B}_{n}^{\prime}}^{P}$
with $g_{{\bf B}_{c}{\bf B}_{c}^{\prime}}^{P}$ in Eq. (IV) and demand
$4g_{2}=5g_{1}$. On the other hand, due to a lack of knowledge of parity-odd
baryons, we impose no further constraints on the $S$-waves in comparison to
the GP scenario.

The experimental data of the ${\bf B}_{c}^{A}$ decays up to date
ParticleDataGroup:2022pth ; Belle:new ; BESIII:new  are collected in Table 1.
By adopting the minimal $\chi^{2}$ fitting, we find

$\footnotesize\left(\tilde{a},\tilde{b},\tilde{b}^{\prime},g_{2}^{\prime},{\cal
C}_{+}\right)=(2.06\pm 0.25,12.51\pm 1.03,-4.01\pm 1.13,0.148\pm
0.075,0.467\pm 0.034)\,,$ (58)

where $(\tilde{a},\tilde{b},\tilde{b}^{\prime})$ are in units of
$10^{-3}G_{F}$GeV3. In the limit of the $SU(4)_{F}$ symmetry, we would expect
$\tilde{b}=\tilde{b}^{\prime}$, but we observe a significant $SU(4)_{F}$
breaking as they differ both in sign and magnitude. It indicates that the
charm quark and the light quarks behave very differently in ${\bf B}^{*}$. We
note that ${\cal C}_{+}$ is twice smaller than the expected value of ${\cal
C}_{+}\approx 1.2$ from the effective color number approach, discussed in
Appendix A.

For comparison, we also update the results of the GP scenario. The free
parameters in Eq. (35) are found to be

$\displaystyle(a_{1},a_{2},a_{3},a_{6})=(3.25\pm 0.11,1.60\pm 0.07,0.58\pm
0.12,1.74\pm 0.22)$ (59) $\displaystyle(b_{1},b_{2},b_{3},b_{6})=(11.66\pm
0.19,-4.96\pm 0.19,2.87\pm 0.27,-0.03\pm 0.36)\,,$

in units of $10^{-2}G_{F}$GeV2. Comparing to the previous values222 With the
experimental data up to 16, May, 2019, Ref. Asymmetries  reported
$(a_{1},a_{2},a_{3},a_{6})=(4.34\pm 0.50,-1.33\pm 0.32,1.25\pm 0.36,-0.26\pm
0.64)$ and $(b_{1},b_{2},b_{3},b_{6})=(9.20\pm 2.09,8.03\pm 1.19,-1.42\pm
1.61,4.05\pm 2.48)$ instead. , we see that the parameters modify
significantly. It is a hint of that the results shall not be trust fully.
Since the $SU(3)_{F}$ symmetry is not exact and too many parameters are
required, it is reasonable that the best fitting solutions are not stable
along with the experimental update.

Table 1: Results of the low-lying and general pole scenarios, denoted with $LP$ and $GP$ in the subscripts, where the parameters are extracted from the current experimental data of ${\cal B}_{exp}$ and $\alpha_{exp}$ collected in the first column ParticleDataGroup:2022pth ; BESIII:new ; Belle:new . Here, the numbers in the parentheses are the uncertainties counting backward in digits, for example, $1.59(8)=1.59\pm 0.08$.  Channels | ${\cal B}_{exp}(\%)$ | $\alpha_{exp}$ | ${\cal B}_{LP}(\%)$ | $\alpha_{LP}$ | ${\cal B}_{GP}(\%)$ | $\alpha_{GP}$
---|---|---|---|---|---|---
$\Lambda_{c}^{+}\to pK_{S}^{0}$ | $1.59(8)$ | $0.18(45)$ | $1.44(7)$ | $-0.68(1)$ | $1.55(6)$ | $-0.81(5)$
$\Lambda_{c}^{+}\to\Lambda\pi^{+}$ | $1.30(6)$ | $-0.755(6)$ | $0.96(16)$ | $-0.75(1)$ | $1.32(5)$ | $-0.75(1)$
$\Lambda_{c}^{+}\to\Sigma^{0}\pi^{+}$ | $1.27(6)$ | $-0.466(18)$ | $1.14(14)$ | $-0.45(4)$ | $1.25(5)$ | $-0.47(1)$
$\Lambda_{c}^{+}\to\Sigma^{+}\pi^{0}$ | $1.25(10)$ | $-0.48(3)$ | $1.14(14)$ | $-0.45(4)$ | $1.25(5)$ | $-0.47(1)$
$\Lambda_{c}^{+}\to\Xi^{0}K^{+}$ | $0.55(7)$ |  | $0.02(2)$ | $-0.02(0)$ | $0.41(3)$ | $0.95(2)$
$\Lambda_{c}^{+}\to\Lambda K^{+}$ | $0.064(3)$ | $-0.585(52)$ | $0.072(7)$ | $-0.76(4)$ | $0.065(3)$ | $-0.56(4)$
$\Lambda_{c}^{+}\to\Sigma^{0}K^{+}$ | $0.0382(25)$ | $-0.54(20)$ | $0.0284(35)$ | $-0.41(5)$ | $0.0393(16)$ | $-1.00(0)$
$\Lambda_{c}^{+}\to n\pi^{+}$ | $0.066(13)$ |  | $0.008(5)$ | $-0.87(11)$ | $0.067(3)$ | $0.53(5)$
$\Lambda_{c}^{+}\to\Sigma^{+}K_{S}^{0}$ | $0.048(14)$ |  | $0.028(3)$ | $-0.41(5)$ | $0.039(2)$ | $-1.00(0)$
$\Lambda_{c}^{+}\to p\pi^{0}$ | $<8\times 10^{-3}$ |  | $0.01(1)$ | $-0.77(24)$ | $0.01(0)$ | $0.92(8)$
333 The experimental branching fractions are not included in the global fit. $\Lambda_{c}^{+}\to p\eta$ | $0.158(12)$ |  | $0.142(8)$ | $-0.70(1)$ | $0.150(8)$ | $-0.45(10)$
${}^{a}\Lambda_{c}^{+}\to\Sigma^{+}\eta$ | $0.312(44)$ |  | $0.13(3)$ | $-0.49(10)$ | $0.35(2)$ | $-0.47(5)$
$\Xi_{c}^{+}\to\Xi^{0}\pi^{+}$ | $1.6(80)$ |  | $0.87(18)$ | $-0.88(7)$ | $0.87(8)$ | $-0.88(4)$
$\Xi_{c}^{0}\to\Lambda K_{S}^{0}$ | $0.32(7)$ |  | $0.54(3)$ | $-0.61(2)$ | $0.68(2)$ | $-0.69(4)$
$\Xi_{c}^{0}\to\Xi^{-}\pi^{+}$ | $1.43(32)$ | $-0.64(5)$ | $2.98(29)$ | $-0.64(2)$ | $2.98(8)$ | $-0.99(0)$
$\Xi_{c}^{0}\to\Xi^{-}K^{+}$ | $0.039(12)$ |  | $0.135(14)$ | $-0.69(1)$ | $0.131(4)$ | $-0.97(0)$
$\Xi_{c}^{0}\to\Sigma^{0}K_{S}^{0}$ | $0.054(16)$ |  | $0.057(13)$ | $-0.91(5)$ | $0.053(16)$ | $0.59(13)$
$\Xi_{c}^{0}\to\Sigma^{+}K^{-}$ | $0.18(4)$ |  | $0.01(2)$ | $-0.27(11)$ | $0.48(3)$ | $1.00(0)$

In regard to the results in Table 1, several comments are in order:

  * •

For $\alpha(\Lambda_{c}^{+}\to pK_{S})$, ${\cal B}(\Xi_{c}^{0}\to\Lambda
K_{S})$ and especially ${\cal B}(\Xi_{c}^{0}\to\Xi^{-}\pi^{+})$, good
accordance is found in two scenarios but both suggest very different values
against the current experimental data. It indicates that the short distance
contributions may play a dominate role in these decays. Experimental revisits
on these channels shall be welcome.

  * •

The results of $\Lambda_{c}^{+}\to\Xi^{0}K^{+}$, $\Lambda_{c}^{+}\to n\pi^{+}$
and $\Xi_{c}^{0}\to\Sigma^{+}K^{-}$ deviate largely between two scenarios. It
implies that the excited states which do not belong to the ${\bf 20}$
$SU(4)_{F}$ multiplets may play an important role in ${\bf B}_{I}$.

  * •

The $P$-wave amplitude of $\Lambda_{c}^{+}\to\Xi^{0}K^{+}$ vanishes naturally
in the LP scenario Groote:2021pxt , resulting in $\alpha_{LP}=0$. Nonetheless,
$\alpha_{GP}=0.95\pm 0.02$ indicates another way round.

  * •

In contrast to the $P$-wave, the $S$-wave does not vanish in general for
$\Lambda_{c}^{+}\to\Xi^{0}K^{+}$ in the LP scenario. However, the current
experimental data prefers a vanishing $S$-wave also, leading to contradiction
against ${\cal B}_{exp}(\Lambda_{c}^{+}\to\Xi^{0}K^{+})$.

  * •

Continuing the above comment, we see that the LP scenario also fails to
explain ${\cal B}_{exp}(\Lambda_{c}^{+}\to n\pi^{+})$, but ${\cal
B}_{GP}(\Lambda_{c}^{+}\to\Xi^{0}K^{+})$ and ${\cal B}_{GP}(\Lambda_{c}^{+}\to
n\pi^{+})$ are consistent with the experimental data.

  * •

The ratio of ${\cal R}_{\pi/K}={\cal B}(\Xi_{c}^{0}\to\Xi^{-}K^{+})/{\cal
B}(\Xi_{c}^{0}\to\Xi^{-}\pi^{+})$ is fixed in the exact $SU(3)_{F}$ symmetry.
From the GP and LP scenarios, we find ${\cal R}_{\pi/K}=4.5\%$ and ${\cal
R}_{\pi/K}=4.4\%$, respectively, which both contradict to the experimental
value of $(2.75\pm 0.51\pm 0.25)\%$ at Belle Belle:2013ntc .

  * •

We do not include ${\cal B}_{exp}(\Lambda_{c}^{+}\to p\eta)$ and ${\cal
B}_{exp}(\Lambda_{c}^{+}\to\Sigma^{+}\eta)$ into the global fit as we do no
consider the $SU(3)_{F}$ singlet in $P$. The results of this work are obtained
by assuming the mixing between $\eta_{0}$ and $\eta_{8}$ is absent.
Surprisingly, the numerical results turn out to be compatible with the current
experimental data.

It is insightful to compare the LP scenario with Ref. Charmed-Cheng  which
computes the $S$-wave amplitudes by the soft meson approximation. Comparisons
for several chosen channels are collected in Table 2. The factorizable
amplitudes with the neutral $P$ agree well as they are fixed by ${\cal
B}_{exp}(\Lambda_{c}^{+}\to p\phi)$. However, for
$\Lambda_{c}^{+}\to\Lambda\pi^{+}$ our $A^{\text{fac}}$ and $B^{\text{fac}}$
are roughly twice smaller than Ref. Charmed-Cheng  as we adopt a much smaller
${\cal C}_{+}$, and we find a sizable $A^{\text{pole}}$ in contrast to
$A^{\text{pole}}=0$ at the soft meson limit. One possible explanation to
reconcile two approaches is that a sizable proportion from exited intermediate
baryons is reabsorbed into ${\cal C}_{+}$, leading to a smaller value of
${\cal C}_{+}=0.469$ against the naïve expectation of ${\cal C}_{+}\approx 1$.
We see that although our sizes of the $S$\- and $P$\- wave amplitudes differ
with Ref. Charmed-Cheng , the signs are consistent for most of the cases. We
point out that good agreements in $\Lambda_{c}^{+}\to p\pi^{0}$ and
$\Lambda_{c}^{+}\to n\pi^{+}$ with Ref. Charmed-Cheng  are found, where large
destructive interference between factorizable and pole amplitudes occurs. It
indicates that the current algebra approach with the soft meson limit is a
good approximation for describing the low-lying poles. However, it shall be
noted that the LP scenario and Ref. Charmed-Cheng  both obtain a much smaller
${\cal B}(\Lambda_{c}^{+}\to n\pi^{+})$ comparing to the experiments444 Ref.
Charmed-Cheng  obtains ${\cal B}(\Lambda_{c}^{+}\to n\pi^{+})=9\times 10^{-5}$
in accordance with $(8\pm 5)\times 10^{-5}$ in the LP scenario.
BESIII:2022bkj .

Table 2: Comparison between the LP scenario and the current algebra approach Charmed-Cheng , where $A$ and $B$ are in units of $10^{-2}G_{F}$GeV2.  Channels | LP scenario | Current algebra Charmed-Cheng
---|---|---
$A^{\text{fac}}$ | $A^{\text{pole}}$ | $B^{\text{fac}}$ | $B^{\text{pole}}$ | $A^{\text{fac}}$ | $A^{\text{pole}}$ | $B^{\text{fac}}$ | $B^{\text{pole}}$
$\Lambda_{c}^{+}\to\Sigma^{+}\pi^{0}$ | $0$ | $-5.82$ | $0$ | $-4.47$ | $0$ | $-7.68$ | $0$ | $-11.34$
$\Lambda_{c}^{+}\to\Sigma^{+}\eta$ | $0$ | $2.16$ | $0$ | $2.04$ | $0$ | $3.10$ | $0$ | $15.54$
$\Lambda_{c}^{+}\to\Sigma^{0}\pi^{+}$ | $0$ | $5.81$ | $0$ | $4.44$ | $0$ | $7.68$ | $0$ | $11.38$
$\Lambda_{c}^{+}\to\Xi^{0}K^{+}$ | $0$ | $-0.79$ | $0$ | $-0.04$ | $0$ | $-4.48$ | $0$ | $12.10$
$\Lambda_{c}^{+}\to p\bar{K}^{0}$ | $3.91$ | $5.31$ | $8.38$ | $0.74$ | $3.45$ | $4.48$ | $6.98$ | $2.06$
$\Lambda_{c}^{+}\to\Lambda\pi^{+}$ | $3.16$ | $1.84$ | $8.18$ | $-1.60$ | $5.34$ | $0$ | $14.11$ | $-3.60$
$\Lambda_{c}^{+}\to p\pi^{0}$ | $0.53$ | $-0.30$ | $1.14$ | $-0.88$ | $0.41$ | $-0.81$ | $0.87$ | $-2.07$
$\Lambda_{c}^{+}\to n\pi^{+}$ | $0.87$ | $-0.43$ | $1.88$ | $-1.24$ | $1.64$ | $-1.15$ | $3.45$ | $-2.93$
$\Xi_{c}^{+}\to\Sigma^{+}\bar{K}^{0}$ | $3.70$ | $-0.76$ | $9.51$ | $-4.46$ | $2.98$ | $-4.48$ | $9.95$ | $-12.28$
$\Xi_{c}^{+}\to\Xi^{0}\pi^{+}$ | $-3.81$ | $0.92$ | $-11.13$ | $5.49$ | $-7.41$ | $5.36$ | $-28.07$ | $14.03$
$\Xi_{c}^{0}\to\Sigma^{+}K^{-}$ | $0$ | $0.79$ | $0$ | $0.33$ | $0$ | $4.42$ | $0$ | $-12.09$
$\Xi_{c}^{0}\to\Sigma^{0}\bar{K}^{0}$ | $2.62$ | $-1.09$ | $6.73$ | $-3.39$ | $2.11$ | $-3.12$ | $7.05$ | $-9.39$
$\Xi_{c}^{0}\to\Xi^{0}\pi^{0}$ | $0$ | $5.15$ | $0$ | $4.62$ | $0$ | $7.58$ | $0$ | $11.79$
$\Xi_{c}^{0}\to\Xi^{0}\eta$ | $0$ | $-3.12$ | $0$ | $-2.41$ | $0$ | $10.80$ | $0$ | $-6.17$
$\Xi_{c}^{0}\to\Xi^{-}\pi^{+}$ | $-3.80$ | $-6.37$ | $-11.16$ | $-1.04$ | $-7.42$ | $-5.36$ | $-28.24$ | $-2.65$
$\Xi_{c}^{0}\to\Lambda\bar{K}^{0}$ | $1.60$ | $4.97$ | $3.80$ | $2.45$ | $1.11$ | $5.41$ | $3.66$ | $6.87$

The numerical results of the ${\bf B}_{c}^{A}$ decay channels, for which there
are no experimental references yet, are collected in Appendix B for use in
future experiments as a basis for verification.

###  V.2 Results of $\Omega_{c}^{0}$ decays

Lacking of experimental input, the GP scenario is not available for
$\Omega_{c}^{0}$ decays. Based on the LP scenario, the predictions of
$\Omega_{c}^{0}\to{\bf B}_{n}P$ are collected in Table 3, where the lifetime
of $\Omega_{c}^{0}$ is taken to be $(273\pm 12)$ fs ParticleDataGroup:2022pth
. It is interesting to see that ${\cal B}(\Omega_{c}\to\Xi^{0}K_{S}^{0})$ and
${\cal B}(\Omega_{c}\to K_{L}^{0})$ deviate significantly, induced by the
interference between the CF and DCS amplitudes.

Table 3: Predictions of the CF, Cabibbo suppressed (CS) and doubly Cabibbo suppressed (DCS) decays with $\Omega_{c}^{0}$ as the initial baryons, where $A$ and $B$ are in units of $10^{-2}G_{F}$GeV2.  CF decays | $A^{\text{fac}}$ | $A^{\text{pole}}$ | $B^{\text{fac}}$ | $B^{\text{pole}}$ | ${\cal B}(\%)$ | $\alpha$
---|---|---|---|---|---|---
$\Omega_{c}^{0}\to\Xi^{0}K_{S}^{0}$ | $-2.43$ | $0.54$ | $2.11$ | $-5.46$ | $0.22(5)$ | $-0.86(10)$
$\Omega_{c}^{0}\to\Xi^{0}K_{L}^{0}$ | $2.19$ | $-1.24$ | $-1.90$ | $5.57$ | $0.11(3)$ | $-0.97^{+0.06}_{-0.03}$
CS decays | $A^{\text{fac}}$ | $A^{\text{pole}}$ | $B^{\text{fac}}$ | $B^{\text{pole}}$ | ${\cal B}(10^{-4})$ | $\alpha$
$\Omega_{c}^{0}\to\Sigma^{+}K^{-}$ | $0$ | $0.31$ | $0$ | $-0.32$ | $0.52^{+1.22}_{-0.52}$ | $0.65(33)$
$\Omega_{c}^{0}\to\Sigma^{0}K_{S/L}^{0}$ | $0$ | $0.16$ | $0$ | $-0.16$ | $0.13^{+0.31}_{-0.13}$ | $0.65(33)$
$\Omega_{c}^{0}\to\Xi^{0}\pi^{0}$ | $0.44$ | $2.54$ | $-0.38$ | $0.35$ | $45.10(3.89)$ | $0.01(1)$
$\Omega_{c}^{0}\to\Xi^{0}\eta$ | $-0.92$ | $-1.24$ | $0.80$ | $-1.24$ | $21.35(5.63)$ | $-0.13(4)$
$\Omega_{c}^{0}\to\Xi^{-}\pi^{+}$ | $-0.73$ | $-3.59$ | $0.64$ | $-0.49$ | $94.59(8.98)$ | $0.02(1)$
$\Omega_{c}^{0}\to\Lambda K_{S/L}^{0}$ | $0$ | $-1.99$ | $0$ | $0.78$ | $18.74(1.96)$ | $0.30(3)$
DCS decays | $A^{\text{fac}}$ | $A^{\text{pole}}$ | $B^{\text{fac}}$ | $B^{\text{pole}}$ | ${\cal B}(10^{-5})$ | $\alpha$
$\Omega_{c}^{0}\to\Sigma^{+}\pi^{-}$ | $0$ | $0.23$ | $0$ | $0.17$ | $2.82(66)$ | $-0.54(12)$
$\Omega_{c}^{0}\to\Sigma^{0}\pi^{0}$ | $0$ | $0.23$ | $0$ | $0.17$ | $2.83(66)$ | $-0.54(12)$
$\Omega_{c}^{0}\to\Sigma^{-}\pi^{+}$ | $0$ | $0.23$ | $0$ | $0.17$ | $2.82(66)$ | $-0.54(12)$
$\Omega_{c}^{0}\to\Xi^{-}K^{+}$ | $-0.20$ | $-0.50$ | $0.18$ | $0.08$ | $23.11(1.84)$ | $0.23(1)$
$\Omega_{c}^{0}\to pK^{-}$ | $0$ | $0.26$ | $0$ | $0.13$ | $3.24(1.13)$ | $-0.43(7)$
$\Omega_{c}^{0}\to nK_{S/L}^{0}$ | $0$ | $-0.18$ | $0$ | $-0.09$ | $1.62(57)$ | $-0.43(7)$
$\Omega_{c}^{0}\to\Lambda\eta$ | $0$ | $-0.27$ | $0$ | $0.31$ | $3.90(88)$ | $0.76(10)$

Up to date, the measurements of the $\Omega_{c}^{0}$ decay ratios are
performed in regard to $\Omega_{c}^{0}\to\Omega^{-}\pi^{+}$. Fortunately,
$\Omega_{c}^{0}\to\Omega^{-}\pi^{+}$ does not receive $W$-exchange
contributions and is color-enhanced. The branching fraction is calculated by

$\Gamma=\frac{p_{f}}{16\pi
M^{2}_{\Omega_{c}}}\left(|H_{+}^{\text{fac}}|^{2}+|H_{-}^{\text{fac}}|^{2}\right)\,,$
(60)

where $H_{+}^{\text{fac}}$ and $H^{\text{fac}}_{-}$ are the factorizable
helicity amplitudes defined as

$H_{\pm}=\frac{G_{F}}{\sqrt{2}}V_{cs}^{*}V_{ud}{\cal
C}_{+}^{\prime}f_{\pi}q^{\mu}\big{\langle}\Omega^{-};\lambda=\pm\frac{1}{2}\big{|}\overline{s}\gamma_{\mu}(1-\gamma_{5})c\big{|}\Omega_{c};J_{z}=\pm\frac{1}{2}\big{\rangle}\,,$
(61)

$q^{\mu}=(q^{0},0,0,-q^{3})$ is the four-momentum of the pion, $\lambda$ and
$J_{z}$ are the helicity and angular momentum of $\Omega^{-}$ and
$\Omega_{c}^{0}$, respectively, and ${\cal C}_{+}^{\prime}$ is the responsible
effective Wilson coefficient. In this work, the baryonic matrix elements in
Eq. (61) are evaluated from the homogeneous bag model CMM .

As $\Omega^{-}$ does not belong to the ${\bf 20}\,\,SU(4)_{F}$ multiplets,
$\Omega_{c}^{0}\to\Omega^{-}\pi^{+}$ does not necessarily share the same
effective Wilson coefficients with ${\cal B}_{c}^{A,S}\to{\cal B}_{n}P$. In
Table 4, we compare the outcomes with various ${\cal C}_{+}^{\prime}$, where

${\cal R}(\Omega_{c}^{0}\to{\bf B}_{n}P):=\frac{{\cal B}(\Omega_{c}^{0}\to{\bf
B}_{n}P)}{{\cal B}(\Omega_{c}^{0}\to\Omega^{-}\pi^{+})}\,,$ (62)

with ${\cal B}(\Omega_{c}\to{\bf B}_{n}P)$ taken from Table 3. We note that
${\cal C}_{+}^{\prime}=1.2,1$ and $0.469$ come from the effective color
scheme, $N_{c}=3$ and ${\bf B}_{c}^{A}\to{\bf B}_{n}P$, respectively. The
scheme of ${\cal C}_{+}^{\prime}=0.469$ is favored by the experiment of ${\cal
R}(\Omega_{c}^{0}\to\Xi^{0}K_{S}^{0})$ but disfavored by the others. On the
other hand, ${\cal R}(\Omega_{c}^{0}\to\Omega^{-}e^{+}\nu_{e})$ suggests
${\cal C}_{+}^{\prime}=1$. One shall bear in mind that these outcomes are
based on the LP scenario and the inconsistencies may disappear in the GP
scenario which is not available due to a lack of experimental input.

Table 4:  Comparisons of the evaluated branching fractions with the experiments ParticleDataGroup:2022pth .  channel | ${\cal C}_{+}^{\prime}=1.20$ | ${\cal C}_{+}^{\prime}=1$ | ${\cal C}_{+}^{\prime}=0.469$ | EXP
---|---|---|---|---
${\cal B}(\Omega_{c}^{0}\to\Omega^{-}\pi^{+})$ | $1.88(15)$ | $1.30(10)$ | $0.29(3)$ | -
${\cal R}(\Omega_{c}^{0}\to\Xi^{0}K_{S}^{0})$ | $0.12(4)$ | $0.17(4)$ | $0.76(25)$ | $0.83(13)$
${\cal R}(\Omega_{c}^{0}\to\Xi^{-}\pi^{+})$ | $0.50(8)$ | $0.73(13)$ | $3.3(6)$ | $0.253(60)$
${\cal R}(\Omega_{c}^{0}\to\Xi^{-}K^{+})$ | $0.012(2)$ | $0.018(3)$ | $0.080(15)$ | $<0.07$
${\cal R}(\Omega_{c}^{0}\to\Omega^{-}e^{+}\nu_{e})$ | $1.35$ | 1.90 | $8.76$ | $1.98(15)$

###  V.3 Results of ${\bf B}_{cc}$ decays

The CF decays of ${\bf B}_{cc}\to{\bf B}_{c}P$ based on the LP scenario are
collected in Table 5, while the others in Appendix C. The lifetimes of the
charmed baryons $(\Xi_{cc}^{++},\Xi_{cc}^{+},\Omega_{cc}^{+})$ are adopted as
$(256,36,136)$ fs, respectively. In analyzing the transition ${\bf
B}^{A}_{c}\to{\bf B}_{n}P$, the fitted value of ${\cal C}_{+}$ is found to be
notably smaller than the naïve expectation. This discrepancy prompts the
consideration of two distinct cases: ${\cal C}_{+}=0.469$ and ${\cal
C}_{+}=1$. All other parameters in this analysis are from Eq. (58).

The branching ratio of ${\cal R}_{\Xi_{cc}}={\cal
B}(\Xi_{cc}^{++}\to\Xi_{c}^{\prime+}\pi^{+})/{\cal
B}(\Xi_{cc}^{++}\to\Xi_{c}^{+}\pi^{+})$ is calculated to be $1.19\pm 0.09$ and
$0.87\pm 0.06$ for ${\cal C}_{+}=0.469$ and $1$, respectively. These results
are roughly consistent with the experimental measurement of $1.41\pm 0.17\pm
0.10$ LHCb:2022rpd . As ${\cal R}_{\Xi_{cc}}$ is not included in the global
fit, it is nontrivial for our outcome to agree with the experiment.
Nevertheless, the calculated branching fraction ${\cal
B}(\Xi_{cc}\to\Xi_{c}^{+}\pi^{+})=(6.24\pm 0.21)\%\ \text{with}\ {\cal
C}_{+}=1$ exceeds the naïve expectation of $(1.33\pm 0.74)\%$, referenced in
Cheng:2020wmk ; Liu:2022igi . Note that
$\Xi_{cc}^{++}\to\Sigma_{cc}^{++}K_{S/L}$ do not receive pole contributions,
and the ratio

$\frac{{\cal B}(\Xi_{cc}^{++}\to\Sigma_{cc}^{++}K_{S}^{0})-{\cal
B}(\Xi_{cc}^{++}\to\Sigma_{cc}^{++}K^{0}_{S})}{{\cal
B}(\Xi_{cc}^{++}\to\Sigma_{cc}^{++}K_{S}^{0})+{\cal
B}(\Xi_{cc}^{++}\to\Sigma_{cc}^{++}K_{L}^{0})}=\frac{2s^{2}_{c}}{1+s_{c}^{4}}\approx
10\%$ (63)

serves as an important prediction of the pole approximation. We emphasize that
the differences between two cases only occur in $A^{\text{fac}}$ and
$B^{\text{fac}}$ with charged $P$, related by a factor of $1/0.469$.

Table 5: Predictions of the CF decays in ${\bf B}_{cc}\to{\bf B}_{c}^{A,S}P$ with ${\cal C}_{+}=0.469$ and $1$, where $A$ and $B$ are in units of $10^{-2}G_{F}$GeV2.  Channels | Results with ${\cal C}_{+}=0.469$ | Results with ${\cal C}_{+}=1$
---|---|---
$A^{\text{fac}}$ | $A^{\text{pole}}$ | $B^{\text{fac}}$ | $B^{\text{pole}}$ | ${\cal B}(\%)$ | $\alpha$ | ${\cal B}(\%)$ | $\alpha$
$\Xi_{cc}^{++}\to\Xi_{c}^{+}\pi^{+}$ | $5.36$ | $-1.22$ | $7.59$ | $-5.43$ | $0.99(21)$ | $-0.19(7)$ | $6.24(21)$ | $-0.38(7)$
$\Xi_{cc}^{++}\to\Sigma_{c}^{++}K_{S}$ | $-3.29$ | $0$ | $-22.12$ | $0$ | $1.34(13)$ | $-0.99(0)$ | $1.34(13)$ | $-0.99(0)$
$\Xi_{cc}^{++}\to\Sigma_{c}^{++}K_{L}$ | $2.96$ | $0$ | $19.89$ | $0$ | $1.08(11)$ | $-0.99(0)$ | $1.08(11)$ | $-0.99(0)$
$\Xi_{cc}^{++}\to\Xi_{c}^{\prime+}\pi^{+}$ | $-2.80$ | $0$ | $-22.31$ | $0$ | $1.18(18)$ | $-0.96(0)$ | $5.41(18)$ | $-0.96(0)$
$\Xi_{cc}^{+}\to\Xi_{c}^{0}\pi^{+}$ | $-5.34$ | $-5.74$ | $-7.60$ | $0$ | $1.00(9)$ | $-0.25(1)$ | $2.44(9)$ | $-0.34(1)$
$\Xi_{cc}^{+}\to\Xi_{c}^{+}\pi^{0}$ | $0$ | $4.92$ | $0$ | $3.84$ | $0.20(3)$ | $-0.29(4)$ | $0.20(3)$ | $-0.29(4)$
$\Xi_{cc}^{+}\to\Xi_{c}^{+}\eta$ | $0$ | $-2.37$ | $0$ | $-1.85$ | $0.04(1)$ | $-0.26(3)$ | $0.04(1)$ | $-0.26(3)$
$\Xi_{cc}^{+}\to\Lambda_{c}^{+}K_{S}$ | $4.61$ | $3.34$ | $5.27$ | $-0.17$ | $0.51(2)$ | $-0.26(0)$ | $0.51(2)$ | $-0.26(0)$
$\Xi_{cc}^{+}\to\Lambda_{c}^{+}K_{L}$ | $-4.15$ | $-3.42$ | $-4.74$ | $-0.17$ | $0.46(2)$ | $-0.27(1)$ | $0.46(2)$ | $-0.27(1)$
$\Xi_{cc}^{+}\to\Sigma_{c}^{++}K^{-}$ | $0$ | $1.87$ | $0$ | $0$ | $0.03(1)$ | $0(0)$ | $0.03(1)$ | $0(0)$
$\Xi_{cc}^{+}\to\Sigma_{c}^{+}K_{S}$ | $-2.33$ | $-0.93$ | $-15.64$ | $0$ | $0.13(1)$ | $-0.98(1)$ | $0.13(1)$ | $-0.98(1)$
$\Xi_{cc}^{+}\to\Sigma_{c}^{+}K_{L}$ | $2.09$ | $0.93$ | $14.07$ | $0$ | $0.11(1)$ | $-0.98(1)$ | $0.11(1)$ | $-0.98(1)$
$\Xi_{cc}^{+}\to\Xi_{c}^{\prime+}\pi^{0}$ | $0$ | $-1.12$ | $0$ | $0$ | $0.01(0)$ | $0(0)$ | $0.01(0)$ | $0(0)$
$\Xi_{cc}^{+}\to\Xi_{c}^{\prime+}\eta$ | $0$ | $-1.62$ | $0$ | $0$ | $0.02(0)$ | $0(0)$ | $0.02(0)$ | $0(0)$
$\Xi_{cc}^{+}\to\Xi_{c}^{\prime 0}\pi^{+}$ | $-2.80$ | $-1.59$ | $-22.31$ | $0$ | $0.25(4)$ | $-0.99(0)$ | $0.92(4)$ | $-1.00(0)$
$\Xi_{cc}^{+}\to\Omega_{c}^{0}K^{+}$ | $0$ | $-1.87$ | $0$ | $0$ | $0.02(1)$ | $0(0)$ | $0.02(1)$ | $0(0)$
$\Omega_{cc}^{+}\to\Xi_{c}^{+}K_{S}$ | $-4.53$ | $0.54$ | $-5.57$ | $3.20$ | $0.49(6)$ | $-0.23(4)$ | $0.49(6)$ | $-0.23(4)$
$\Omega_{cc}^{+}\to\Xi_{c}^{+}K_{L}$ | $4.08$ | $-0.90$ | $5.01$ | $-3.20$ | $0.31(4)$ | $-0.22(5)$ | $0.31(4)$ | $-0.22(5)$
$\Omega_{cc}^{+}\to\Xi_{c}^{\prime+}K_{S}$ | $-2.40$ | $-0.05$ | $-16.36$ | $0$ | $0.40(0)$ | $-0.99(0)$ | $0.40(0)$ | $-0.99(0)$
$\Omega_{cc}^{+}\to\Xi_{c}^{\prime+}K_{L}$ | $2.16$ | $-0.05$ | $14.72$ | $0$ | $0.31(0)$ | $-0.98(0)$ | $0.31(0)$ | $-0.98(0)$
$\Omega_{cc}^{+}\to\Omega_{c}^{0}\pi^{+}$ | $-4.11$ | $0$ | $-32.96$ | $0$ | $1.41(21)$ | $-0.96(0)$ | $6.47(21)$ | $-0.96(0)$

Due to the smallness of the $\Xi_{cc}^{+}$ lifetime, the branching fractions
of $\Xi_{cc}^{+}$ are systematically smaller, but the predicted ${\cal
B}(\Xi_{cc}^{+}\to\Xi_{c}^{0}\pi^{+})$ is still huge. Particularly, with
${\cal C}_{+}=1$, we find ${\cal
B}(\Xi_{cc}^{+}\to\Xi_{c}^{0}\pi^{+}\to\Xi^{-}\pi^{+}\pi^{+}\pi^{+}\pi^{-})=(1.1\pm
0.6)\times 10^{-3}$, where ${\cal
B}(\Xi_{c}^{0}\to\Xi^{-}\pi^{+}\pi^{+}\pi^{-})=(4.8\pm 2.3)\%$ is used. As the
final state particles are all charged, searches of
$\Xi_{cc}^{+}\to\Xi^{-}\pi^{+}\pi^{+}\pi^{+}\pi^{-}$ are recommended. In
addition ${\cal B}(\Omega_{cc}^{+}\to\Omega_{c}^{0}\pi^{+})$ consists solely
of factorizable contributions and is predicted to be notably large. It is also
recommended for future experimental investigations.

##  VI Summary

We have analyzed the two-body nonleptonic weak decays of charmed baryons using
the pole approximation in conjunction with the $SU(3)_{F}$ symmetry. We have
shown that the KPW theorem demands that $O_{+}^{qq^{\prime}}$ and $P$ form a
${\bf 3}$ representation in the $SU(3)_{F}$ group, reducing the numbers of the
free parameters significantly. In particular, Eqs. (36), (37) and (38) are
given for the first time. With the GP scenario, most of the experimental data
of ${\bf B}_{c}^{A}\to{\bf B}_{n}P$ can be explained, but inconsistencies with
the experiments have been found in $\alpha(\Lambda_{c}^{+}\to pK_{S}^{0})$,
${\cal B}(\Xi_{c}^{0}\to\Xi^{-}\pi^{+})$ and ${\cal
B}(\Xi_{c}^{0}\to\Xi^{-}K^{+})$. These inconsistencies are recommended to be
revisited in future experiments.

Furthermore, by assuming the dominance of the low-lying intermediate baryons,
we have obtained the ability to make several predictions for
$\Omega_{c}^{0}\to{\bf B}_{n}P$ and ${\bf B}_{cc}\to{\bf B}_{c}^{A,S}P$ based
on the experimental input of ${\bf B}_{c}^{A}\to{\bf B}_{n}P$. However, the
fitted value ${\cal C}_{+}=0.469$ is too small in comparison to the naïve
expectation of ${\cal C}_{+}\approx 1$. In addition the LP scenario fails to
explain ${\cal B}_{exp}(\Lambda_{c}^{+}\to\Xi^{0}K^{+})$ and ${\cal
B}_{exp}(\Lambda_{c}\to n\pi^{+})$ though consistencies have been found with
the soft meson limit Charmed-Cheng . To search for the evidence of
$\Xi_{cc}^{+}$, we have recommended the decay channel of
$\Xi_{cc}^{+}\to\Xi_{c}^{0}\pi^{+}\to\Xi^{-}\pi^{+}\pi^{+}\pi^{+}\pi^{-}$, of
which the branching fraction is found to be $(1.1\pm 0.6)\times 10^{-3}$. The
predictions for the nonleptonic weak decay channels have been collected in the
appendices, to be used as a reference for future experiments seeking
verification

##  Appendix A Effective color number

The decay of $\Lambda_{c}^{+}\to p\phi$ does not receive the $W$-exchange
contributions. From LQCD, the decay constant of $\phi$ is found to be
$f_{\phi}=0.241(9)$ GeV Chen:2020qma  and the form factors of
$\Lambda_{c}^{+}\to p$ are latn

$(f_{1},f_{2},g_{1},g_{2})=(0.939,0.524,0.785,-0.050)\,,$ (64)

at $q^{2}=M_{p}^{2}$. Combing with ${\cal B}_{exp}(\Lambda_{c}\to
p\phi)=(1.06\pm 0.14)\times 10^{-3}$, we find

${\cal C}_{0}=c_{2}+\frac{c_{1}}{N_{c}^{eff}}=0.36\pm 0.04\,,$ (65)

where $N_{c}^{eff}$ is the effective color number. The formalism of the decay
width and the definitions of $f_{1,2}$ and $g_{1,2}$ can be found in Ref.
Charmed-Cheng . In the effective color number approach, one assume ${\cal
C}_{+}=c_{1}+c_{2}/N_{c}^{eff}$ with $N_{c}^{eff}$ fitted by Eq. (65). The
values of ${\cal C}_{+}$ at different energy scales are collected in Table 6,
where the naïve expectations with $N_{c}^{eff}=N_{c}=3$ are also listed.

In the naïve factorization approach, though ${\cal C}_{+}$ behaves stably,
${\cal C}_{0}$ varies heavily according to the energy scale and flip sign at
the next-to-leading order (NLO). It is a sign that the naïve factorization
approach cannot be trusted. On the other hand, the effective color approach
provides a much stable value of ${\cal C}_{+}$.

Table 6:  The effective Wilson coefficient, where ${\cal C}_{+}(N_{c}^{eff})$ is fitted from Eq. (65). The values of $c_{1,2}$ are from Ref. Buchalla:1995vs .  |  $\mu$[GeV] | $~{}~{}~{}~{}c_{1}~{}~{}~{}~{}$ | $~{}~{}~{}~{}c_{2}~{}~{}~{}~{}$ |  ${\cal C}_{+}(N_{c}^{eff})$ | ${\cal C}_{+}(N_{c}=3)$ | ${\cal C}_{0}(N_{c}=3)$
---|---|---|---|---|---|---
LO | 1.0 | 1.422 | $-0.742$ | $1.23\pm 0.01$ | $1.175$ | $-0.268$
1.5 | 1.298 | $-0.565$ | $1.22\pm 0.01$ | $1.110$ | $-0.132$
2.0 | 1.239 | $-0.475$ | $1.20\pm 0.01$ | $1.081$ | $-0.062$
NLO | 1.0 | 1.275 | $-0.510$ | $1.22\pm 0.01$ | $1.105$ | $-0.085$
1.5 | 1.188 | $-0.378$ | $1.19\pm 0.01$ | $1.062$ | $0.018$
2.0 | 1.148 | $-0.311$ | $1.17\pm 0.01$ | $1.044$ | $0.072$

##  Appendix B Predictions of the LP scenario for ${\bf B}_{c}^{A}\to{\bf
B}_{n}P$

Tables 7, 8, and 9 compile the numerical predictions for the
$\mathbf{B}_{c}^{A}$ decays from the LP scenario. These predictions can be
used as a reference in future experiments for validation and testing.

Table 7: Predictions of the LP scenario for the CF decays of ${\bf B}_{c}^{A}\to{\bf B}_{n}P$, where $A$ and $B$ are in units of $10^{-2}G_{F}$GeV2.  Channels | $A^{\text{fac}}$ | $A^{\text{pole}}$ | $B^{\text{fac}}$ | $B^{\text{pole}}$ | ${\cal B}(\%)$ | $\alpha$
---|---|---|---|---|---|---
$\Lambda_{c}^{+}\to pK_{L}^{0}$ | $2.62$ | $3.78$ | $5.61$ | $0.65$ | $1.33(7)$ | $-0.67(2)$
$\Xi_{c}^{+}\to\Sigma^{+}K_{S}^{0}$ | $-2.75$ | $0.34$ | $-7.08$ | $3.14$ | $0.52(7)$ | $-0.83(6)$
$\Xi_{c}^{+}\to\Sigma^{+}K_{L}^{0}$ | $2.48$ | $-0.74$ | $6.37$ | $-3.17$ | $0.28(5)$ | $-0.88(7)$
$\Xi_{c}^{0}\to\Sigma^{0}K_{L}^{0}$ | $1.76$ | $-0.91$ | $4.51$ | $-2.41$ | $0.03(1)$ | $-0.98(4)$
$\Xi_{c}^{0}\to\Xi^{0}\pi^{0}$ | $0$ | $5.15$ | $0$ | $4.62$ | $0.73(9)$ | $-0.51(6)$
$\Xi_{c}^{0}\to\Xi^{0}\eta$ | $0$ | $-3.12$ | $0$ | $-2.41$ | $0.22(5)$ | $-0.40(5)$
$\Xi_{c}^{0}\to\Lambda K_{L}^{0}$ | $1.07$ | $3.64$ | $2.55$ | $1.89$ | $0.57(4)$ | $-0.60(3)$
Table 8: Predictions of the LP scenario for the CS decays of ${\bf B}_{c}^{A}\to{\bf B}_{n}P$, where $A$ and $B$ are in units of $10^{-2}G_{F}$GeV2.  Channels | $A^{\text{fac}}$ | $A^{\text{pole}}$ | $B^{\text{fac}}$ | $B^{\text{pole}}$ | ${\cal B}(10^{-4})$ | $\alpha$
---|---|---|---|---|---|---
$\Lambda_{c}^{+}\to\Sigma^{+}K_{L}^{0}$ | $0$ | $0.99$ | $0$ | $0.75$ | $2.84(35)$ | $-0.41(5)$
$\Xi_{c}^{+}\to\Sigma^{+}\pi^{0}$ | $0.50$ | $1.19$ | $1.29$ | $0.26$ | $24.21(2.59)$ | $-0.58(3)$
$\Xi_{c}^{+}\to\Sigma^{+}\eta$ | $-1.04$ | $-0.28$ | $-2.68$ | $0.77$ | $14.34(1.04)$ | $-0.76(2)$
$\Xi_{c}^{+}\to\Sigma^{0}\pi^{+}$ | $-0.59$ | $-1.19$ | $-1.51$ | $-0.26$ | $27.11(2.38)$ | $-0.62(4)$
$\Xi_{c}^{+}\to\Xi^{0}K^{+}$ | $-1.05$ | $0.36$ | $-3.07$ | $1.11$ | $5.34(1.92)$ | $-0.97(2)$
$\Xi_{c}^{+}\to pK_{S/L}^{0}$ | $0$ | $0.99$ | $0$ | $0.71$ | $7.34(88)$ | $-0.56(7)$
$\Xi_{c}^{+}\to\Lambda\pi^{+}$ | $0.36$ | $-0.68$ | $0.85$ | $-0.91$ | $0.80(38)$ | $-0.14(31)$
$\Xi_{c}^{0}\to\Sigma^{+}\pi^{-}$ | $0$ | $-0.22$ | $0$ | $-0.22$ | $0.14(19)$ | $-0.63(23)$
$\Xi_{c}^{0}\to\Sigma^{0}\pi^{0}$ | $0.36$ | $0.62$ | $0.91$ | $-0.04$ | $2.69(41)$ | $-0.57(5)$
$\Xi_{c}^{0}\to\Sigma^{0}\eta$ | $-0.74$ | $-0.20$ | $-1.90$ | $0.55$ | $2.40(17)$ | $-0.76(2)$
$\Xi_{c}^{0}\to\Sigma^{-}\pi^{+}$ | $0.83$ | $1.47$ | $2.13$ | $0.15$ | $15.08(1.48)$ | $-0.62(2)$
$\Xi_{c}^{0}\to\Xi^{0}K_{S/L}^{0}$ | $0$ | $-1.12$ | $0$ | $-0.92$ | $2.97(38)$ | $-0.44(4)$
$\Xi_{c}^{0}\to pK^{-}$ | $0$ | $0.18$ | $0$ | $0.18$ | $0.09(12)$ | $-0.73(25)$
$\Xi_{c}^{0}\to nK_{S}^{0}$ | $0$ | $-1.12$ | $0$ | $-0.84$ | $3.14(40)$ | $-0.58(5)$
$\Xi_{c}^{0}\to nK_{L}^{0}$ | $0$ | $1.12$ | $0$ | $0.84$ | $3.14(40)$ | $-0.58(5)$
$\Xi_{c}^{0}\to\Lambda\pi^{0}$ | $0.22$ | $-0.48$ | $0.52$ | $-0.64$ | $0.18(9)$ | $-0.35(25)$
$\Xi_{c}^{0}\to\Lambda\eta$ | $-0.45$ | $-0.52$ | $-1.07$ | $0.02$ | $2.45(29)$ | $-0.66(4)$
Table 9: Predictions of the LP scenario for the DCS decays of ${\bf B}_{c}^{A}\to{\bf B}_{n}P$, where $A$ and $B$ are in units of $10^{-2}G_{F}$GeV2.  Channels | $A^{\text{fac}}$ | $A^{\text{pole}}$ | $B^{\text{fac}}$ | $B^{\text{pole}}$ | ${\cal B}(10^{-5})$ | $\alpha$
---|---|---|---|---|---|---
$\Lambda_{c}^{+}\to nK^{+}$ | $0.24$ | $-0.04$ | $0.52$ | $-0.18$ | $1.65(29)$ | $-0.92(4)$
$\Xi_{c}^{+}\to\Sigma^{0}K^{+}$ | $-0.16$ | $-0.20$ | $-0.42$ | $-0.02$ | $10.41(1.08)$ | $-0.68(2)$
$\Xi_{c}^{+}\to p\pi^{0}$ | $0$ | $0.04$ | $0$ | $0.05$ | $0.13(14)$ | $-0.93(22)$
$\Xi_{c}^{+}\to p\eta$ | $0$ | $0.28$ | $0$ | $0.20$ | $5.72(59)$ | $-0.55(6)$
$\Xi_{c}^{+}\to n\pi^{+}$ | $0$ | $0.05$ | $0$ | $0.08$ | $0.27(29)$ | $-0.93(22)$
$\Xi_{c}^{+}\to\Lambda K^{+}$ | $0.1$ | $-0.18$ | $0.23$ | $-0.22$ | $0.48(25)$ | $0.1(27)$
$\Xi_{c}^{0}\to\Sigma^{-}K^{+}$ | $0.23$ | $0.28$ | $0.59$ | $0.03$ | $6.95(72)$ | $-0.68(2)$
$\Xi_{c}^{0}\to p\pi^{-}$ | $0$ | $-0.05$ | $0$ | $-0.08$ | $0.09(10)$ | $-0.93(22)$
$\Xi_{c}^{0}\to n\pi^{0}$ | $0$ | $0.04$ | $0$ | $0.05$ | $0.04(5)$ | $-0.93(22)$
$\Xi_{c}^{0}\to n\eta$ | $0$ | $-0.28$ | $0$ | $-0.20$ | $1.90(20)$ | $-0.55(6)$

##  Appendix C Predictions of the LP scenario for ${\bf B}_{cc}\to{\bf
B}_{c}^{A,S}P$

Tables 10 and 11 compile the CS and DCS predictions for the $\mathbf{B}_{cc}$
decays from the LP scenario. These predictions are intended for validation and
testing in future experiments.

Table 10: Predictions of the LP scenario for the CS decays of ${\bf B}_{cc}\to{\bf B}_{c}^{A,S}P$ with ${\cal C}_{+}=0.469$ and $1$, where $A$ and $B$ are in units of $10^{-2}G_{F}$GeV2.  Channels |  Results with ${\cal C}_{+}=0.469$ |  Results with ${\cal C}_{+}=1$
---|---|---
$A^{\text{fac}}$ | $A^{\text{pole}}$ | $B^{\text{fac}}$ | $B^{\text{pole}}$ | ${\cal B}(10^{-4})$ | $\alpha$ | ${\cal B}(10^{-4})$ | $\alpha$
$\Xi_{cc}^{++}\to\Xi_{c}^{+}K^{+}$ | $1.48$ | $-0.23$ | $2.10$ | $-1.04$ | $8.17(1.41)$ | $-0.28(4)$ | $46.12(1.41)$ | $-0.39(4)$
$\Xi_{cc}^{++}\to\Lambda_{c}^{+}\pi^{+}$ | $1.39$ | $-0.28$ | $1.58$ | $-1.25$ | $7.47(1.46)$ | $-0.14(7)$ | $45.23(1.46)$ | $-0.35(7)$
$\Xi_{cc}^{++}\to\Sigma_{c}^{++}\pi^{0}$ | $0.48$ | $0$ | $3.22$ | $0$ | $5.45(55)$ | $-0.97(0)$ | $5.45(55)$ | $-0.97(0)$
$\Xi_{cc}^{++}\to\Sigma_{c}^{++}\eta$ | $-1.00$ | $0$ | $-6.69$ | $0$ | $18.06(18)$ | $-0.99(0)$ | $18.06(18)$ | $-0.99(0)$
$\Xi_{cc}^{++}\to\Sigma_{c}^{+}\pi^{+}$ | $0.70$ | $0$ | $4.70$ | $0$ | $7.43(1.12)$ | $-0.97(0)$ | $34.03(1.12)$ | $-0.97(0)$
$\Xi_{cc}^{++}\to\Xi_{c}^{\prime+}K^{+}$ | $-0.77$ | $0$ | $-6.16$ | $0$ | $6.88(1.04)$ | $-0.99(0)$ | $31.51(1.04)$ | $-0.99(0)$
$\Xi_{cc}^{+}\to\Xi_{c}^{0}K^{+}$ | $-1.47$ | $-1.10$ | $-2.10$ | $0$ | $4.89(48)$ | $-0.27(1)$ | $13.55(48)$ | $-0.35(1)$
$\Xi_{cc}^{+}\to\Xi_{c}^{+}K_{S/L}$ | $0$ | $-0.94$ | $0$ | $-0.74$ | $0.66(8)$ | $-0.26(3)$ | $0.66(8)$ | $-0.26(3)$
$\Xi_{cc}^{+}\to\Lambda_{c}^{+}\pi^{0}$ | $-0.67$ | $0.20$ | $-0.77$ | $0.88$ | $0.35(7)$ | $-0.05(7)$ | $0.35(7)$ | $-0.05(7)$
$\Xi_{cc}^{+}\to\Lambda_{c}^{+}\eta$ | $1.40$ | $0.80$ | $1.59$ | $-0.42$ | $5.09(20)$ | $-0.25(1)$ | $5.09(20)$ | $-0.25(1)$
$\Xi_{cc}^{+}\to\Sigma_{c}^{++}\pi^{-}$ | $0$ | $-0.52$ | $0$ | $0$ | $0.21(6)$ | $0(0)$ | $0.21(6)$ | $0(0)$
$\Xi_{cc}^{+}\to\Sigma_{c}^{+}\pi^{0}$ | $0.34$ | $0.52$ | $2.28$ | $0$ | $0.95(11)$ | $-0.87(4)$ | $0.95(11)$ | $-0.87(4)$
$\Xi_{cc}^{+}\to\Sigma_{c}^{+}\eta$ | $-0.70$ | $0$ | $-4.73$ | $0$ | $1.27(0)$ | $-0.99(0)$ | $1.27(0)$ | $-0.99(0)$
$\Xi_{cc}^{+}\to\Sigma_{c}^{0}\pi^{+}$ | $0.99$ | $0.52$ | $6.65$ | $0$ | $3.13(51)$ | $-0.99(0)$ | $11.55(51)$ | $-1.00(0)$
$\Xi_{cc}^{+}\to\Xi_{c}^{\prime+}K_{S/L}$ | $0$ | $-0.21$ | $0$ | $0$ | $0.03(1)$ | $0(0)$ | $0.03(1)$ | $0(0)$
$\Xi_{cc}^{+}\to\Xi_{c}^{\prime 0}K^{+}$ | $-0.77$ | $0.30$ | $-6.16$ | $0$ | $0.71(10)$ | $-0.81(3)$ | $3.82(10)$ | $-0.93(3)$
$\Omega_{cc}^{+}\to\Xi_{c}^{0}\pi^{+}$ | $-1.36$ | $-1.32$ | $-1.68$ | $0$ | $24.00(2.23)$ | $-0.26(1)$ | $60.65(2.23)$ | $-0.34(1)$
$\Omega_{cc}^{+}\to\Xi_{c}^{+}\pi^{0}$ | $0.66$ | $0.93$ | $0.81$ | $0$ | $10.32(51)$ | $-0.24(1)$ | $10.32(51)$ | $-0.24(1)$
$\Omega_{cc}^{+}\to\Xi_{c}^{+}\eta$ | $-1.37$ | $-0.26$ | $-1.69$ | $0.85$ | $11.75(75)$ | $-0.24(2)$ | $11.75(75)$ | $-0.24(2)$
$\Omega_{cc}^{+}\to\Lambda_{c}^{+}K_{S/L}$ | $0$ | $0.94$ | $0$ | $0.74$ | $2.96(38)$ | $-0.35(4)$ | $2.96(38)$ | $-0.35(4)$
$\Omega_{cc}^{+}\to\Sigma_{c}^{++}K^{-}$ | $0$ | $0.43$ | $0$ | $0$ | $0.56(14)$ | $0(0)$ | $0.56(14)$ | $0(0)$
$\Omega_{cc}^{+}\to\Sigma_{c}^{+}K_{S/L}$ | $0$ | $-0.21$ | $0$ | $0$ | $0.14(4)$ | $0(0)$ | $0.14(4)$ | $0(0)$
$\Omega_{cc}^{+}\to\Xi_{c}^{\prime+}\pi^{0}$ | $0.35$ | $-0.26$ | $2.38$ | $0$ | $1.09(6)$ | $-0.58(9)$ | $1.09(6)$ | $-0.58(9)$
$\Omega_{cc}^{+}\to\Xi_{c}^{\prime+}\eta$ | $-0.73$ | $-0.37$ | $-4.95$ | $0$ | $7.66(39)$ | $-0.98(1)$ | $7.66(39)$ | $-0.98(1)$
$\Omega_{cc}^{+}\to\Xi_{c}^{\prime 0}\pi^{+}$ | $0.72$ | $-0.36$ | $4.92$ | $0$ | $3.10(45)$ | $-0.67(5)$ | $16.78(45)$ | $-0.88(5)$
$\Omega_{cc}^{+}\to\Omega_{c}^{0}K^{+}$ | $-1.13$ | $-0.43$ | $-9.10$ | $0$ | $11.54(1.87)$ | $-0.99(0)$ | $44.67(1.87)$ | $-1.00(0)$
Table 11: Predictions of the LP scenario for the DCS decays of ${\bf B}_{cc}\to{\bf B}_{c}^{A,S}P$ with ${\cal C}_{+}=0.469$ and $1$, where $A$ and $B$ are in units of $10^{-2}G_{F}$GeV2.  Channels | Results with ${\cal C}_{+}=0.469$ | Results with ${\cal C}_{+}=1$
---|---|---
$A^{\text{fac}}$ | $A^{\text{pole}}$ | $B^{\text{fac}}$ | $B^{\text{pole}}$ | ${\cal B}(10^{-5})$ | $\alpha$ | ${\cal B}(10^{-5})$ | $\alpha$
$\Xi_{cc}^{++}\to\Lambda_{c}^{+}K^{+}$ | $0.38$ | $-0.05$ | $0.44$ | $-0.24$ | $6.18(1.03)$ | $-0.25(4)$ | $34.09(1.03)$ | $-0.37(4)$
$\Xi_{cc}^{++}\to\Sigma_{c}^{+}K^{+}$ | $0.19$ | $0$ | $1.30$ | $0$ | $4.60(70)$ | $-0.99(0)$ | $21.07(70)$ | $-0.99(0)$
$\Xi_{cc}^{+}\to\Sigma_{c}^{0}K^{+}$ | $0.27$ | $0$ | $1.84$ | $0$ | $1.29(20)$ | $-0.99(0)$ | $5.93(20)$ | $-0.99(0)$
$\Omega_{cc}^{+}\to\Xi_{c}^{0}K^{+}$ | $-0.38$ | $-0.25$ | $-0.46$ | $0$ | $12.22(1.23)$ | $-0.28(1)$ | $34.98(1.23)$ | $-0.35(1)$
$\Omega_{cc}^{+}\to\Lambda_{c}^{+}\eta$ | $0$ | $0.25$ | $0$ | $0.20$ | $2.05(25)$ | $-0.35(4)$ | $2.05(25)$ | $-0.35(4)$
$\Omega_{cc}^{+}\to\Sigma_{c}^{++}\pi^{-}$ | $0$ | $-0.12$ | $0$ | $0$ | $0.46(12)$ | $0(0)$ | $0.46(12)$ | $0(0)$
$\Omega_{cc}^{+}\to\Sigma_{c}^{+}\pi^{0}$ | $0$ | $0.12$ | $0$ | $0$ | $0.46(12)$ | $0(0)$ | $0.46(12)$ | $0(0)$
$\Omega_{cc}^{+}\to\Sigma_{c}^{0}\pi^{+}$ | $0$ | $0.12$ | $0$ | $0$ | $0.46(12)$ | $0(0)$ | $0.46(12)$ | $0(0)$
$\Omega_{cc}^{+}\to\Xi_{c}^{\prime 0}K^{+}$ | $0.20$ | $0.07$ | $1.36$ | $0$ | $3.67(59)$ | $-0.99(0)$ | $14.36(59)$ | $-1.00(0)$

###### Acknowledgements.

The author extends gratitude to Hai-Yang Cheng for the insightful discussions.
This research was supported by the National Natural Science Foundation of
China under Grant No. 12205063.

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