††institutetext: Department of Physics, Jinan University,
Guangzhou 510632, P.R. China

# Weak Decays of Antitriplet Charmed Baryons from the Perspective of Flavor
Symmetry

Huiling Zhong  Fanrong Xu  Qiaoyi Wen, Yu Gu<EMAIL_ADDRESS>
###### Abstract

In this work, we carry out a global fit for the two-body weak decays of
antitriplet charmed baryons in both SU(3) respected and broken scenarios
incorporating all the available data up to date. In the SU(3) irreducible
representation approach (IRA), more amplitudes for irreducible representation
terms are taken into account and the ranges for their coefficients in each
scenarios are predicted. By a comparison among various fitting schemes in this
work, experimental data prefer the SU(3) symmetry breaking scenario.
Observables of interest, branching fractions and decay asymmetries of all the
Cabibbo-favored (CF), singly Cabibbo-suppressed (SCS) and doubly Cabibbo-
suppressed (DCS) channels are calculated in the chosen fitting scheme. Most of
our predictions are consistent well with experimental data. We further propose
more ways to explore the SU(3) symmetry in charmed baryon decays: (i) A clear
measurement of branching fraction on CF mode $\Xi_{c}^{0}\to\Xi^{0}\pi^{0}$ as
a large difference exists between SU(3) respected and broken scenarios. (ii)
The decay asymmetries of the five yet-to-be measured CF decay modes,
$\Xi_{c}^{0}\to\Lambda^{0}K_{S},\Sigma^{0}K_{S},\Xi^{0}\pi^{0},\Xi^{0}\eta,\Xi^{0}\eta^{\prime}$,
for their opposite signs in the two different cases. (iii) The future
measurement on branching fraction ratios indicated from Eq. (13) and presented
on Table 7. An improved measurement for $\alpha(\Lambda_{c}^{+}\to pK_{S})$ is
called for since predictions from both fittings, including the one in current
work and earlier works, and the pole model calculation prefer an opposite sign
from previous experimental value. Given branching fractions of most of the CF
modes and part of the SCS modes being measured, predictions for the remaining
channels, including the DCS modes as well as more decay asymmetries, are
anticipated to be checked by the upcoming experiments in BESIII, Belle/Belle-
II and LHCb.

###### Keywords:

Charmed baryons, weak decays, flavor symmetry

##  1 Introduction

In recent years, more and more attentions have been paid on two-body hadronic
weak decays of charmed baryons both experimentally and theoretically.
Especially for the branching fractions and decay asymmetries of antitriplet
charmed baryon decays, continuous progresses have been made in summer 2022.
For the first time, branching fractions of the following channels have been
measured by BESIII BESIII:2022bkj ; BESIII:2022izy ; BESIII:2022tnm ;
BESIII:2022wxj

$\begin{split}&\mathcal{B}(\Lambda_{c}^{+}\to n\pi^{+})=(6.6\pm 1.2\pm
0.4)\times 10^{-4},\\\ &\mathcal{B}(\Lambda_{c}^{+}\to
p\eta^{\prime})=(5.62^{+2.46}_{-2.04}\pm 0.26)\times 10^{-4},\\\
&\mathcal{B}(\Lambda_{c}^{+}\to\Lambda^{0}K^{+})=(6.21\pm 0.44\pm 0.26\pm
0.34)\times 10^{-4},\\\
&\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{+}K_{S}^{0})=(4.8\pm 1.4\pm 0.2\pm
0.3)\times 10^{-4},\\\ &\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{0}K^{+})=(4.7\pm
0.9\pm 0.1\pm 0.3)\times 10^{-4}.\end{split}$ (1)

Meanwhile, Belle has made independent contributions in the measurements of
both branching fractions and decay asymmetries, giving Belle:2021vyq ;
Belle:2022uod ; Belle:2022bsi

$\begin{split}&\mathcal{B}(\Lambda_{c}^{+}\to p\eta^{\prime})=(4.73\pm 0.82\pm
0.47\pm 0.24)\times 10^{-4},\\\
&\mathcal{B}(\Lambda_{c}^{+}\to\Lambda^{0}K^{+})=(6.57\pm 0.17\pm 0.11\pm
0.35)\times 10^{-4},\\\
&\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{0}K^{+})=(3.58\pm 0.19\pm 0.06\pm
0.19)\times 10^{-4},\\\
&\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{+}\eta)=(3.14\pm 0.35\pm 0.11\pm
0.25)\times 10^{-3},\\\
&\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{+}\eta^{\prime})=(4.16\pm 0.75\pm
0.21\pm 0.33)\times 10^{-3},\end{split}$ (2)

and

$\begin{split}&\alpha(\Lambda_{c}^{+}\to\Lambda^{0}K^{+})=-0.585\pm 0.049\pm
0.018,\\\ &\alpha(\Lambda_{c}^{+}\to\Sigma^{0}K^{+})=-0.55\pm 0.18\pm 0.09,\\\
&\alpha(\Lambda_{c}^{+}\to\Sigma^{+}\eta)=-0.99\pm 0.03\pm 0.05,\\\
&\alpha(\Lambda_{c}^{+}\to\Sigma^{+}\eta^{\prime})=-0.46\pm 0.06\pm
0.03.\end{split}$ (3)

In addition, the branching fraction measurements of
$\Lambda_{c}\to\Sigma^{+}K_{S}$ and $\Lambda_{c}\to p\eta^{\prime}$ have been
improved by BESIII and Belle, respectively. Moreover, decay asymmetries of
$\Lambda_{c}^{+}\to\Lambda^{0}\pi^{+},\Sigma^{0}\pi^{+}$ and
$\Sigma^{+}\pi^{0}$ have been measured more precisely.

The dynamics of charmed baryon weak decays, especially for antitriplet charmed
baryon, had a long research history for several decades Korner:1992wi ;
Xu:1992sw ; Cheng:1993gf ; Ivanov:1997ra and recently has been studied
extensively Cheng:2018hwl ; Zou:2019kzq ; Niu:2020gjw ; Groote:2021pxt
promoted by rapid developments in experiment, which has been reviewed in
Cheng:2021qpd . These phenomenological studies are model dependent, due to the
lack of a consistent QCD-inspired framework to deal with charmed baryon decays
so far, still require further checks from experiments. On the other hand,
without involving the non-perturbative details, the fitting methodology
provides a tool as a complementary way to interpret data and further predict
the yet-to-be measured channels. Two different ways, IRA (irreducible
representation approach) and TDA (topological diagram approach), have been
developed to carry out the fitting analysis. In IRA, different decay channels
can be connected Lu:2016ogy ; Geng:2019xbo ; Huang:2021aqu via irreducible
representations for the SU(3) flavor symmetry. Another attempt, TDA, is to
take topological diagrams as basic elements to implement the fitting scheme
Zhao:2018mov . The topic that whether an equivalence between the two
approaches exists or not, has been discussed recently for mesons Wang:2020gmn
; He:2018php ; He:2018joe and for baryons Hsiao:2021nsc , and further efforts
are still required.

In this work, we adopt the IRA to carry out such a global fitting analysis.
Meanwhile in charmed baryon decays whether the SU(3) flavor symmetry is
strictly respected or broken is still a question worth answering. Hence the
two scenarios, SU(3) symmetry being kept strictly and broken, are both taken
in the following study by incorporating the latest experimental data. We will
give explicitly the relations among different channels at the amplitude level.
Coefficients of amplitudes in terms of irreducible representations will be
fitted after comparing with different fitting schemes. Branching fractions and
decay asymmetries of all the two-body decays, including CF, SCS and DCS
channels, will be predicted based on the coefficients in the fixed scheme.
Armed with the explicit predicted values, we propose ways to discriminate
SU(3) symmetry between its keeping and breaking scenarios, including the focus
on branching fraction of special modes ($\Xi_{c}^{0}\to\Xi^{0}\pi^{0}$ as an
example), decay asymmetries of particular modes (like
$\Xi_{c}^{0}\to\Lambda^{0}K_{S}$) as well as branching fraction ratios of some
certain channels ( of which an example is $\mathcal{B}(\Xi_{c}^{0}\to
p\pi^{-})/\mathcal{B}(\Xi_{c}^{0}\to pK^{-})$). We point out that theoretical
predictions from both fittings and model calculations as well as experimental
measuring provide complementary information to explore charmed baryon weak
decays from different aspects. In general, opportunities are indicated in
these inconsistent channels, of which an example is $\alpha(\Lambda_{c}^{+}\to
pK_{S})$. By combing predictions in different independent ways, some
suggestions for experiments are also given.

The remaining part of this paper is organized as follows. In Section 2, we set
up the framework of the entire fitting work, including theoretical formalism
and a summary of experimental data. In both scenarios, connections among decay
amplitudes are also presented in this section. Then fitting schemes are
introduced in the first part of Sec. 3, while in the second part we present
all the numerical results and related discussion. A conclusion is made in Sec.
4. For more details of choosing fitting schemes, results for a comparison are
displayed in Appendix A.

##  2 Formalism

###  2.1 The irreducible representation and kinematics

The state-of-the-art effective Hamiltonian approach is adopted to describe
charmed baryon decays. At quark level, the effective Hamiltonian to depict
quark decay process $c\to\bar{q}_{1}q_{2}u$ is

$\begin{split}&\mathcal{H}_{\rm{eff}}=\frac{G_{F}}{\sqrt{2}}V_{q_{1}c}^{*}V_{uq_{2}}(c_{+}\mathcal{O}_{+}+c_{-}\mathcal{O}_{-})+h.c.\\\
&\mathcal{O}_{+}=\frac{1}{2}\left(\mathcal{O}_{1}+\mathcal{O}_{2}\right),\quad\mathcal{O}_{-}=\frac{1}{2}\left(\mathcal{O}_{1}-\mathcal{O}_{2}\right)\end{split}$
(4)

in which
$\mathcal{O}_{1}=(\bar{q}_{1}c)(\bar{u}q_{2}),\mathcal{O}_{2}=(\bar{u}c)(\bar{q}_{1}q_{2})$,
and the bilinear operator is defined as
$(\bar{q}_{1}q_{2})=\bar{q}_{1}\gamma^{\mu}(1-\gamma_{5})q_{2}$. In
particular, the quark flavors of the four-quark operator are denoted as
$(q_{1},q_{2})=(s,d),\\{(s,s),(d,d)\\},(d,s)$, corresponding to Cabibbo-
favored (CF), singly Cabibbo-suppressed (SCS) and doubly Cabibbo-suppressed
(DCS) processes, respectively.

From the point of view of SU(3) flavor symmetry, the four-quark operators
$\mathcal{O}_{\pm}$ in the weak Hamiltonian fall into two irreducible
representations $(\bm{6},\overline{\bm{15}})$ Savage:1989qr ; Geng:2019xbo ;
Huang:2021aqu , given by

$\begin{split}&H(\overline{15})_{k}^{ij}=\left(\left(\begin{array}[]{ccc}0&0&0\\\
0&0&0\\\ 0&0&0\end{array}\right),\quad\left(\begin{array}[]{ccc}0&s_{c}&1\\\
s_{c}&0&0\\\
1&0&0\end{array}\right),\quad\left(\begin{array}[]{ccc}0&-s_{c}^{2}&-s_{c}\\\
-s^{2}_{c}&0&0\\\ -s_{c}&0&0\end{array}\right)\right),\\\
&H(6)_{ij}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&2&-2s_{c}\\\
0&-2s_{c}&2s_{c}^{2}\end{array}\right)\end{split}$ (5)

with $s_{c}=\sin\theta_{c}=\sqrt{0.23}$ Workman:2022ynf and $(i,j,k)=1,2,3$,
in which traceless $H(\overline{{15}})$ is symmetric for the superscript
indices while $H({{6}})$ is symmetric for the subscript indices. For the two-
body decays of antitriplet charmed baryon $B_{c}$ into hadrons in the baryon
and meson octets $B_{n}$ and $M$, the amplitudes are separated as $S$\- and
$P$-waves Geng:2019xbo , given

$\mathcal{M}=\langle
M\bm{B}_{n}|\mathcal{H}_{\rm{eff}}|\bm{B}_{c}\rangle=i\bar{u}_{f}(A-B\gamma_{5})u_{i}.$
(6)

If the $S$\- and $P$-wave amplitudes ($A$ and $B$ terms) are invariant under
SU(3) symmetry, generically they can further be parameterized by the
irreducible representations of SU(3) group,

$\displaystyle A_{0}$ $\displaystyle=$ $\displaystyle
a_{0}H(6)_{ij}({\bm{B}}^{\prime}_{c})^{ik}({\bm{B}}_{n})^{j}_{k}(M)^{\ell}_{\ell}+a_{1}H(6)_{ij}({\bm{B}}^{\prime}_{c})^{ik}({\bm{B}}_{n})^{\ell}_{k}(M)^{j}_{\ell}+a_{2}H(6)_{ij}({\bm{B}}^{\prime}_{c})^{ik}(M)^{\ell}_{k}({\bm{B}}_{n})^{j}_{\ell}$
$\displaystyle+a_{3}H(6)_{ij}({\bm{B}}_{n})^{i}_{k}(M)^{j}_{\ell}({\bm{B}}^{\prime}_{c})^{k\ell}+a^{\prime}_{0}({\bm{B}}_{n})^{i}_{j}(M)^{\ell}_{\ell}H(\overline{15})_{i}^{jk}({\bm{B}}_{c})_{k}+a_{4}H(\overline{15})_{k}^{\ell
i}({\bm{B}}_{c})_{j}(M)_{i}^{j}({\bm{B}}_{n})_{\ell}^{k}$
$\displaystyle+a_{5}({\bm{B}}_{n})^{i}_{j}(M)^{\ell}_{i}H(\overline{15})_{\ell}^{jk}({\bm{B}}_{c})_{k}+a_{6}({\bm{B}}_{n})^{j}_{i}(M)_{\ell}^{m}H(\overline{15})_{m}^{\ell
i}({\bm{B}}_{c})_{j}+a_{7}({\bm{B}}_{n})_{i}^{\ell}(M)^{i}_{j}H(\overline{15})_{\ell}^{jk}({\bm{B}}_{c})_{k}$
$\displaystyle B_{0}$ $\displaystyle=$ $\displaystyle A_{0}\Big{|}_{a_{i}\to
b_{i}},$ (7)

where

$\begin{split}&\bm{B}_{c}=\left(\Xi_{c}^{0},-\Xi_{c}^{+},\Lambda_{c}^{+}\right),\\\
&\bm{B}_{n}=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{6}}\Lambda^{0}+\frac{1}{\sqrt{2}}\Sigma^{0}&\Sigma^{+}&p\\\
\Sigma^{-}&\frac{1}{\sqrt{6}}\Lambda^{0}-\frac{1}{\sqrt{2}}\Sigma^{0}&n\\\
\Xi^{-}&\Xi^{0}&-\sqrt{\frac{2}{3}}\Lambda^{0}\end{array}\right),\\\
&M=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}(\pi^{0}+c_{\phi}\eta+s_{\phi}\eta^{\prime})&\pi^{+}&K^{+}\\\
\pi^{-}&\frac{1}{\sqrt{2}}(-\pi^{0}+c_{\phi}\eta+s_{\phi}\eta^{\prime})&K^{0}\\\
K^{-}&\overline{K}^{0}&-s_{\phi}\eta+c_{\phi}\eta^{\prime}\end{array}\right),\end{split}$
(8)

and $({\bm{B}}^{\prime}_{c})^{ij}\equiv\epsilon^{ijk}({\bm{B}}_{c})_{k}$. Here
the mixing of $\eta$ and $\eta^{\prime}$ have also been considered in which
$(c_{\phi},s_{\phi})=(\cos\phi,\sin\phi)$ with $\phi=39.3^{\circ}$
Feldmann:1998vh ; Feldmann:1998sh . The coefficients $a_{i}$ and $b_{i}$ will
be determined by experimental data. Thanks to the more and more accumulated
data, here we incorporate the terms characterized by
$a_{0}^{\prime},a_{4,5,7}$ (and the corresponding terms in $P$-wave), which
were neglected in Geng:2019xbo and recently have been recollected in
Huang:2021aqu .

It is known that in D meson sector, SU(3) breaking effect takes up a certain
contribution in the decays Cheng:2010ry ; Li:2012cfa ; Cheng:2021yrn . To
incorporate the analogous SU(3) breaking effect in the decays of charmed
baryons is a reasonable and interesting attempt. In the earlier studies, it
has been shown that there are several SU(3) representations to describe the
breaking effect Savage:1991wu ; Pirtskhalava:2011va . In this work, we
continue to take the assumption that $\overline{3}$ is the dominant source of
SU(3) breaking Geng:2018bow and the corresponding amplitudes can be
parameterized as

$\displaystyle A^{\prime}$ $\displaystyle=$ $\displaystyle
u_{1}({\bm{B}}_{c})_{i}H(\overline{3})^{i}({\bm{B}}_{n})^{j}_{k}(M)^{k}_{j}+u_{2}({\bm{B}}_{c})_{i}H(\overline{3})^{j}({\bm{B}}_{n})^{i}_{k}(M)^{k}_{j}$
$\displaystyle+u_{3}({\bm{B}}_{c})_{i}H(\overline{3})^{j}({\bm{B}}_{n})^{k}_{j}(M)^{i}_{k}$
$\displaystyle B^{\prime}$ $\displaystyle=$ $\displaystyle
A^{\prime}\Big{|}_{u_{i}\to v_{i}}$ (9)

with

$H(\overline{3})=(s_{c},0,0),$ (10)

which manifests in the SCS processes.111 Although this occurs under the
$\overline{3}$ dominance assumption, the feature is in agreement with the fit
in terms of topological diagrams Cheng:2010ry ; Cheng:2021yrn .

Then in terms of the total $S$\- and $P$-wave amplitudes
$A=A_{0}+A^{\prime},B=B_{0}+B^{\prime}$, the observables of interest, decay
width $\Gamma$ and decay asymmetry $\alpha$, are given as

$\begin{split}&\Gamma=\frac{p_{c}}{8\pi}\left(\frac{(m_{i}+m_{f})^{2}-m_{P}^{2}}{m_{i}^{2}}|A|^{2}+\frac{(m_{i}-m_{f})^{2}-m_{P}^{2}}{m_{i}^{2}}|B|^{2}\right)\\\
&\alpha=\frac{2\kappa{\textrm{Re}}(A^{*}B)}{|A|^{2}+\kappa^{2}|B|^{2}}\end{split}$
(11)

where $m_{i},m_{f},m_{P}$ stand for the masses of initial state baryons, final
state baryons and mesons with $p_{c}$ being the c.m. three-momentum in the
rest frame of initial baryon. And the auxiliary parameter $\kappa$ is defined
as $\kappa=p_{c}/(E_{f}+m_{f})=\sqrt{(E_{f}-m_{f})/(E_{f}+m_{f})}$.

###  2.2 Decay amplitudes

Table 1: The S-wave amplitudes for Cabibbo-favored decays of
${\bm{B}}_{c}\to{\bm{B}}_{n}M$ in terms of coefficients of irreducible
representation amplitudes (IRA). The P-wave amplitudes can be obtained by
replacing $a_{i}$ by $b_{i}$ accordingly.

Channel | $A$ | Channel | $A$
---|---|---|---
$\Lambda_{c}^{+}\to\Lambda^{0}\pi^{+}$ | $\frac{\sqrt{6}}{6}(-2a_{1}-2a_{2}-2a_{3}+a_{5}-2a_{6}+a_{7})$ | $\Xi_{c}^{+}\to\Xi^{0}\pi^{+}$ | $-2a_{3}-a_{4}-a_{6}$
$\Lambda_{c}^{+}\to p\overline{K}^{0}$ | $-2a_{1}+a_{5}+a_{6}$ | $\Xi_{c}^{0}\to\Lambda^{0}\overline{K}^{0}$ | $\frac{\sqrt{6}}{6}(-4a_{1}+2a_{2}+2a_{3}-2a_{5}+a_{6}+a_{7})$
$\Lambda_{c}^{+}\to\Sigma^{0}\pi^{+}$ | $\frac{\sqrt{2}}{2}(-2a_{1}+2a_{2}+2a_{3}+a_{5}-a_{7})$ | $\Xi_{c}^{0}\to\Sigma^{0}\overline{K}^{0}$ | $\frac{\sqrt{2}}{2}(-2a_{2}-2a_{3}+a_{6}-a_{7})$
$\Lambda_{c}^{+}\to\Sigma^{+}\pi^{0}$ | $\frac{\sqrt{2}}{2}(2a_{1}-2a_{2}-2a_{3}-a_{5}+a_{7})$ | $\Xi_{c}^{0}\to\Sigma^{+}K^{-}$ | $2a_{2}+a_{4}+a_{7}$
$\Lambda_{c}^{+}\to\Sigma^{+}\eta$ |  $\frac{\sqrt{2}}{6}c_{\phi}(-12a_{0}-6a_{1}-6a_{2}+6a_{3}+6a^{\prime}_{0}+3a_{5}+3a_{7})$ $+s_{\phi}(2a_{0}-a^{\prime}_{0}-a_{4})$ | $\Xi_{c}^{0}\to\Xi^{0}\pi^{0}$ | $\frac{\sqrt{2}}{2}(-2a_{1}+2a_{3}+a_{4}-a_{5})$
$\Lambda_{c}^{+}\to\Sigma^{+}\eta^{\prime}$ |  $\frac{\sqrt{2}}{6}s_{\phi}(-12a_{0}-6a_{1}-6a_{2}+6a_{3}+6a^{\prime}_{0}+3a_{5}+3a_{7})$ $-c_{\phi}(2a_{0}-a^{\prime}_{0}-a_{4})$ | $\Xi_{c}^{0}\to\Xi^{0}\eta$ |  $\frac{\sqrt{2}}{6}c_{\phi}(12a_{0}+6a_{1}-6a_{3}+6a^{\prime}_{0}+3a_{4}+3a_{5})$ $+\frac{1}{3}s_{\phi}(-6a_{0}-6a_{2}-3a^{\prime}_{0}-3a_{7})$
$\Lambda_{c}^{+}\to\Xi^{0}K^{+}$ | $-2a_{2}+a_{4}+a_{7}$ | $\Xi_{c}^{0}\to\Xi^{0}\eta^{\prime}$ |  $\frac{\sqrt{2}}{6}s_{\phi}(12a_{0}+6a_{1}-6a_{3}+6a^{\prime}_{0}+3a_{4}+3a_{5})$ $-\frac{1}{3}c_{\phi}(-6a_{0}-6a_{2}-3a^{\prime}_{0}-3a_{7})$
$\Xi_{c}^{+}\to\Sigma^{+}\overline{K}^{0}$ | $2a_{3}-a_{4}-a_{6}$ | $\Xi_{c}^{0}\to\Xi^{-}\pi^{+}$ | $2a_{1}+a_{5}+a_{6}$

Table 2: The same as Table 1, except for singly Cabibbo-suppressed (SCS)
decays of ${\bm{B}}_{c}\to{\bm{B}}_{n}M$.

Channel | $s_{c}^{-1}A$ | Channel | $s_{c}^{-1}A$
---|---|---|---
$\Lambda_{c}^{+}\to\Lambda^{0}K^{+}$ |  $\frac{\sqrt{6}}{6}(2a_{1}-4a_{2}+2a_{3}+3a_{4}-a_{5}+2a_{6}+2a_{7}$ $-2u_{2}+u_{3})$ | $\Xi_{c}^{+}\to\Xi^{0}K^{+}$ | $2a_{2}+2a_{3}+a_{6}-a_{7}-u_{2}$
$\Lambda_{c}^{+}\to p\pi^{0}$ | $\frac{\sqrt{2}}{2}(2a_{2}+2a_{3}-a_{6}-a_{7}+u_{2})$ | $\Xi_{c}^{0}\to\Lambda^{0}\pi^{0}$ |  $\frac{\sqrt{3}}{6}(-2a_{1}-2a_{2}+4a_{3}+3a_{4}-a_{5}-a_{6}-a_{7}$ $+u_{2}+u_{3})$
$\Lambda_{c}^{+}\to p\eta$ |  $\frac{\sqrt{2}}{2}c_{\phi}(4a_{0}+2a_{2}-2a_{3}-2a^{\prime}_{0}+a_{6}-a_{7}+u_{2})$ $+s_{\phi}(-2a_{0}-2a_{1}+a^{\prime}_{0}+a_{4}+a_{5}+a_{6}-u_{3})$ | $\Xi_{c}^{0}\to\Lambda^{0}\eta$ |  $\frac{\sqrt{3}}{6}c_{\phi}(12a_{0}+2a_{1}+2a_{2}-4a_{3}+6a^{\prime}_{0}+3a_{4}+a_{5}$ $+a_{6}+a_{7}+2u_{1}+u_{2}+u_{3})$ $+\frac{\sqrt{6}}{6}s_{\phi}(-6a_{0}-4a_{1}-4a_{2}+2a_{3}-3a^{\prime}_{0}-2a_{5}$ $+a_{6}-2a_{7}+2u_{1})$
$\Lambda_{c}^{+}\to p\eta^{\prime}$ |  $\frac{\sqrt{2}}{2}s_{\phi}(4a_{0}+2a_{2}-2a_{3}-2a^{\prime}_{0}+a_{6}-a_{7}+u_{2})$ $-c_{\phi}(-2a_{0}-2a_{1}+a^{\prime}_{0}+a_{4}+a_{5}+a_{6}-u_{3})$ | $\Xi_{c}^{0}\to\Lambda^{0}\eta^{\prime}$ |  $\frac{\sqrt{3}}{6}s_{\phi}(12a_{0}+2a_{1}+2a_{2}-4a_{3}+6a^{\prime}_{0}+3a_{4}+a_{5}$ $+a_{6}+a_{7}+2u_{1}+u_{2}+u_{3})$ $-\frac{\sqrt{6}}{6}c_{\phi}(-6a_{0}-4a_{1}-4a_{2}+2a_{3}-3a^{\prime}_{0}-2a_{5}$ $+a_{6}-2a_{7}+2u_{1})$
$\Lambda_{c}^{+}\to n\pi^{+}$ | $2a_{2}+2a_{3}+a_{6}-a_{7}+u_{2}$ | $\Xi_{c}^{0}\to pK^{-}$ | $-2a_{2}-a_{4}-a_{7}+u_{1}+u_{3}$
$\Lambda_{c}^{+}\to\Sigma^{0}K^{+}$ | $\frac{\sqrt{2}}{2}(2a_{1}-2a_{3}-a_{4}-a_{5}+u_{3})$ | $\Xi_{c}^{0}\to n\overline{K}^{0}$ | $2a_{1}-2a_{2}-2a_{3}+a_{5}-a_{7}+u_{1}$
$\Lambda_{c}^{+}\to\Sigma^{+}K^{0}$ | $2a_{1}-2a_{3}+a_{4}-a_{5}+u_{3}$ | $\Xi_{c}^{0}\to\Sigma^{0}\pi^{0}$ |  $\frac{1}{2}(2a_{1}+2a_{2}-a_{4}+a_{5}-a_{6}+a_{7}$ $+2u_{1}+u_{2}+u_{3})$
$\Xi_{c}^{+}\to\Lambda^{0}\pi^{+}$ |  $\frac{\sqrt{6}}{6}(2a_{1}+2a_{2}-4a_{3}-3a_{4}-a_{5}-a_{6}-a_{7}$ $-u_{2}-u_{3})$ | $\Xi_{c}^{0}\to\Sigma^{0}\eta$ |  $\frac{1}{2}c_{\phi}(-4a_{0}-2a_{1}-2a_{2}-2a^{\prime}_{0}-a_{4}-a_{5}$ $+a_{6}-a_{7}+u_{2}+u_{3})$ $+\frac{\sqrt{2}}{2}s_{\phi}(2a_{0}-2a_{3}+a^{\prime}_{0}+a_{6})$
$\Xi_{c}^{+}\to p\overline{K}^{0}$ | $2a_{1}-2a_{3}+a_{4}-a_{5}-u_{3}$ | $\Xi_{c}^{0}\to\Sigma^{0}\eta^{\prime}$ |  $\frac{1}{2}s_{\phi}(-4a_{0}-2a_{1}-2a_{2}-2a^{\prime}_{0}-a_{4}-a_{5}$ $+a_{6}-a_{7}+u_{2}+u_{3})$ $-\frac{\sqrt{2}}{2}c_{\phi}(2a_{0}-2a_{3}+a^{\prime}_{0}+a_{6})$
$\Xi_{c}^{+}\to\Sigma^{0}\pi^{+}$ | $\frac{\sqrt{2}}{2}(2a_{1}-2a_{2}+a_{4}-a_{5}+a_{6}+a_{7}+u_{2}-u_{3})$ | $\Xi_{c}^{0}\to\Xi^{0}K^{0}$ | $-2a_{1}+2a_{2}+2a_{3}-a_{5}+a_{7}+u_{1}$
$\Xi_{c}^{+}\to\Sigma^{+}\pi^{0}$ | $\frac{\sqrt{2}}{2}(-2a_{1}+2a_{2}+a_{4}+a_{5}+a_{6}-a_{7}-u_{2}+u_{3})$ | $\Xi_{c}^{0}\to\Sigma^{+}\pi^{-}$ | $2a_{2}+a_{4}+a_{7}+u_{1}+u_{3}$
$\Xi_{c}^{+}\to\Sigma^{+}\eta$ |  $\frac{\sqrt{2}}{2}c_{\phi}(4a_{0}+2a_{1}+2a_{2}-2a^{\prime}_{0}-a_{4}-a_{5}$ $-a_{6}-a_{7}-u_{2}-u_{3})$ $+s_{\phi}(-2a_{0}+2a_{3}+a^{\prime}_{0}-a_{6})$ | $\Xi_{c}^{0}\to\Sigma^{-}\pi^{+}$ | $2a_{1}+a_{5}+a_{6}+u_{1}+u_{2}$
$\Xi_{c}^{+}\to\Sigma^{+}\eta^{\prime}$ |  $\frac{\sqrt{2}}{2}s_{\phi}(4a_{0}+2a_{1}+2a_{2}-2a^{\prime}_{0}-a_{4}-a_{5}$ $-a_{6}-a_{7}-u_{2}-u_{3})$ $-c_{\phi}(-2a_{0}+2a_{3}+a^{\prime}_{0}-a_{6})$ | $\Xi_{c}^{0}\to\Xi^{-}K^{+}$ | $-2a_{1}-a_{5}-a_{6}+u_{1}+u_{2}$

Without involving the dynamics of decay details, decay amplitudes containing
SU(3) breaking effect can be expressed by coefficients of irreducible
representations by expanding Eqs. (7) and (9). From the detailed expressions
exhibited in Tables 1, 2 and 3, the relations of amplitudes (here $S$-wave
amplitudes are taken as an illustration) between some of the CF and DCS decays
can be established, giving

$\begin{split}&A(\Lambda_{c}^{+}\to\Sigma^{0}\pi^{+})=-A(\Lambda_{c}^{+}\to\Sigma^{+}\pi^{0})\\\
&A(\Lambda_{c}^{+}\to
nK^{+})=\sin^{2}\theta_{c}A(\Xi_{c}^{+}\to\Xi^{0}\pi^{+})\\\ &A(\Xi_{c}^{+}\to
n\pi^{+})=\sin^{2}\theta_{c}A(\Lambda_{c}^{+}\to\Xi^{0}K^{+})\\\
&A(\Xi_{c}^{+}\to\Sigma^{+}{K}^{0})=\sin^{2}\theta_{c}A(\Lambda_{c}^{+}\to
p\overline{K}^{0})\\\ &A(\Lambda_{c}^{+}\to
p{K}^{0})={\sin^{2}\theta_{c}}A(\Xi_{c}^{+}\to\Sigma^{+}\overline{K}^{0})\\\
&A(\Xi_{c}^{0}\to\Sigma^{-}{K}^{+})=-\sin^{2}\theta_{c}A(\Xi_{c}^{0}\to\Xi^{-}\pi^{+})\\\
&A(\Xi_{c}^{0}\to
p\pi^{-})=-\sin^{2}\theta_{c}A(\Xi_{c}^{0}\to\Sigma^{+}K^{-}).\end{split}$
(12)

Except the first relation which holds between the two CF modes, other
connections are established between CF and DCS modes. In addition to the
relations in Eq. (12), more relations can be recovered if SU(3) symmetry is
strictly respected,222It can be easily obtained by setting $u_{i}$ and $v_{i}$
zero in Table 2.

$\begin{split}&A(\Lambda_{c}^{+}\to\Sigma^{+}K^{0})=A(\Xi_{c}^{+}\to
p\overline{K}^{0})\\\ &A(\Lambda_{c}^{+}\to
n\pi^{+})=A(\Xi_{c}^{+}\to\Xi^{0}K^{+})\\\ &A(\Xi_{c}^{0}\to
n\overline{K}^{0})=-A(\Xi_{c}^{0}\to\Xi^{0}K^{0})\\\ &A(\Xi_{c}^{0}\to
p\pi^{-})=\sin\theta_{c}A(\Xi_{c}^{0}\to
pK^{-})=-\sin\theta_{c}A(\Xi_{c}^{0}\to\Sigma^{+}\pi^{-})=-\sin^{2}\theta_{c}A(\Xi_{c}^{0}\to\Sigma^{+}K^{-})\\\
&A(\Xi_{c}^{0}\to\Sigma^{-}{K}^{+})=\sin\theta_{c}A(\Xi_{c}^{0}\to\Xi^{-}K^{+})=-\sin\theta_{c}A(\Xi_{c}^{0}\to\Sigma^{-}\pi^{+})=-\sin^{2}\theta_{c}A(\Xi_{c}^{0}\to\Xi^{-}\pi^{+})\end{split}$
(13)

most of which manifest in SCS modes. These relations displayed in Eqs. (12)
and (13) can be confirmed in Huang:2021aqu up to a sign in some channels. 333
Note the sign difference originates from the convention in irreducible
representations of $\overline{15}$ and $6$, which is unphysical.  It is
interesting to notice that any violation of the relations in Eq. (13) is a
signal for SU(3) symmetry breaking in charmed baryon decays.

Table 3: The same as Table 1, except for doubly Cabibbo-suppressed (DCS)
processes.

Channel | $s_{c}^{-2}A$ | Channel | $s_{c}^{-2}A$
---|---|---|---
$\Lambda_{c}^{+}\to pK^{0}$ | $2a_{3}-a_{4}-a_{6}$ | $\Xi_{c}^{+}\to\Sigma^{+}K^{0}$ | $-2a_{1}+a_{5}+a_{6}$
$\Lambda_{c}^{+}\to nK^{+}$ | $-2a_{3}-a_{4}-a_{6}$ | $\Xi_{c}^{0}\to\Lambda^{0}K^{0}$ | $\frac{\sqrt{6}}{6}(-2a_{1}+4a_{2}+4a_{3}-a_{5}-a_{6}+2a_{7})$
$\Xi_{c}^{+}\to\Lambda^{0}K^{+}$ | $\frac{\sqrt{6}}{6}(-2a_{1}+4a_{2}+4a_{3}+a_{5}+a_{6}-2a_{7})$ | $\Xi_{c}^{0}\to p\pi^{-}$ | $-2a_{2}-a_{4}-a_{7}$
$\Xi_{c}^{+}\to p\pi^{0}$ | $\frac{\sqrt{2}}{2}(-2a_{2}-a_{4}+a_{7})$ | $\Xi_{c}^{0}\to n\pi^{0}$ | $\frac{\sqrt{2}}{2}(2a_{2}-a_{4}+a_{7})$
$\Xi_{c}^{+}\to p\eta$ |  $\frac{\sqrt{2}}{2}c_{\phi}(-4a_{0}-2a_{2}+2a^{\prime}_{0}+a_{4}+a_{7})$ $+s_{\phi}(2a_{0}+2a_{1}-2a_{3}-a^{\prime}_{0}-a_{5})$ | $\Xi_{c}^{0}\to n\eta$ |  $\frac{\sqrt{2}}{2}c_{\phi}(-4a_{0}-2a_{2}-2a^{\prime}_{0}-a_{4}-a_{7})$ $+s_{\phi}(2a_{0}+2a_{1}-2a_{3}+a^{\prime}_{0}+a_{5})$
$\Xi_{c}^{+}\to p\eta^{\prime}$ |  $\frac{\sqrt{2}}{2}s_{\phi}(-4a_{0}-2a_{2}+2a^{\prime}_{0}+a_{4}+a_{7})$ $-c_{\phi}(2a_{0}+2a_{1}-2a_{3}-a^{\prime}_{0}-a_{5})$ | $\Xi_{c}^{0}\to n\eta^{\prime}$ |  $\frac{\sqrt{2}}{2}s_{\phi}(-4a_{0}-2a_{2}-2a^{\prime}_{0}-a_{4}-a_{7})$ $-c_{\phi}(2a_{0}+2a_{1}-2a_{3}+a^{\prime}_{0}+a_{5})$
$\Xi_{c}^{+}\to n\pi^{+}$ | $-2a_{2}+a_{4}+a_{7}$ | $\Xi_{c}^{0}\to\Sigma^{0}K^{0}$ | $\frac{\sqrt{2}}{2}(2a_{1}+a_{5}-a_{6})$
$\Xi_{c}^{+}\to\Sigma^{0}K^{+}$ | $\frac{\sqrt{2}}{2}(-2a_{1}+a_{5}-a_{6})$ | $\Xi_{c}^{0}\to\Sigma^{-}K^{+}$ | $-2a_{1}-a_{5}-a_{6}$

##  3 Numerical Analysis

###  3.1 Fitting schemes

To fit the existed data and make predictions to the unmeasured channels, a
$\chi^{2}$ function is defined by incorporating all the up to date
observables, giving

$\chi^{2}=\sum_{i}\frac{(\mathcal{B}^{\rm{th}}_{i}-\mathcal{B}_{i}^{\rm{exp}})^{2}}{\delta_{1i}^{2}}+\sum_{i}\frac{({\mathcal{R}}^{\rm{th}}_{i}-{\mathcal{R}}_{i}^{\rm{exp}})^{2}}{\delta_{2i}^{2}}+\sum_{i}\frac{({\alpha}^{\rm{th}}_{i}-{\alpha}_{i}^{\rm{exp}})^{2}}{\sigma_{i}^{2}},$
(14)

in which $\mathcal{B}^{\rm{th}},\mathcal{R}^{\rm{th}},\alpha^{\rm{th}}$
($\mathcal{B}^{\rm{exp}},\mathcal{R}^{\rm{exp}},\alpha^{\rm{exp}}$) are
theoretical formulas (experimental central values) for branching fractions,
their ratios and decay asymmetries while $\delta_{1},\delta_{2},\sigma$ are
experimental errors for corresponding quantities.

Table 4: A summary of current experimental results, including branching
fractions, branching fraction ratios and decay asymmetries. All the data
without references is quoted from PDG Workman:2022ynf .

Channel | Expt. | Channel | Expt.
---|---|---|---
$10^{2}\mathcal{B}(\Lambda_{c}^{+}\to\Lambda^{0}\pi^{+})$ | $1.30\pm 0.07$ | $\mathcal{R}_{1}=\frac{\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{+}\eta)}{\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{+}\pi^{0})}$ |  $0.25\pm 0.03$ Belle:2022bsi
$10^{2}\mathcal{B}(\Lambda_{c}^{+}\to pK_{S})$ | $1.59\pm 0.08$ | $\mathcal{R}_{2}=\frac{\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{+}\eta^{\prime})}{\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{+}\pi^{0})}$ |  $0.33\pm 0.06$ Belle:2022bsi
$10^{2}\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{0}\pi^{+})$ | $1.29\pm 0.07$ | $\mathcal{R}_{3}=\frac{\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{+}\eta^{\prime})}{\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{+}\eta)}$ |  $1.34\pm 0.29$ Belle:2022bsi
$10^{2}\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{+}\pi^{0})$ | $1.25\pm 0.10$ | $\mathcal{R}_{4}=\frac{\mathcal{B}(\Lambda_{c}^{+}\to\Lambda^{0}K^{+})}{\mathcal{B}(\Lambda_{c}^{+}\to\Lambda^{0}\pi^{+})}$ |  $(4.68\pm 0.39)\times 10^{-2}$ BESIII:2022tnm
$10^{2}\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{+}\eta)$ | $0.44\pm 0.20$ |  |  $(5.05\pm 0.16)\times 10^{-2}$ Belle:2022uod
|  $0.314\pm 0.044$ Belle:2022bsi |  | $(4.7\pm 0.9)\times 10^{-2}$
$10^{2}\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{+}\eta^{\prime})$ | $1.50\pm 0.60$ | $\mathcal{R}_{5}=\frac{\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{0}K^{+})}{\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{0}\pi^{+})}$ |  $(3.61\pm 0.73)\times 10^{-2}$ BESIII:2022wxj
|  $0.416\pm 0.085$ Belle:2022bsi |  |  $(2.78\pm 0.16)\times 10^{-2}$ Belle:2022uod
$10^{2}\mathcal{B}(\Lambda_{c}^{+}\to\Xi^{0}K^{+})$ | $0.55\pm 0.07$ |  | $(4.0\pm 0.6)\times 10^{-2}$
$10^{3}\mathcal{B}(\Lambda_{c}^{+}\to p\eta)$ | $1.42\pm 0.12$ | $\mathcal{R}_{6}=\frac{\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{0}\pi^{+})}{\mathcal{B}(\Lambda_{c}^{+}\to\Lambda^{0}\pi^{+})}$ | $0.98\pm 0.05$
$10^{4}\mathcal{B}(\Lambda_{c}^{+}\to p\eta^{\prime})$ |  $4.73\pm 0.97$ Belle:2021vyq | $\mathcal{R}_{7}=\frac{\mathcal{B}(\Xi_{c}^{0}\to\Lambda^{0}K_{S})}{\mathcal{B}(\Xi_{c}^{0}\to\Xi^{-}\pi^{+})}$ | $0.225\pm 0.013$
|  $5.62_{-2.04}^{+2.46}\pm 0.26$ BESIII:2022izy | $\mathcal{R}_{8}=\frac{\mathcal{B}(\Xi_{c}^{0}\to\Sigma^{0}K_{S})}{\mathcal{B}(\Xi_{c}^{0}\to\Xi^{-}\pi^{+})}$ | $(3.8\pm 0.72)\times 10^{-2}$
$10^{4}\mathcal{B}(\Lambda_{c}^{+}\to\Lambda^{0}K^{+})$ |  $6.21\pm 0.61$ BESIII:2022tnm | $\mathcal{R}_{9}=\frac{\mathcal{B}(\Xi_{c}^{0}\to\Sigma^{+}K^{-})}{\mathcal{B}(\Xi_{c}^{0}\to\Xi^{-}\pi^{+})}$ | $0.123\pm 0.0122$
|  $6.57\pm 0.40$ Belle:2022uod | $\mathcal{R}_{10}=\frac{\mathcal{B}(\Xi_{c}^{0}\to\Xi^{-}K^{+})}{\mathcal{B}(\Xi_{c}^{0}\to\Xi^{-}\pi^{+})}$ | $(2.75\pm 0.57)\times 10^{-2}$
$10^{4}\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{0}K^{+})$ |  $4.7\pm 0.95$ BESIII:2022wxj | $\alpha(\Lambda_{c}^{+}\to\Lambda^{0}\pi^{+})$ | $-0.84\pm 0.09$
|  $3.58\pm 0.28$ Belle:2022uod |  |  $-0.755\pm 0.006$ Belle:2022uod
$10^{4}\mathcal{B}(\Lambda_{c}^{+}\to n\pi^{+})$ |  $6.6\pm 1.3$ BESIII:2022bkj | $\alpha(\Lambda_{c}^{+}\to pK_{S})$ | $0.18\pm 0.45$
$10^{4}\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{+}K_{S})$ |  $4.8\pm 1.4$ BESIII:2022wxj | $\alpha(\Lambda_{c}^{+}\to\Sigma^{0}\pi^{+})$ | $-0.73\pm 0.18$
$10^{4}\mathcal{B}(\Lambda_{c}^{+}\to p\pi^{0})$ |  $<0.80$ Belle:2021mvw |  |  $-0.463\pm 0.018$ Belle:2022uod
$10^{2}\mathcal{B}(\Xi_{c}^{0}\to\Xi^{-}\pi^{+})$ | $1.43\pm 0.32$ | $\alpha(\Lambda_{c}^{+}\to\Sigma^{+}\pi^{0})$ | $-0.55\pm 0.11$
$10^{3}\mathcal{B}(\Xi_{c}^{0}\to\Xi^{-}K^{+})$ | $0.38\pm 0.12$ |  |  $-0.48\pm 0.03$ Belle:2022bsi
$10^{3}\mathcal{B}(\Xi_{c}^{0}\to\Lambda^{0}K_{S})$ | $3.34\pm 0.67$ | $\alpha(\Xi_{c}^{0}\to\Xi^{-}\pi^{+})$ | $-0.64\pm 0.05$
$10^{3}\mathcal{B}(\Xi_{c}^{0}\to\Sigma^{0}K_{S})$ | $0.69\pm 0.24$ | $\alpha(\Lambda_{c}^{+}\to\Sigma^{+}\eta)$ |  $-0.99\pm 0.06$ Belle:2022bsi
$10^{3}\mathcal{B}(\Xi_{c}^{0}\to\Sigma^{+}K^{-})$ | $1.8\pm 0.4$ | $\alpha(\Lambda_{c}^{+}\to\Sigma^{+}\eta^{\prime})$ |  $-0.46\pm 0.07$ Belle:2022bsi
$10^{2}\mathcal{B}(\Xi_{c}^{+}\to\Xi^{0}\pi^{+})$ | $1.6\pm 0.8$ | $\alpha(\Lambda_{c}^{+}\to\Lambda^{0}K^{+})$ |  $-0.585\pm 0.052$ Belle:2022uod
|  | $\alpha(\Lambda_{c}^{+}\to\Sigma^{0}K^{+})$ |  $-0.55\pm 0.20$ Belle:2022uod

We use Eq. (11), together with explicit amplitudes shown in Tables 1, 2, 3 to
calculate various theoretical quantities. In practice, two sets of amplitude
formulas with or without SU(3) symmetry breaking effects are adopted inside
Eq. (11), corresponding to the two different scenarios. As for the
experimental measurements, in Table 4 we have collected all the available
results up to date, including 20 branching fractions, 10 ratios and 9 decay
asymmetries. To make a reasonable choice of experimental values as part of
inputs of the $\chi^{2}$ function, we make a comparison among 3 groups of
fittings with different choices of experimental data shown in Appendix A.

The detailed numerical calculation in Appendix A helps interpret experimental
data and make a choice of inputs. A common feature of all the 6 fitting
results is that the one containing SU(3) breaking has a better goodness than
the corresponding fit without symmetry breaking terms. This SU(3) breaking
preference is also indicated from the measurement of $\mathcal{R}_{10}$ shown
in Table 4. The branching fraction of $\Lambda_{c}^{+}\to p\pi^{0}$ is the
main challenge for the first group of fitting (Fit-I and Fit-I’), no matter to
include SU(3) symmetry breaking or not. To incorporate both $\mathcal{R}_{i}$
and single decay branching fraction make the observables double counting, and
the values of $\chi^{2}_{\rm min}/{\rm d.o.f.}$ are not the best among all the
3 groups as shown in Table 9. So in Fit-II and Fit-II’ all the ratios of
branching fractions $\mathcal{R}_{i}$ are not incorporated in the definition
of $\chi^{2}$. Moreover, it is easy to find that in all the 6 fitting schemes
the sign of $\alpha(\Lambda_{c}\to pK_{S})$ is negative while BESIII provided
a positive sign before. The similar situation occurs in the branching fraction
of $\Xi_{c}^{+}\to\Xi^{0}\pi^{+}$. Considering a correlation between
$\Xi_{c}^{+}\to\Xi^{0}\pi^{+}$ and $\Xi^{0}\to\Xi^{-}\pi^{+}$, the three
observables mentioned above are removed from Fit-II/II’, marked as Fit-
III/III’. Apparently, the numbers of deviations from experimental results with
more than $1\sigma$ is larger in Fit-II compared with Fit-III. Though
$\mathcal{R}_{i}$ is not taken as input in the definition of $\chi^{2}$ in the
scheme Fit-III, more predictions for $\mathcal{R}_{i}$ are consistent well
with experiments. Therefore, we choose the group of Fit-III and Fit-III’ as a
benchmarking combination of input in the following analysis.

###  3.2 Results and discussion

In total there are $18$ fitting parameters to be determined in the SU(3)
respected situation while the number increases to $24$ containing SU(3)
breaking terms. Throughout this work, we rely on the package IMINUIT iminuit
to perform all the fitting analysis. A local minimum of $\chi^{2}$, from a
certain starting point, can be obtained by combing Newton steps and gradient-
descents within the algorithm of the function ‘Minuit’. To achieve a global
minimum, we further take the obtained local minimum as input fixing one of the
parameters to different values. We finally get a set of solution of which most
of the parameters can be ensured that its central value locates at the minimum
region.

The fitting results of IRA amplitude coefficients are listed in Table 5. In
the flavor SU(3) respected framework, the coefficients $a_{1,2,3,6}$ and
$b_{2,5}$ are well fitted as their errors are controlled less than $10\%$
comparing with their corresponding central values. With an around $20\%\sim
50\%$ relative error, $a_{5,7}$ and $b_{1,4,6}$ can provide useful
information. However, the errors are too large for $a^{\prime}_{0},a_{0,4}$
and $b^{\prime}_{0},b_{0,3,7}$. Keeping other $a_{i}$ similar precision, the
fitting precision of $a_{5}$ is improved with a price of large uncertainty for
$u_{1}$ in SU(3) breaking framework. For the $P$-wave amplitude coefficients,
the errors of $b_{1,2,4,6}$ are less than $50\%$, while the errors of
$b_{5},v_{2,3}$ are between $50\%$ and $100\%$. The others are with large
uncertainty. We should also keep in mind that the errors of each coefficients
are also correlated. The correlation matrix $R$ are given as

${\tiny
R=\left(\begin{array}[]{cccccccccccccccccc}1&-0.00&-0.02&0.03&0.99&-0.02&0.00&0.00&0.00&-0.00&0.00&0.01&0.00&0.00&0.01&-0.01&-0.00&0.00\\\
-0.00&1&0.09&0.09&0.00&-0.01&0.89&-0.13&0.11&0.00&-0.25&0.02&0.09&-0.00&-0.09&-0.67&-0.05&0.07\\\
-0.02&0.09&1&-0.42&0.02&0.28&0.08&0.04&0.78&0.01&0.09&-0.50&0.12&-0.01&-0.05&-0.05&-0.14&-0.40\\\
0.03&0.09&-0.42&1&-0.03&-0.39&-0.05&-0.39&-0.10&-0.00&0.22&0.17&0.24&0.00&-0.01&0.09&-0.26&0.27\\\
0.99&0.00&0.02&-0.03&1&0.02&-0.00&-0.00&-0.00&0.00&-0.00&-0.01&0.00&-0.00&-0.01&0.00&0.00&-0.00\\\
-0.02&-0.01&0.28&-0.39&0.02&1&-0.11&-0.37&-0.19&0.01&0.17&-0.20&0.09&-0.01&0.12&0.03&-0.13&-0.25\\\
0.00&0.89&0.08&-0.05&-0.00&-0.11&1&0.21&0.17&0.00&-0.48&0.00&-0.11&-0.00&-0.12&-0.74&0.22&0.08\\\
0.00&-0.13&0.04&-0.39&-0.00&-0.37&0.21&1&0.27&0.00&-0.54&-0.11&-0.47&-0.00&0.01&-0.01&0.48&-0.02\\\
0.00&0.11&0.78&-0.10&-0.00&-0.19&0.17&0.27&1&-0.00&-0.02&-0.41&0.10&0.00&-0.13&-0.13&-0.04&-0.29\\\
-0.00&0.00&0.01&-0.00&0.00&0.01&0.00&0.00&-0.00&1&-0.01&-0.02&0.01&0.99&-0.01&0.01&0.00&0.01\\\
0.00&-0.25&0.09&0.22&-0.00&0.17&-0.48&-0.54&-0.02&-0.01&1&0.00&0.78&0.01&0.03&0.38&-0.89&-0.07\\\
0.00&0.02&-0.50&0.17&-0.01&-0.20&0.00&-0.11&-0.41&-0.02&0.00&1&-0.10&0.02&-0.10&-0.07&0.05&0.63\\\
0.00&0.09&0.12&0.24&0.00&0.09&-0.11&-0.47&0.10&0.01&0.78&-0.10&1&-0.01&-0.32&-0.02&-0.83&0.22\\\
0.00&-0.00&-0.01&0.00&-0.001&-0.01&-0.00&-0.00&0.00&0.99&0.01&0.02&-0.01&1&0.01&-0.01&-0.00&-0.01\\\
0.01&-0.09&-0.05&-0.01&-0.01&0.12&-0.12&0.01&-0.13&-0.01&0.03&-0.10&-0.32&0.01&1&0.15&-0.06&-0.55\\\
-0.00&-0.67&-0.05&0.09&0.00&0.03&-0.74&-0.01&-0.13&0.01&0.38&-0.07&-0.02&-0.01&0.15&1&-0.14&-0.01\\\
-0.00&-0.05&-0.14&-0.26&0.00&-0.13&0.22&0.48&-0.04&0.00&-0.89&0.05&-0.83&-0.00&-0.06&-0.14&1&0.06\\\
0.00&0.07&-0.40&0.27&-0.00&-0.25&0.08&-0.02&-0.29&0.01&-0.07&0.63&0.22&-0.01&-0.55&-0.01&0.06&1\\\
\end{array}\right).}$ (15)

Here we only present the $18\times 18$ correlation matrix in the exact SU(3)
respected scenario as an illustration, and in practical calculation another
$24\times 24$ correlation matrix for SU(3) broken case is also needed.

Combing the errors in Table 5 and the correlation matrices, Eq. (15) as an
example, one can straightforwardly calculate the physical quantities of
interest, decay branching fractions and decay asymmetries of all the two-body
weak decays of antitriplet charmed baryons. The results of the total 16 CF, 26
CSC and 16 DCS channels, together with their $S$\- and $P$-wave amplitudes,
are displayed in Table 6. Though 6 more fitting parameters are introduced in
the case of SU(3) breaking, as shown in Table 9 the goodness is better for
$\chi_{\rm min}^{2}/{\rm d.o.f.}=1.27$ comparing with $\chi_{\rm min}^{2}/{\rm
d.o.f.}=2.15$ for the symmetry keeping case.

So far all the CF modes of antitriplet charmed baryons weak decays have been
measured except the channels $\Xi_{c}^{+}\to\Sigma^{+}K_{S}$ and
$\Xi_{c}^{0}\to\Xi^{0}\pi^{0},\Xi^{0}\eta,\Xi^{0}\eta^{\prime}$. The
prediction in current work,
$\mathcal{B}(\Xi_{c}^{+}\to\Sigma^{+}K_{S})=(0.85^{+0.15}_{-0.16})\times
10^{-2}$ and corresponding $\alpha=0.65\pm 0.25$, is consistent with the one
in Huang:2021aqu but differs the sign of $\alpha$ from the pole model
calculation Zou:2019kzq . Suppose that the two modes
$\Xi_{c}^{+,0}\to\Xi^{0,-}\pi^{+}$ can be improved by Belle/Belle-II in the
near future, we have removed them from fitting input lists in current scheme
and find other predictions for the CF measured modes are consistent well in
the two scenarios. To discriminate whether SU(3) symmetry is kept or broken
from the aspect of branching fraction, the CF channel
$\Xi_{c}^{0}\to\Xi^{0}\pi^{0}$ may play a role as its central value is around
$7$ times larger in the SU(3) breaking case. In addition, the decay
asymmetries of the yet-to-be measured modes
$\Xi_{c}^{0}\to\Lambda^{0}K_{S},\Sigma^{0}K_{S},\Xi^{0}\pi^{0},\Xi^{0}\eta,\Xi^{0}\eta^{\prime}$
provide alternative options to explore the flavor symmetry since the
corresponding signs of $\alpha$ differ from each other in the two scenarios.
Moreover, from the relations exhibited in Eq. (13) the branching fraction
ratios between corresponding modes should be a good window to observe SU(3)
symmetry in charmed baryon decays. We have defined several ratios as
$\mathcal{R}_{1i}$ and calculated them based on the fitted parameters in both
scenarios presented in Table 7. In principle, once the relative size between
$S$\- and $P$-wave amplitudes of corresponding channel are known, the ratios
can be calculated theoretically. The predictions of
$\mathcal{R}_{1,2,\ldots,10}$ have been mostly confirmed by experimental data
from the Fit-III column of Table 10, hence a further confirmation of
predictions in Table 7 is highly expected. For the SCS modes, there are only 7
of 26

Table 5:  The fitting results of amplitude coefficients in irreducible
amplitude approach (see Eq. (7) and (9)) in the unit of $10^{-2}G_{F}{\rm
GeV}^{2}$.

SU(3) respected (Fit-III’) | SU(3) breaking (Fit-III)
---|---
Coefficient | Value | Coefficient | Value | Coefficient | Value | Coefficient | Value
$a_{0}$ | $-0.10\pm 1.00$ | $b_{0}$ | $-0.50\pm 2.70$ | $a_{0}$ | $-1.20\pm 1.00$ | $b_{0}$ | $-0.70\pm 3.30$
$a_{1}$ | $-3.75_{-0.17}^{+0.20}$ | $b_{1}$ | $3.50_{-0.90}^{+1.00}$ | $a_{1}$ | $-3.50_{-0.13}^{+0.21}$ | $b_{1}$ | $8.60_{-0.90}^{+1.30}$
$a_{2}$ | $1.10_{-0.16}^{+0.17}$ | $b_{2}$ | $4.04_{-0.47}^{+0.33}$ | $a_{2}$ | $1.45_{-0.25}^{+0.16}$ | $b_{2}$ | $4.00_{-1.60}^{+0.90}$
$a_{3}$ | $-1.66_{-0.10}^{+0.12}$ | $b_{3}$ | $0.80_{-0.90}^{+1.00}$ | $a_{3}$ | $-1.98_{-0.13}^{+0.15}$ | $b_{3}$ | $-0.80_{-0.70}^{+1.20}$
$a^{\prime}_{0}$ | $1.60\pm 2.00$ | $b^{\prime}_{0}$ | $2.00\pm 5.00$ | $a^{\prime}_{0}$ | $0.10\pm 2.00$ | $b^{\prime}_{0}$ | $2.00\pm 7.00$
$a_{4}$ | $0.04_{-0.21}^{+0.18}$ | $b_{4}$ | $-1.20_{-0.70}^{+0.60}$ | $a_{4}$ | $0.23_{-0.28}^{+0.29}$ | $b_{4}$ | $-3.40\pm 0.90$
$a_{5}$ | $1.36_{-0.37}^{+0.45}$ | $b_{5}$ | $-9.10_{-0.70}^{+0.60}$ | $a_{5}$ | $2.14_{-0.23}^{+0.26}$ | $b_{5}$ | $3.00_{-2.20}^{+1.70}$
$a_{6}$ | $1.26_{-0.22}^{+0.16}$ | $b_{6}$ | $8.00_{-1.70}^{+1.60}$ | $a_{6}$ | $1.56_{-0.20}^{+0.14}$ | $b_{6}$ | $10.10_{-2.30}^{+1.50}$
$a_{7}$ | $-1.04_{-0.29}^{+0.33}$ | $b_{7}$ | $0.60_{-1.10}^{+0.90}$ | $a_{7}$ | $-0.59_{-0.42}^{+0.33}$ | $b_{7}$ | $-0.90_{-3.20}^{+1.90}$
|  |  |  | $u_{1}$ | $0.00\pm 110$ | $v_{1}$ | $13.00_{-36}^{+47}$
|  |  |  | $u_{2}$ | $1.20_{-0.70}^{+0.80}$ | $v_{2}$ | $-3.00_{-2.00}^{+1.70}$
|  |  |  | $u_{3}$ | $2.80\pm 0.70$ | $v_{3}$ | $4.80_{-1.60}^{+1.40}$

branching fractions and 2 of 26 decay asymmetries have been measured, with
them predictions in Fit-III are consistent well. In the case with exact SU(3)
symmetry (Fit-III’) the predicted $\alpha(\Lambda_{c}^{+}\to\Sigma^{0}K^{+})$
has a deviation from Belle measured value Belle:2022uod originated mainly from
the difference in $P$-wave amplitude. Among all the 16 DCS channels, the two
modes $\Xi_{c}^{+}\to\Sigma^{0}K^{+}$ and $\Xi_{c}^{+}\to\Sigma^{+}K^{0}$ are
most possibly to access in next few years with branching ratio at the order of
$10^{-4}$ and negative decay asymmetries in both scenarios, which are also in
agreement with the theoretical prediction combing the pole model and MIT bag
model Meng:2020euv .

The comparison with experimental values combing fitting results from other
groups Geng:2019xbo ; Huang:2021aqu ; Zhao:2018mov as well as the model
calculationZou:2019kzq ; Meng:2020euv , see Table 8, is meaningful. As a
typical example, the sign of decay asymmetry of $\Lambda_{c}^{+}\to pK_{S}$
helps discriminate theoretical calculations, fitting calculations and
experimental measurement. In all the 6 different fitting schemes of this work
(see also Table 9), the predictions provide a negative sign, which is also
supported by the earlier fitting Geng:2019xbo as well as the model calculation
Zou:2019kzq . However, the central value of the same quantity provided by
BESIII, together with a previous fitting analysis Huang:2021aqu , prefers a
positive sign (though there is still a possibility that the sign is negative
considering the errors). Currently the error is still too large to determine
the sign of $\alpha$ yet, when more data accumulated BESIII, Belle/Belle-II
and LHCb are supposed to give the final word. The branching fraction of
$\Lambda_{c}^{+}\to\Sigma^{+}\eta^{\prime}$ was firstly measured in 2019 by
BESIII measured, given $(1.50\pm 0.60)\times 10^{-2}$ Workman:2022ynf , which
is supported by earlier fitting results Geng:2019xbo ; Huang:2021aqu .
However, the recent independent measurement from Belle gives a smaller

Table 6: The fitting results of all the weak decays of antitriplet charmed
baryons into octet pseudoscalar mesons in both SU(3) flavor symmetry respected
and breaking scenarios, including Cabibbo-favored, singly Cabibbo-suppressed
and doubly Cabibbo-suppressed processes.

| SU(3) respected | SU(3) broken
---|---|---
Channel(CF) | $A(10^{-1}G_{F})$ | $B(10^{-1}G_{F})$ | $10^{2}\mathcal{B}$ | $\alpha$ | $A(10^{-1}G_{F})$ | $B(10^{-1}G_{F})$ | $10^{2}\mathcal{B}$ | $\alpha$
$\Lambda_{c}^{+}\to\Lambda^{0}\pi^{+}$ | $0.26\pm 0.01$ | $-1.68\pm{0.06}$ | $1.27_{-0.08}^{+0.09}$ | $-0.75\pm{0.01}$ | $0.26\pm{0.01}$ | $-1.70_{-0.10}^{+0.09}$ | $1.30_{-0.14}^{+0.12}$ | $-0.75\pm{0.01}$
$\Lambda_{c}^{+}\to p\overline{K}^{0}$ | $1.01_{-0.04}^{+0.03}$ | $-0.81\pm{0.34}$ | $3.18\pm{0.20}$ | $-0.57\pm{0.21}$ | $1.07_{-0.03}^{+0.04}$ | $-0.41_{-0.42}^{+0.36}$ | $3.30\pm{0.21}$ | $-0.29\pm{0.24}$
$\Lambda_{c}^{+}\to\Sigma^{0}\pi^{+}$ | $0.62_{-0.01}^{+0.02}$ | $-0.50_{-0.03}^{+0.04}$ | $1.30\pm{0.06}$ | $-0.47\pm{0.03}$ | $0.61\pm{0.02}$ | $-0.49_{-0.07}^{+0.06}$ | $1.27\pm{0.09}$ | $-0.47\pm{0.03}$
$\Lambda_{c}^{+}\to\Sigma^{+}\pi^{0}$ | $-0.62_{-0.01}^{+0.02}$ | $0.50_{-0.03}^{+0.04}$ | $1.30\pm{0.06}$ | $-0.47\pm{0.03}$ | $-0.61\pm{0.02}$ | $0.49_{-0.07}^{+0.06}$ | $1.27\pm{0.09}$ | $-0.47\pm{0.03}$
$\Lambda_{c}^{+}\to\Sigma^{+}\eta$ | $0.21\pm{0.03}$ | $-0.99_{-0.12}^{+0.11}$ | $0.314_{-0.051}^{+0.049}$ | $-0.96\pm{0.05}$ | $0.19\pm{0.03}$ | $-0.98_{-0.15}^{+0.14}$ | $0.297_{-0.067}^{+0.060}$ | $-0.95\pm{0.06}$
$\Lambda_{c}^{+}\to\Sigma^{+}\eta^{\prime}$ | $0.41\pm{0.04}$ | $-0.58\pm{0.11}$ | $0.241_{-0.054}^{+0.052}$ | $-0.43\pm{0.07}$ | $0.51\pm{0.05}$ | $-0.80\pm{0.15}$ | $0.386_{-0.080}^{+0.079}$ | $-0.47\pm{0.05}$
$\Lambda_{c}^{+}\to\Xi^{0}K^{+}$ | $-0.32\pm{0.02}$ | $-0.87_{-0.08}^{+0.06}$ | $0.381_{-0.050}^{+0.040}$ | $0.91_{-0.04}^{+0.03}$ | $-0.33_{-0.07}^{+0.05}$ | $-1.23_{-0.18}^{+0.12}$ | $0.501_{-0.085}^{+0.063}$ | $0.99\pm{0.01}$
$\Xi_{c}^{+}\to\Sigma^{+}\overline{K}^{0}$ | $-0.46\pm{0.04}$ | $-0.52\pm{0.35}$ | $1.69_{-0.44}^{+0.43}$ | $0.65_{-0.31}^{+0.32}$ | $-0.58\pm{0.04}$ | $-0.83\pm{0.39}$ | $2.82\pm{0.54}$ | $0.77\pm{0.20}$
$\Xi_{c}^{+}\to\Xi^{0}\pi^{+}$ | $0.20\pm{0.02}$ | $-0.84\pm 0.08$ | $0.813_{-0.106}^{+0.111}$ | $-0.97\pm{0.03}$ | $0.22\pm{0.04}$ | $-0.51_{-0.15}^{+0.14}$ | $0.547_{-0.202}^{+0.195}$ | $-0.94\pm{0.05}$
$\Xi_{c}^{0}\to\Lambda^{0}\overline{K}^{0}$ | $0.46_{-0.06}^{+0.07}$ | $0.92\pm{0.21}$ | $0.758_{-0.123}^{+0.132}$ | $0.94_{-0.10}^{+0.11}$ | $0.39\pm{0.05}$ | $-1.01_{-0.29}^{+0.27}$ | $0.668_{-0.196}^{+0.195}$ | $-0.996\pm{0.017}$
$\Xi_{c}^{0}\to\Sigma^{0}\overline{K}^{0}$ | $0.24\pm{0.04}$ | $-0.16_{-0.27}^{+0.26}$ | $0.144_{-0.049}^{+0.051}$ | $-0.41_{-0.64}^{+0.63}$ | $0.23_{-0.06}^{+0.05}$ | $0.33_{-0.47}^{+0.34}$ | $0.147_{-0.057}^{+0.050}$ | $0.76_{-0.39}^{+0.38}$
$\Xi_{c}^{0}\to\Sigma^{+}K^{-}$ | $0.12_{-0.06}^{+0.07}$ | $0.75_{-0.17}^{+0.13}$ | $0.174_{-0.051}^{+0.041}$ | $0.79_{-0.33}^{+0.32}$ | $0.25_{-0.08}^{+0.06}$ | $0.37_{-0.62}^{+0.38}$ | $0.186_{-0.071}^{+0.045}$ | $0.77_{-0.31}^{+0.27}$
$\Xi_{c}^{0}\to\Xi^{0}\pi^{0}$ | $0.20_{-0.05}^{+0.06}$ | $0.18\pm{0.11}$ | $0.113_{-0.049}^{+0.059}$ | $0.50_{-0.35}^{+0.37}$ | $0.080_{-0.040}^{+0.050}$ | $-1.78_{-0.27}^{+0.29}$ | $0.774_{-0.232}^{+0.252}$ | $-0.29_{-0.17}^{+0.20}$
$\Xi_{c}^{0}\to\Xi^{0}\eta$ | $-0.16\pm{0.19}$ | $-0.77_{-0.52}^{+0.51}$ | $0.156\pm{0.192}$ | $0.97\pm{0.31}$ | $-0.29\pm{0.19}$ | $0.58_{-0.70}^{+0.65}$ | $0.243_{-0.290}^{+0.279}$ | $-0.84_{-0.51}^{+0.50}$
$\Xi_{c}^{0}\to\Xi^{0}\eta^{\prime}$ | $0.20\pm{0.67}$ | $0.62\pm{1.77}$ | $0.0683_{-0.3268}^{+0.3272}$ | $0.84\pm{2.16}$ | $-0.24_{-0.67}^{+0.66}$ | $1.47_{-2.26}^{+2.22}$ | $0.163_{-0.514}^{+0.509}$ | $-0.99\pm{0.09}$
$\Xi_{c}^{0}\to\Xi^{-}\pi^{+}$ | $-0.49_{-0.07}^{+0.09}$ | $0.59_{-0.13}^{+0.14}$ | $0.698_{-0.217}^{+0.248}$ | $-0.64\pm{0.07}$ | $-0.33_{-0.05}^{+0.06}$ | $3.03_{-0.42}^{+0.38}$ | $2.43_{-0.64}^{+0.60}$ | $-0.64_{-0.09}^{+0.11}$
Channel(SCS) | $A(10^{-2}G_{F})$ | $B(10^{-2}G_{F})$ | $10^{4}\mathcal{B}$ | $\alpha$ | $A(10^{-2}G_{F})$ | $B(10^{-2}G_{F})$ | $10^{4}\mathcal{B}$ | $\alpha$
$\Lambda_{c}^{+}\to\Lambda^{0}K^{+}$ | $-1.47\pm{0.04}$ | $1.39_{-0.18}^{+0.16}$ | $6.62_{-0.37}^{+0.35}$ | $-0.55\pm{0.06}$ | $-1.45\pm{0.05}$ | $1.43_{-0.29}^{+0.23}$ | $6.53_{-0.49}^{+0.42}$ | $-0.57\pm{0.06}$
$\Lambda_{c}^{+}\to p\pi^{0}$ | $-0.21\pm{0.06}$ | $0.17_{-0.52}^{+0.53}$ | $0.157_{-0.088}^{+0.089}$ | $-0.60_{-1.57}^{+1.60}$ | $-0.13_{-0.13}^{+0.14}$ | $-0.92_{-0.59}^{+0.57}$ | $0.513_{-0.610}^{+0.592}$ | $0.61_{-0.25}^{+0.26}$
$\Lambda_{c}^{+}\to p\eta$ | $2.22\pm{0.09}$ | $0.07\pm{0.91}$ | $13.56_{-1.16}^{+1.10}$ | $0.025\pm{0.316}$ | $2.15\pm{0.09}$ | $0.06_{-1.11}^{+1.03}$ | $12.75_{-1.06}^{+1.09}$ | $0.022\pm{0.328}$
$\Lambda_{c}^{+}\to p\eta^{\prime}$ | $-1.65\pm{0.10}$ | $1.89_{-0.35}^{+0.34}$ | $5.93_{-0.73}^{+0.71}$ | $-0.63\pm{0.09}$ | $-1.32_{-0.15}^{+0.16}$ | $2.63\pm_{-0.52}^{+0.46}$ | $4.65_{-0.77}^{+0.79}$ | $-0.89\pm{0.06}$
$\Lambda_{c}^{+}\to n\pi^{+}$ | $0.26\pm{0.03}$ | $3.83_{-0.18}^{+0.17}$ | $8.15_{-0.69}^{+0.68}$ | $0.32\pm{0.05}$ | $0.51_{-0.16}^{+0.19}$ | $3.23_{-0.49}^{+0.41}$ | $6.47_{-1.33}^{+1.55}$ | $0.67\pm{0.05}$
$\Lambda_{c}^{+}\to\Sigma^{0}K^{+}$ | $-0.89_{-0.03}^{+0.04}$ | $2.49_{-0.13}^{+0.14}$ | $3.56_{-0.22}^{+0.23}$ | $-0.97_{-0.01}^{+0.02}$ | $-0.41\pm{0.11}$ | $3.81_{-0.26}^{+0.25}$ | $3.71_{-0.36}^{+0.39}$ | $-0.67\pm{0.05}$
$\Lambda_{c}^{+}\to\Sigma^{+}K^{0}$ | $-1.23\pm{0.09}$ | $2.99_{-0.33}^{+0.34}$ | $6.22_{-0.67}^{+0.69}$ | $-0.93\pm{0.05}$ | $-0.48\pm{0.20}$ | $3.86_{-0.48}^{+0.49}$ | $3.98_{-0.92}^{+0.97}$ | $-0.74_{-0.10}^{+0.11}$
$\Xi_{c}^{+}\to\Lambda^{0}\pi^{+}$ | $-0.033_{-0.044}^{+0.045}$ | $1.46_{-0.19}^{+0.18}$ | $2.34_{-0.60}^{+0.57}$ | $-0.12\pm{0.16}$ | $-0.36\pm{0.21}$ | $2.25_{-0.65}^{+0.56}$ | $6.55_{-3.81}^{+3.37}$ | $-0.73\pm{0.08}$
$\Xi_{c}^{+}\to p\overline{K}^{0}$ | $-1.23\pm{0.09}$ | $2.99_{-0.33}^{+0.34}$ | $21.87_{-2.69}^{+2.75}$ | $-0.999\pm{0.005}$ | $-1.74\pm{0.15}$ | $1.71_{-0.52}^{+0.55}$ | $24.61_{-3.62}^{+3.85}$ | $-0.72\pm{0.11}$
$\Xi_{c}^{+}\to\Sigma^{0}\pi^{+}$ | $-1.71\pm{0.03}$ | $2.45_{-0.12}^{+0.11}$ | $28.32_{-1.09}^{+1.05}$ | $-0.79\pm{0.02}$ | $-1.98_{-0.17}^{+0.16}$ | $0.67_{-0.52}^{+0.39}$ | $30.60_{-5.27}^{+5.17}$ | $-0.23\pm{0.04}$
$\Xi_{c}^{+}\to\Sigma^{+}\pi^{0}$ | $2.13\pm{0.07}$ | $-0.29_{-0.57}^{+0.56}$ | $35.12_{-2.16}^{+2.09}$ | $-0.094_{-0.187}^{+0.181}$ | $2.54_{-0.15}^{+0.14}$ | $1.46_{-0.66}^{+0.57}$ | $52.08_{-6.18}^{+5.86}$ | $0.38_{-0.14}^{+0.13}$
$\Xi_{c}^{+}\to\Sigma^{+}\eta$ | $-1.69_{-0.10}^{+0.09}$ | $0.84_{-0.88}^{+0.89}$ | $19.90_{-1.92}^{+1.82}$ | $-0.31\pm{0.32}$ | $-2.45\pm{0.21}$ | $-0.22_{-1.26}^{+1.13}$ | $40.90_{-7.00}^{+6.90}$ | $0.058_{-0.227}^{+0.229}$
$\Xi_{c}^{+}\to\Sigma^{+}\eta^{\prime}$ | $-0.57\pm{0.09}$ | $1.67_{-0.34}^{+0.35}$ | $2.28_{-0.73}^{+0.65}$ | $-0.94\pm 0.07$ | $-1.13\pm{0.21}$ | $2.23\pm{0.53}$ | $7.24_{-2.58}^{+2.56}$ | $-0.77\pm{0.08}$
$\Xi_{c}^{+}\to\Xi^{0}K^{+}$ | $0.26\pm{0.03}$ | $3.83_{-0.18}^{+0.17}$ | $8.26\pm{0.63}$ | $0.47\pm{0.07}$ | $-0.025_{-0.193}^{+0.184}$ | $4.58_{-0.59}^{+0.51}$ | $11.11_{-2.89}^{+2.52}$ | $-0.039\pm{0.054}$
$\Xi_{c}^{0}\to\Lambda^{0}\pi^{0}$ | $-0.18_{-0.08}^{+0.09}$ | $-0.97_{-0.27}^{+0.28}$ | $0.431_{-0.212}^{+0.205}$ | $0.80_{-0.27}^{+0.28}$ | $-0.15_{-0.17}^{+0.16}$ | $-3.18_{-0.70}^{+0.60}$ | $3.75_{-1.60}^{+1.36}$ | $0.24_{-0.14}^{+0.12}$
$\Xi_{c}^{0}\to\Lambda^{0}\eta$ | $0.79\pm 0.51$ | $0.31\pm 1.40$ | $1.46\pm 1.84$ | $0.27\pm 1.22$ | $0.45\pm{23.69}$ | $1.83_{-8.04}^{+10.18}$ | $1.40_{-72.22}^{+74.29}$ | $0.94\pm{0.02}$
$\Xi_{c}^{0}\to\Lambda^{0}\eta^{\prime}$ | $0.21\pm{1.85}$ | $1.03_{-4.88}^{+4.87}$ | $0.206\pm{1.432}$ | $0.95\pm{5.67}$ | $-0.77\pm 6.79$ | $4.09_{-6.55}^{+6.69}$ | $3.10_{-15.05}^{+15.54}$ | $-0.93\pm{1.87}$
$\Xi_{c}^{0}\to pK^{-}$ | $-0.27_{-0.14}^{+0.15}$ | $-1.68_{-0.38}^{+0.29}$ | $1.38_{-0.47}^{+0.35}$ | $0.65_{-0.32}^{+0.34}$ | $0.058_{-24.571}^{+24.570}$ | $3.17_{-8.23}^{+10.45}$ | $4.31_{-23.17}^{+29.53}$ | $0.086\pm{0.031}$
$\Xi_{c}^{0}\to n\overline{K}^{0}$ | $-0.89_{-0.19}^{+0.22}$ | $-2.78_{-0.59}^{+0.54}$ | $5.15_{-1.10}^{+1.18}$ | $0.96\pm{0.10}$ | $-0.72_{-24.57}^{+24.56}$ | $6.22_{-8.28}^{+10.53}$ | $17.73_{-246.37}^{+257.95}$ | $-0.50_{-0.03}^{+0.04}$
$\Xi_{c}^{0}\to\Sigma^{0}\pi^{0}$ | $-0.71_{-0.11}^{+0.12}$ | $-0.025_{-0.472}^{+0.447}$ | $1.29_{-0.40}^{+0.45}$ | $0.024_{-0.464}^{+0.440}$ | $-0.038_{-24.565}^{+24.566}$ | $5.43_{-8.28}^{+10.51}$ | $9.26_{-168.81}^{+176.25}$ | $-0.040_{-0.049}^{+0.051}$
$\Xi_{c}^{0}\to\Sigma^{0}\eta$ | $0.90\pm{0.31}$ | $0.80\pm{1.01}$ | $1.99_{-1.34}^{+1.35}$ | $0.53\pm 0.57$ | $1.41_{-0.33}^{+0.32}$ | $0.090_{-1.337}^{+1.248}$ | $4.54_{-2.09}^{+2.08}$ | $0.041_{-0.456}^{+0.455}$
$\Xi_{c}^{0}\to\Sigma^{0}\eta^{\prime}$ | $-0.49\pm{1.06}$ | $-0.87\pm{2.81}$ | $0.446_{-1.575}^{+1.576}$ | $0.71\pm 2.12$ | $0.49\pm{1.06}$ | $-2.44_{-3.54}^{+3.52}$ | $0.91\pm{2.26}$ | $-0.98\pm 0.28$
$\Xi_{c}^{0}\to\Sigma^{+}\pi^{-}$ | $0.27_{-0.14}^{+0.15}$ | $1.68_{-0.38}^{+0.29}$ | $1.08_{-0.33}^{+0.25}$ | $0.76\pm 0.33$ | $1.20\pm{24.57}$ | $4.83_{-8.34}^{+10.50}$ | $11.08_{-11.00}^{+16.60}$ | $0.95\pm{0.18}$
$\Xi_{c}^{0}\to\Sigma^{-}\pi^{+}$ | $-1.10\pm{0.19}$ | $1.32\pm{0.30}$ | $3.66_{-1.15}^{+1.31}$ | $-0.71\pm 0.07$ | $-0.47\pm 24.57$ | $9.05_{-8.32}^{+10.55}$ | $25.95_{-241.78}^{+254.15}$ | $-0.30_{-0.04}^{+0.05}$
$\Xi_{c}^{0}\to\Xi^{0}K^{0}$ | $0.89_{-0.19}^{+0.22}$ | $2.78_{-0.59}^{+0.54}$ | $3.21_{-0.76}^{+0.67}$ | $0.99\pm 0.06$ | $0.72_{-24.56}^{+24.57}$ | $-0.38_{-8.25}^{+10.43}$ | $1.22_{-38.19}^{+37.82}$ | $-0.29_{-0.80}^{+0.77}$
$\Xi_{c}^{0}\to\Xi^{-}K^{+}$ | $1.10_{-0.17}^{+0.19}$ | $-1.32\pm{0.30}$ | $3.07_{-0.95}^{+1.09}$ | $-0.60\pm 0.07$ | $1.01_{-24.58}^{+24.57}$ | $-4.56_{-10.38}^{+8.13}$ | $5.99_{-6.68}^{+10.32}$ | $-0.98_{-0.26}^{+0.30}$
Channel(DCS) | $A(10^{-3}G_{F})$ | $B(10^{-3}G_{F})$ | $10^{5}\mathcal{B}$ | $\alpha$ | $A(10^{-3}G_{F})$ | $B(10^{-3}G_{F})$ | $10^{5}\mathcal{B}$ | $\alpha$
$\Lambda_{c}^{+}\to pK^{0}$ | $-2.33_{-0.21}^{+0.20}$ | $-2.62_{-1.75}^{+1.79}$ | $1.83_{-0.56}^{+0.55}$ | $0.74\pm 0.32$ | $-2.90_{-0.19}^{+0.20}$ | $-4.18_{-1.94}^{+1.99}$ | $3.14_{-0.75}^{+0.76}$ | $0.86\pm{0.18}$
$\Lambda_{c}^{+}\to nK^{+}$ | $1.02\pm{0.09}$ | $-4.23_{-0.39}^{+0.41}$ | $1.08_{-0.15}^{+0.16}$ | $-0.89\pm 0.05$ | $1.09_{-0.19}^{+0.21}$ | $-2.57_{-0.82}^{+0.71}$ | $0.627_{-0.255}^{+0.234}$ | $-0.997\pm{0.011}$
$\Xi_{c}^{+}\to\Lambda^{0}K^{+}$ | $2.05_{-0.06}^{+0.07}$ | $2.07_{-0.21}^{+0.23}$ | $3.33_{-0.16}^{+0.19}$ | $0.63\pm 0.06$ | $2.01_{-0.10}^{+0.08}$ | $2.16_{-0.44}^{+0.34}$ | $3.25_{-0.29}^{+0.22}$ | $0.67\pm{0.06}$
$\Xi_{c}^{+}\to p\pi^{0}$ | $-1.17_{-0.14}^{+0.13}$ | $-2.24_{-0.50}^{+0.42}$ | $1.78_{-0.43}^{+0.39}$ | $0.99_{-0.04}^{+0.03}$ | $-1.33_{-0.17}^{+0.14}$ | $-1.96_{-0.72}^{+0.50}$ | $1.89_{-0.53}^{+0.40}$ | $0.92_{-0.10}^{+0.09}$
$\Xi_{c}^{+}\to p\eta$ | $-2.23\pm{0.12}$ | $2.93_{-0.34}^{+0.38}$ | $4.43_{-0.46}^{+0.48}$ | $-0.85\pm 0.06$ | $-1.97_{-0.13}^{+0.15}$ | $2.44_{-0.38}^{+0.48}$ | $3.36_{-0.49}^{+0.53}$ | $-0.82_{-0.07}^{+0.08}$
$\Xi_{c}^{+}\to p\eta^{\prime}$ | $2.95\pm{0.21}$ | $-5.09\pm{0.57}$ | $6.33\pm 0.85$ | $-0.90\pm 0.04$ | $3.39\pm{0.24}$ | $-6.08_{-0.82}^{+0.79}$ | $8.52_{-1.28}^{+1.26}$ | $-0.92\pm{0.04}$
$\Xi_{c}^{+}\to n\pi^{+}$ | $-1.61\pm{0.08}$ | $-4.38_{-0.38}^{+0.30}$ | $4.82_{-0.53}^{+0.43}$ | $0.98\pm 0.02$ | $-1.64_{-0.25}^{+0.20}$ | $-6.20_{-0.91}^{+0.62}$ | $7.78_{-1.55}^{+1.03}$ | $0.88\pm{0.03}$
$\Xi_{c}^{+}\to\Sigma^{0}K^{+}$ | $2.71\pm{0.05}$ | $-8.59\pm{0.26}$ | $10.56_{-0.43}^{+0.45}$ | $-0.9996\pm{0.0007}$ | $2.70_{-0.07}^{+0.08}$ | $-8.66_{-0.41}^{+0.42}$ | $10.63_{-0.69}^{+0.70}$ | $-0.999\pm{0.001}$
$\Xi_{c}^{+}\to\Sigma^{+}K^{0}$ | $5.10_{-0.18}^{+0.17}$ | $-4.08_{-1.71}^{+1.69}$ | $19.42_{-1.09}^{+1.04}$ | $-0.49\pm 0.19$ | $5.39_{-0.16}^{+0.18}$ | $-2.07_{-2.34}^{+1.60}$ | $20.65_{-1.21}^{+1.30}$ | $-0.25\pm{0.20}$
$\Xi_{c}^{0}\to\Lambda^{0}K^{0}$ | $0.12_{-0.21}^{+0.29}$ | $3.02_{-1.10}^{+1.03}$ | $0.273_{-0.194}^{+0.182}$ | $0.21_{-0.49}^{+0.53}$ | $0_{-0.32}^{+0.25}$ | $-3.97_{-3.10}^{+1.90}$ | $0.468_{-0.555}^{+0.379}$ | $0_{-0.37}^{+0.40}$
$\Xi_{c}^{0}\to p\pi^{-}$ | $-0.60_{-0.32}^{+0.34}$ | $-3.77_{-0.85}^{+0.65}$ | $0.810_{-0.279}^{+0.206}$ | $0.63\pm{0.33}$ | $-1.28_{-0.40}^{+0.29}$ | $-1.87_{-2.21}^{+1.36}$ | $0.588_{-0.410}^{+0.234}$ | $0.92_{-0.24}^{+0.21}$
$\Xi_{c}^{0}\to n\pi^{0}$ | $0.40_{-0.22}^{+0.23}$ | $3.52_{-0.62}^{+0.78}$ | $0.666_{-0.268}^{+0.215}$ | $0.48\pm{0.27}$ | $0.74_{-0.34}^{+0.26}$ | $3.74_{-1.64}^{+0.99}$ | $0.845_{-0.764}^{+0.464}$ | $0.74_{-0.20}^{+0.19}$
$\Xi_{c}^{0}\to n\eta$ | $-1.56_{-0.97}^{+0.98}$ | $-3.48_{-2.56}^{+2.54}$ | $1.05_{-0.95}^{+0.96}$ | $0.999\pm{0.062}$ | $-0.45\pm{0.95}$ | $5.80_{-3.40}^{+3.27}$ | $1.43_{-1.65}^{+1.59}$ | $-0.36_{-0.71}^{+0.72}$
$\Xi_{c}^{0}\to n\eta^{\prime}$ | $-0.35_{-3.38}^{+3.39}$ | $-1.09_{-8.94}^{+8.93}$ | $0.0488_{-0.4541}^{+0.4542}$ | $0.99\pm 4.19$ | $1.71\pm{3.35}$ | $-9.85_{-11.29}^{+11.22}$ | $2.77_{-5.50}^{+5.47}$ | $-0.78\pm 1.02$
$\Xi_{c}^{0}\to\Sigma^{0}K^{0}$ | $-2.64_{-0.26}^{+0.30}$ | $-3.69_{-1.33}^{+1.32}$ | $1.95_{-0.36}^{+0.38}$ | $0.74\pm{0.20}$ | $-2.29_{-0.20}^{+0.25}$ | $3.60_{-1.51}^{+1.52}$ | $1.55_{-0.35}^{+0.39}$ | $-0.81\pm 0.17$
$\Xi_{c}^{0}\to\Sigma^{-}K^{+}$ | $2.46_{-0.37}^{+0.43}$ | $-2.97_{-0.67}^{+0.68}$ | $1.64_{-0.51}^{+0.58}$ | $-0.68\pm{0.07}$ | $1.66_{-0.23}^{+0.31}$ | $-15.28_{-2.11}^{+1.93}$ | $6.37_{-1.68}^{+1.58}$ | $-0.61_{-0.09}^{+0.11}$

Table 7: Predictions of branching fraction ratios between some special modes in both scenarios on SU(3) flavor symmetry.  | SU(3) respected | SU(3) broken
---|---|---
${\mathcal{R}}_{11}\equiv\frac{\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{+}K^{0})}{\mathcal{B}(\Xi_{c}^{+}\to pK_{S})}$ | $0.57\pm{0.03}$ | $0.32\pm{0.09}$
${\mathcal{R}}_{12}\equiv\frac{\mathcal{B}(\Lambda_{c}^{+}\to n\pi^{+})}{\mathcal{B}(\Xi_{c}^{+}\to\Xi^{0}K^{+})}$ | $0.99\pm{0.01}$ | $0.58_{-0.24}^{+0.20}$
${\mathcal{R}}_{13}\equiv\frac{\mathcal{B}(\Xi_{c}^{0}\to nK_{S})}{\mathcal{B}(\Xi_{c}^{0}\to\Xi^{0}K_{S})}$ | $1.60_{-0.25}^{+0.26}$ | $14.53_{-7931.44}^{+7812.23}$
${\mathcal{R}}_{14}\equiv\frac{\mathcal{B}(\Xi_{c}^{0}\to p\pi^{-})}{\mathcal{B}(\Xi_{c}^{0}\to pK^{-})}$ | $(5.87\pm{0.07})10^{-2}$ | $(1.36_{-18.84}^{+19.62})10^{-2}$
${\mathcal{R}}_{15}\equiv\frac{\mathcal{B}(\Xi_{c}^{0}\to\Sigma^{-}K^{+})}{\mathcal{B}(\Xi_{c}^{0}\to\Xi^{-}K^{+})}$ | $(5.34\pm{0.05})10^{-2}$ | $(10.63_{-29.87}^{+46.06})10^{-2}$
${\mathcal{R}}_{16}\equiv\frac{\mathcal{B}(\Xi_{c}^{0}\to\Xi^{-}K^{+})}{\mathcal{B}(\Xi_{c}^{0}\to\Sigma^{-}\pi^{+})}$ | $0.84\pm{0.01}$ | $0.23_{-1.15}^{+1.32}$

central value with decreased error, $(0.416\pm 0.085)\times
10^{-2}$Belle:2022bsi , which locates in the predicted region in current work.

For the decay of $\Lambda_{c}^{+}\to p\eta^{\prime}$, predictions of the two
fittings in SU(3) IRA Geng:2019xbo ; Huang:2021aqu are both larger than recent
measured results from both BESIII BESIII:2022izy and Belle Belle:2021vyq ,
while the two scenarios in current work as well as the fitting based on
topological diagram Zhao:2018mov can give consistent predictions. Although the
branching ratio of $\Lambda_{c}^{+}\to p\pi^{0}$ haven’t been measured yet, an
upper limit has been updated by Belle Belle:2021mvw , which is supported also
by current fitting in both two scenarios consistent with the fittings in
Geng:2019xbo and Zhao:2018mov . For the recent measured decay asymmetry of
$\Lambda_{c}\to\Sigma^{0}K^{+}$ Belle:2022uod , both current fitting in Fit-
III and the pole model calculation Zou:2019kzq converge well.

Table 8: The comparison of theoretical predictions for branching fractions and
decay asymmetries in different methods and the latest measured experimental
values.

channel | Fit-III | Fit-III’ | GLT Geng:2019xbo | HXH Huang:2021aqu | ZWHY Zhao:2018mov | ZXMC Zou:2019kzq ; Meng:2020euv | Expt.
---|---|---|---|---|---|---|---
$10^{2}\mathcal{B}(\Lambda_{c}^{+}\to\Lambda^{0}\pi^{+})$ | $1.30_{-0.14}^{+0.12}$ | $1.27_{-0.09}^{+0.08}$ | $1.30\pm 0.07$ | $1.307\pm 0.069$ | $1.32\pm 0.34$ | $1.30$ | $1.30\pm 0.07$
$10^{2}\mathcal{B}(\Lambda_{c}^{+}\to pK_{S})$ | $1.65\pm{0.11}$ | $1.59\pm{0.10}$ | $1.57\pm 0.08$ | $1.587\pm 0.077$ | $1.57\pm 0.05$ | $1.06$ | $1.59\pm 0.08$
$10^{2}\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{0}\pi^{+})$ | $1.27\pm{0.09}$ | $1.30\pm{0.06}$ | $1.27\pm 0.06$ | $1.272\pm 0.056$ | $1.26\pm 0.32$ | $2.24$ | $1.29\pm 0.07$
$10^{2}\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{+}\pi^{0})$ | $1.27\pm{0.09}$ | $1.30\pm{0.06}$ | $1.27\pm 0.06$ | $1.283\pm 0.057$ | $1.23\pm 0.17$ | $2.24$ | $1.25\pm 0.10$
$10^{2}\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{+}\eta)$ | $0.30_{-0.07}^{+0.06}$ | $0.31\pm{0.05}$ | $0.32\pm 0.13$ | $0.45\pm 0.19$ | $0.47\pm 0.22$ | $0.74$ | $0.44\pm 0.20$
|  |  |  |  |  |  |  $0.314\pm 0.044$ Belle:2022bsi
$10^{2}\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{+}\eta^{\prime})$ | $0.39\pm{0.08}$ | $0.24\pm{0.05}$ | $1.44\pm 0.56$ | $1.5\pm 0.60$ | $0.93\pm 0.28$ | - | $1.50\pm 0.60$
|  |  |  |  |  |  |  $0.416\pm 0.085$ Belle:2022bsi
$10^{2}\mathcal{B}(\Lambda_{c}^{+}\to\Xi^{0}K^{+})$ | $0.50_{-0.09}^{+0.06}$ | $0.38\pm{0.03}$ | $0.56\pm 0.09$ | $0.548\pm 0.068$ | $0.59\pm 0.17$ | $0.73$ | $0.55\pm 0.07$
$10^{3}\mathcal{B}(\Lambda_{c}^{+}\to p\eta)$ | $1.27\pm{0.11}$ | $1.36_{-0.12}^{+0.11}$ | $1.15\pm 0.27$ | $1.27\pm 0.24$ | $1.14\pm 0.35$ | $1.28$ | $1.42\pm 0.12$
$10^{4}\mathcal{B}(\Lambda_{c}^{+}\to p\eta^{\prime})$ | $4.65_{-0.77}^{+0.79}$ | $5.93_{-0.71}^{+0.73}$ | $24.5\pm 14.6$ | $27\pm 38$ | $7.1\pm 1.4$ | - |  $4.73\pm 0.97$ Belle:2021vyq
|  |  |  |  |  |  |  $5.62^{+2.46}_{-2.04}\pm 0.26$ BESIII:2022izy
$10^{4}\mathcal{B}(\Lambda_{c}^{+}\to\Lambda^{0}K^{+})$ | $6.54_{-0.49}^{+0.42}$ | $6.62_{-0.22}^{+0.23}$ | $6.5\pm 1.0$ | $6.4\pm 1.0$ | $5.9\pm 1.7$ | $10.7$ |  $6.21\pm 0.61$ BESIII:2022tnm
|  |  |  |  |  |  |  $6.57\pm 0.40$ Belle:2022uod
$10^{4}\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{0}K^{+})$ | $3.71_{-0.36}^{+0.39}$ | $3.56_{-0.69}^{+0.68}$ | $5.4\pm 0.7$ | $5.04\pm 0.56$ | $5.5\pm 1.6$ | $7.2$ |  $4.7\pm 0.95$ BESIII:2022wxj
|  |  |  |  |  |  |  $3.58\pm 0.28$ Belle:2022uod
$10^{4}\mathcal{B}(\Lambda_{c}^{+}\to n\pi^{+})$ | $6.47_{-1.55}^{+1.33}$ | $8.15_{-0.67}^{+0.69}$ | $8.5\pm 2.0$ | $3.5\pm 1.1$ | $7.7\pm 2.0$ | - |  $6.6\pm 1.3$ BESIII:2022bkj
$10^{4}\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{+}K_{S})$ | $1.99_{-0.46}^{+0.49}$ | $3.11_{-0.34}^{+0.35}$ | $5.45\pm 0.75$ | $1.03\pm 0.42$ | $9.55\pm 2.4$ | $7.2$ |  $4.8\pm 1.4$ BESIII:2022wxj
$10^{4}\mathcal{B}(\Lambda_{c}^{+}\to p\pi^{0})$ | $0.51_{-0.61}^{+0.59}$ | $0.16\pm{0.09}$ | $1.2\pm 1.2$ | $44.5\pm 8.5$ | $0.8_{-0.8}^{+0.9}$ | $1.26$ |  $<0.80$ Belle:2021mvw
$10^{2}\mathcal{B}(\Xi_{c}^{0}\to\Xi^{-}\pi^{+})$ | $2.43_{-0.64}^{+0.60}$ | $0.70_{-0.22}^{+0.25}$ | $2.21\pm 0.14$ | $1.21\pm 0.21$ | $1.93\pm 0.28$ | $6.47$ | $1.43\pm 0.32$
$10^{3}\mathcal{B}(\Xi_{c}^{0}\to\Xi^{-}K^{+})$ | $0.60_{-0.67}^{+1.03}$ | $0.31_{-0.09}^{+0.11}$ | $0.98\pm 0.06$ | $0.47\pm 0.083$ | $0.56\pm 0.08$ | $3.90$ | $0.38\pm 0.12$
$10^{3}\mathcal{B}(\Xi_{c}^{0}\to\Lambda^{0}K_{S})$ | $3.34_{-0.98}^{+0.97}$ | $3.79_{-0.61}^{+0.66}$ | $5.25\pm 0.3$ | $3.34\pm 0.65$ | $4.16\pm 2.51$ | $6.65$ | $3.34\pm 0.67$
$10^{3}\mathcal{B}(\Xi_{c}^{0}\to\Sigma^{0}K_{S})$ | $0.74_{-0.30}^{+0.25}$ | $0.73_{-0.25}^{+0.25}$ | $0.4\pm 0.4$ | $0.69\pm 0.24$ | $3.96\pm 0.25$ | $0.2$ | $0.69\pm 0.24$
$10^{3}\mathcal{B}(\Xi_{c}^{0}\to\Sigma^{+}K^{-})$ | $1.86_{-0.71}^{+0.45}$ | $1.74_{-0.51}^{+0.41}$ | $5.9\pm 1.1$ | $2.21\pm 0.68$ | $22.0\pm 5.7$ | $4.6$ | $1.8\pm 0.4$
$10^{2}\mathcal{B}(\Xi_{c}^{+}\to\Xi^{0}\pi^{+})$ | $0.55_{-0.20}^{+0.19}$ | $0.81\pm{0.11}$ | $0.38\pm 0.20$ | $0.54\pm 0.18$ | $0.93\pm 0.36$ | $1.72$ | $1.6\pm 0.8$
$\alpha(\Lambda_{c}^{+}\to\Lambda^{0}\pi^{+})$ | $-0.75\pm{0.01}$ | $-0.75\pm{0.01}$ | $-0.87\pm 0.10$ | $-0.841\pm 0.083$ | - | $-0.93$ | $-0.84\pm 0.09$
|  |  |  |  |  |  |  $-0.755\pm 0.006$ Belle:2022uod
$\alpha(\Lambda_{c}^{+}\to pK_{S})$ | $-0.29\pm{0.24}$ | $-0.57\pm{0.21}$ | $-0.90^{+0.22}_{-0.10}$ | $0.19\pm 0.41$ | - | $-0.75$ | $0.18\pm 0.45$
$\alpha(\Lambda_{c}^{+}\to\Sigma^{0}\pi^{+})$ | $-0.47\pm{0.03}$ | $-0.47\pm{0.03}$ | $-0.35\pm 0.27$ | $-0.605\pm 0.088$ | - | $-0.76$ | $-0.73\pm 0.18$
|  |  |  |  |  |  |  $-0.463\pm 0.018$ Belle:2022uod
$\alpha(\Lambda_{c}^{+}\to\Sigma^{+}\pi^{0})$ | $-0.47\pm{0.03}$ | $-0.47\pm{0.03}$ | $-0.35\pm 0.27$ | $-0.603\pm 0.088$ | - | $-0.76$ | $-0.55\pm 0.11$
|  |  |  |  |  |  |  $-0.48\pm 0.03$ Belle:2022bsi
$\alpha(\Xi_{c}^{0}\to\Xi^{-}\pi^{+})$ | $-0.64_{-0.09}^{+0.11}$ | $-0.64\pm{0.07}$ | $-0.98^{+0.07}_{-0.02}$ | $-0.56\pm 0.32$ | - | $-0.95$ | $-0.64\pm 0.05$
$\alpha(\Lambda_{c}^{+}\to\Sigma^{+}\eta)$ | $-0.95\pm{0.06}$ | $-0.96\pm{0.05}$ | $-0.40\pm 0.47$ | $0.3\pm 3.8$ | - | $-0.95$ |  $-0.99\pm 0.06$ Belle:2022bsi
$\alpha(\Lambda_{c}^{+}\to\Sigma^{+}\eta^{\prime})$ | $-0.47\pm{0.05}$ | $-0.43\pm{0.07}$ | $1.00_{-0.17}^{+0.00}$ | $0.8\pm 1.9$ | - | - |  $-0.46\pm 0.07$ Belle:2022bsi
$\alpha(\Lambda_{c}^{+}\to\Lambda^{0}K^{+})$ | $-0.57\pm{0.06}$ | $-0.55\pm{0.06}$ | $0.32\pm 0.32$ | $-0.24\pm 0.15$ | - | $-0.96$ |  $-0.585\pm 0.052$ Belle:2022uod
$\alpha(\Lambda_{c}^{+}\to\Sigma^{0}K^{+})$ | $-0.67\pm{0.05}$ | $-0.97_{-0.01}^{+0.02}$ | $-1.00_{-0.00}^{+0.06}$ | $-0.953\pm 0.040$ | - | $-0.73$ |  $-0.55\pm 0.20$ Belle:2022uod
$\alpha(\Lambda_{c}^{+}\to\Xi^{0}K^{+})$ | $0.99\pm{0.01}$ | $0.91_{-0.04}^{+0.03}$ | $0.94_{-0.11}^{+0.06}$ | $0.866\pm 0.090$ | - | $0.90$ |

The dynamical theoretical calculation, the kinetic fitting calculation and the
experimental measurement provide complementary ways to explore two-body weak
decays of charmed baryon from different aspects. Each of them can help the
other two methodologies to improve their predictions/measured values. For
example, by taking more experimental values into account as inputs, more
reliable predictions can be produced by the fitting methodology. On the other
hand, calculation from both fitting methods and a model calculation provide a
strong motivation to revise $\alpha(\Lambda_{c}^{+}\to pK_{S})$ in experiment.
Now except $\Xi_{c}^{+}\to\Sigma^{+}\overline{K}^{0}$ (or
$\Xi_{c}^{+}\to\Sigma^{+}{K}_{S}$ ) all the branching fractions of CF modes as
well as part of SCS modes have been measured. It is highly anticipated that
more observables and modes can be accessible, which provide more evidence to
confirm and improve fitting calculation in current work, in the coming years.

##  4 Summary and conclusion

In this work, based on SU(3) flavor symmetry we have carried out a global
fitting analysis for the two-body weak decays of antitriplet charmed baryons
incorporating the latest experimental data. Two scenarios for fitting, both in
the flavor symmetry strictly respected case and the SU(3) breaking case, are
considered. In the framework of keeping exact SU(3) symmetry, more amplitudes
in terms of SU(3) irreducible representation have been taken into account
benefited from new data. As for the SU(3) breaking effect, in current working
scenario only SCS processes receive changes while CF and DCS processes keep
unchanged.

Making use of the data of the measured 20 branching fractions and 9 decay
asymmetries, we obtain the ranges of 18 parameters in SU(3) keeping scenario
as well as the 24 ones in SU(3) breaking case, though some of which are with
large uncertainties. In general, the goodness is better by incorporating SU(3)
symmetry breaking effect. In both scenarios, we have calculated branching
fractions and decay asymmetries for all the CF, SCS and DCS modes of
antitriplet charmed baryon weak decays. Most of the predictions for branching
fractions and decay asymmetries are consistent well with experimental
measurements. For more details, we have

  * •

Among the branching fractions of all the 16 CF channels, there are still 4 of
them unmeasured. It is highly anticipated to be measured in the upcoming years
for the four modes
$\Xi_{c}^{+}\to\Sigma^{+}K_{S},\Xi_{c}^{0}\to\Xi^{0}\pi^{0},\Xi^{0}\eta,\Xi^{0}\eta^{\prime}$.

  * •

Current data prefers SU(3) flavor symmetry breaking in charmed baryon weak
decays.

  * •

We propose three ways to further explore SU(3) symmetry. (i) Measuring the
branching fraction of CF decay $\Xi_{c}^{0}\to\Xi^{0}\pi^{0}$ for the large
difference exists in two scenarios. (ii) Measuring the decay asymmetries in
the CF decays
$\Xi_{c}^{0}\to\Lambda^{0}K_{S},\Sigma^{0}K_{S},\Xi^{0}\pi^{0},\Xi^{0}\eta,\Xi^{0}\eta^{\prime}$
as their signs are opposite correspondingly in the two cases. (iii) Measuring
series of branching fraction ratios, some of which have been presented in
Table 7.

  * •

Fittings in different schemes of current work, as well as independent fittings
by other groups all indicate a 2 or 3 times smaller value for the branching
fraction of $\Xi_{c}^{+}\to\Xi^{0}\pi^{+}$. Hence we remove this mode as well
the correlated mode $\Xi_{c}^{0}\to\Xi^{-}\pi^{+}$ from the inputs of fitting.
We hope experiments in the near future can help clarify the two modes.

  * •

Predictions from most fitting works and pole model calculation prefer a
negative $\alpha(\Lambda_{c}^{+}\to pK_{S})$ while BESIII provided a positive
central value with large uncertainty. Future measurements from both BESIII and
other experiments are expected to make a further clarification.

  * •

Both fittings (in IRA and TDA) and pole model calculation give a positive and
large value for $\alpha(\Lambda_{c}^{+}\to\Xi^{0}K^{+})$, then a check from
experiment would be interesting.

Together with other results presented in Table 6, all the predictions of
current work are expected to be checked and improved by the upcoming
experiments with the more accumulated data.

###### Acknowledgements.

The authors benefited by discussions with Prof. H.-Y. Cheng, C.-Q. Geng and
C.-W. Liu. The communication with Prof. C. D. Lu, X.R. Lyu, F. -S. Yu, P.-R.
Li and T. Luo are acknowledged. This work is supported by NSFC under Grant
Nos. U1932104, 12142502 and 12047503, and by Guangdong Provincial Key
Laboratory of Nuclear Science with No. 2019B121203010. Note Added All the
authors contribute equally and they are co-first authors, while F. Xu is the
corresponding author. Appendix

##  Appendix A The comparison among different schemes

We present 3 groups of fitting schemes here, in which those with a prime stand
for SU(3) symmetry being strictly respected while those without a prime
receiving symmetry breaking terms. In particular, the first group (Fit-I/I’)
contains all the values of all the observables in Table 4 while the second
group (Fit-II/II’) delete all the ratios between branching fractions in the
definition of $\chi^{2}$ in Fit-I/I’. Then the

Table 9: Predictions of branching fractions and decay asymmetries in different
schemes, in which the scheme with a prime stands for the SU(3) respected
scenario while these without a prime containing SU(3) breaking effect. The
corresponding experimental values for a comparison are given in the last
column.

Channel | Fit-I | Fit-I’ | Fit-II | Fit-II’ | Fit-III | Fit-III’ | Expt.
---|---|---|---|---|---|---|---
$\qquad(\chi_{\rm{min}}^{2}/{\rm{d.o.f.}}$ | 1.16 | 2.60 | 1.98 | 2.20 | 1.27 | 2.15) |
$10^{2}\mathcal{B}(\Lambda_{c}^{+}\to\Lambda\pi^{+})$ | $1.32_{-0.14}^{+0.12}$ | $1.32_{-0.14}^{+0.12}$ | $1.30_{-0.14}^{+0.12}$ | $1.28_{-0.13}^{+0.12}$ | $1.30_{-0.14}^{+0.12}$ | $1.27_{-0.09}^{+0.08}$ | $1.30\pm 0.07$
$10^{2}\mathcal{B}(\Lambda_{c}^{+}\to pK_{S})$ | $1.59_{-0.38}^{+0.33}$ | $1.58_{-0.30}^{+0.25}$ | $1.66_{-0.17}^{+0.15}$ | $1.56_{-0.13}^{+0.12}$ | $1.65\pm{0.11}$ | $1.59\pm{0.10}$ | $1.59\pm 0.08$
$10^{2}\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{0}\pi^{+})$ | $1.28\pm{0.09}$ | $1.30\pm{0.09}$ | $1.27\pm{0.09}$ | $1.31\pm{0.09}$ | $1.27\pm{0.09}$ | $1.30\pm{0.06}$ | $1.29\pm 0.07$
$10^{2}\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{+}\pi^{0})$ | $1.28\pm{0.09}$ | $1.30\pm{0.09}$ | $1.27\pm{0.09}$ | $1.31\pm{0.09}$ | $1.27\pm{0.09}$ | $1.30\pm{0.06}$ | $1.25\pm 0.10$
$10^{2}\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{+}\eta)$ | $0.30\pm{0.06}$ | $0.29\pm{0.06}$ | $0.28\pm{0.06}$ | $0.32_{-0.07}^{+0.06}$ | $0.30_{-0.07}^{+0.06}$ | $0.31\pm{0.05}$ | $0.44\pm 0.20$
|  |  |  |  |  |  |  $0.314\pm 0.044$ Belle:2022bsi
$10^{2}\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{+}\eta^{\prime})$ | $0.40\pm{0.08}$ | $0.06_{-0.74}^{+0.73}$ | $0.42\pm{0.08}$ | $0.26\pm{0.06}$ | $0.39\pm{0.08}$ | $0.24\pm{0.05}$ | $1.50\pm 0.60$
|  |  |  |  |  |  |  $0.416\pm 0.085$ Belle:2022bsi
$10^{2}\mathcal{B}(\Lambda_{c}^{+}\to\Xi^{0}K^{+})$ | $0.60_{-0.09}^{+0.07}$ | $0.38_{-0.08}^{+0.06}$ | $0.50_{-0.08}^{+0.06}$ | $0.38_{-0.08}^{+0.06}$ | $0.50_{-0.09}^{+0.06}$ | $0.38\pm{0.03}$ | $0.55\pm 0.07$
$10^{3}\mathcal{B}(\Lambda_{c}^{+}\to p\eta)$ | $1.31_{-0.25}^{+0.23}$ | $1.27_{-0.22}^{+0.21}$ | $1.23_{-0.17}^{+0.16}$ | $1.36\pm{0.12}$ | $1.27\pm{0.11}$ | $1.36_{-0.12}^{+0.11}$ | $1.42\pm 0.12$
$10^{4}\mathcal{B}(\Lambda_{c}^{+}\to p\eta^{\prime})$ | $5.04_{-0.78}^{+0.75}$ | $3.90_{-0.94}^{+0.97}$ | $4.95\pm{0.77}$ | $6.14_{-0.91}^{+0.95}$ | $4.65_{-0.77}^{+0.79}$ | $5.93_{-0.71}^{+0.73}$ |  $4.73\pm 0.97$ Belle:2021vyq
|  |  |  |  |  |  |  $5.62^{+2.46}_{-2.04}\pm 0.26$ BESIII:2022izy
$10^{4}\mathcal{B}(\Lambda_{c}^{+}\to\Lambda K^{+})$ | $6.55_{-0.50}^{+0.42}$ | $6.62_{-0.49}^{+0.42}$ | $6.68_{-0.50}^{+0.42}$ | $6.66_{-0.49}^{+0.42}$ | $6.54_{-0.49}^{+0.42}$ | $6.62_{-0.22}^{+0.23}$ |  $6.21\pm 0.61$ BESIII:2022tnm
|  |  |  |  |  |  |  $6.57\pm 0.40$ Belle:2022uod
$10^{4}\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{0}K^{+})$ | $3.63_{-0.36}^{+0.38}$ | $3.65_{-0.45}^{+0.49}$ | $3.78_{-0.36}^{+0.39}$ | $3.58_{-0.45}^{+0.49}$ | $3.71_{-0.36}^{+0.39}$ | $3.56_{-0.69}^{+0.68}$ |  $4.7\pm 0.95$ BESIII:2022wxj
|  |  |  |  |  |  |  $3.58\pm 0.28$ Belle:2022uod
$10^{4}\mathcal{B}(\Lambda_{c}^{+}\to n\pi^{+})$ | $4.67_{-1.38}^{+1.17}$ | $8.43_{-1.97}^{+1.66}$ | $5.99_{-1.48}^{+1.28}$ | $8.19_{-1.92}^{+1.62}$ | $6.47_{-1.55}^{+1.33}$ | $8.15_{-0.67}^{+0.69}$ |  $6.6\pm 1.3$ BESIII:2022bkj
$10^{4}\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{+}K_{S})$ | $1.75_{-0.48}^{+0.50}$ | $2.67_{-0.64}^{+0.66}$ | $1.83_{-0.44}^{+0.46}$ | $3.14_{-0.72}^{+0.74}$ | $1.99_{-0.46}^{+0.49}$ | $3.11_{-0.34}^{+0.35}$ |  $4.8\pm 1.4$ BESIII:2022wxj
$10^{4}\mathcal{B}(\Lambda_{c}^{+}\to p\pi^{0})$ | $2.01_{-1.11}^{+1.08}$ | $1.45_{-1.04}^{+1.01}$ | $0.02\pm{0.10}$ | $0.15_{-0.17}^{+0.19}$ | $0.51_{-0.61}^{+0.59}$ | $0.16\pm{0.09}$ |  $<0.80$ Belle:2021mvw
$10^{2}\mathcal{B}(\Xi_{c}^{0}\to\Xi^{-}\pi^{+})$ | $1.40_{-0.28}^{+0.32}$ | $1.27_{-0.29}^{+0.32}$ | $1.01_{-0.24}^{+0.27}$ | $0.92_{-0.23}^{+0.26}$ | $2.43_{-0.64}^{+0.60}$ | $0.70_{-0.22}^{+0.25}$ | $1.43\pm 0.32$
$10^{3}\mathcal{B}(\Xi_{c}^{0}\to\Xi^{-}K^{+})$ | $0.36_{-12.22}^{+12.44}$ | $0.56_{-17.01}^{+17.17}$ | $0.39_{-13.95}^{+13.88}$ | $0.40_{-14.53}^{+14.65}$ | $0.60_{-0.67}^{+1.03}$ | $0.31_{-0.09}^{+0.11}$ | $0.38\pm 0.12$
$10^{3}\mathcal{B}(\Xi_{c}^{0}\to\Lambda K_{S})$ | $3.25_{-0.59}^{+0.68}$ | $3.15_{-0.58}^{+0.66}$ | $3.96_{-0.68}^{+0.76}$ | $4.25_{-0.91}^{+0.93}$ | $3.34_{-0.98}^{+0.97}$ | $3.79_{-0.61}^{+0.66}$ | $3.34\pm 0.67$
$10^{3}\mathcal{B}(\Xi_{c}^{0}\to\Sigma^{0}K_{S})$ | $0.53_{-0.50}^{+0.40}$ | $0.57_{-0.80}^{+0.58}$ | $0.78_{-0.40}^{+0.32}$ | $0.79_{-0.42}^{+0.31}$ | $0.74_{-0.30}^{+0.25}$ | $0.73_{-0.25}^{+0.25}$ | $0.69\pm 0.24$
$10^{3}\mathcal{B}(\Xi_{c}^{0}\to\Sigma^{+}K^{-})$ | $1.71_{-2.18}^{+1.43}$ | $1.51_{-1.08}^{+0.61}$ | $1.74_{-2.03}^{+1.35}$ | $1.71_{-2.05}^{+1.22}$ | $1.86_{-0.71}^{+0.45}$ | $1.74_{-0.51}^{+0.41}$ | $1.8\pm 0.4$
$10^{2}\mathcal{B}(\Xi_{c}^{+}\to\Xi^{0}\pi^{+})$ | $0.66\pm{0.21}$ | $0.83_{-0.27}^{+0.24}$ | $0.51\pm{0.19}$ | $0.83_{-0.26}^{+0.24}$ | $0.55_{-0.20}^{+0.19}$ | $0.81\pm{0.11}$ | $1.6\pm 0.8$
$\alpha(\Lambda_{c}^{+}\to\Lambda\pi^{+})$ | $-0.75\pm{0.01}$ | $-0.76\pm{0.01}$ | $-0.76\pm{0.01}$ | $-0.75\pm{0.01}$ | $-0.75\pm{0.01}$ | $-0.75\pm{0.01}$ | $-0.84\pm 0.09$
|  |  |  |  |  |  |  $-0.755\pm 0.006$ Belle:2022uod
$\alpha(\Lambda_{c}^{+}\to pK_{S})$ | $-0.95\pm{0.06}$ | $-0.98\pm{0.04}$ | $-0.65_{-0.19}^{+0.18}$ | $-0.50\pm{0.22}$ | $-0.29\pm{0.24}$ | $-0.57\pm{0.21}$ | $0.18\pm 0.45$
$\alpha(\Lambda_{c}^{+}\to\Sigma^{0}\pi^{+})$ | $-0.47\pm{0.03}$ | $-0.47\pm{0.03}$ | $-0.48\pm{0.03}$ | $-0.47\pm{0.03}$ | $-0.47\pm{0.03}$ | $-0.47\pm{0.03}$ | $-0.73\pm 0.18$
|  |  |  |  |  |  |  $-0.463\pm 0.018$ Belle:2022uod
$\alpha(\Lambda_{c}^{+}\to\Sigma^{+}\pi^{0})$ | $-0.47\pm{0.03}$ | $-0.48\pm{0.03}$ | $-0.48\pm{0.03}$ | $-0.47\pm{0.03}$ | $-0.47\pm{0.03}$ | $-0.47\pm{0.03}$ | $-0.55\pm 0.11$
|  |  |  |  |  |  |  $-0.48\pm 0.03$ Belle:2022bsi
$\alpha(\Xi_{c}^{0}\to\Xi^{-}\pi^{+})$ | $-0.65\pm{0.05}$ | $-0.71\pm{0.05}$ | $-0.65\pm{0.06}$ | $-0.66\pm{0.06}$ | $-0.64_{-0.09}^{+0.11}$ | $-0.64\pm{0.07}$ | $-0.64\pm 0.05$
$\alpha(\Lambda_{c}^{+}\to\Sigma^{+}\eta)$ | $-0.97\pm{0.06}$ | $-0.39\pm{0.09}$ | $-0.95\pm{0.06}$ | $-0.96\pm{0.05}$ | $-0.95\pm{0.06}$ | $-0.96\pm{0.05}$ |  $-0.99\pm 0.06$ Belle:2022bsi
$\alpha(\Lambda_{c}^{+}\to\Sigma^{+}\eta^{\prime})$ | $-0.50\pm{0.05}$ | $-0.88\pm{0.06}$ | $-0.48\pm{0.05}$ | $-0.41\pm{0.07}$ | $-0.47\pm{0.05}$ | $-0.43\pm{0.07}$ |  $-0.46\pm 0.07$ Belle:2022bsi
$\alpha(\Lambda_{c}^{+}\to\Lambda K^{+})$ | $-0.61_{-0.06}^{+0.05}$ | $-0.55\pm{0.06}$ | $-0.58_{-0.06}^{+0.05}$ | $-0.56\pm{0.06}$ | $-0.57\pm{0.06}$ | $-0.55\pm{0.06}$ |  $-0.585\pm 0.052$ Belle:2022uod
$\alpha(\Lambda_{c}^{+}\to\Sigma^{0}K^{+})$ | $-0.63\pm{0.05}$ | $-0.98\pm{0.01}$ | $-0.66\pm{0.05}$ | $-0.97\pm{0.02}$ | $-0.67\pm{0.05}$ | $-0.97_{-0.01}^{+0.02}$ |  $-0.55\pm 0.20$ Belle:2022uod
$\alpha(\Lambda_{c}^{+}\to\Xi^{0}K^{+})$ | $0.98\pm{0.01}$ | $0.92_{-0.04}^{+0.03}$ | $0.996_{-0.006}^{+0.005}$ | $0.90_{-0.04}^{+0.03}$ | $0.99\pm{0.01}$ | $0.91_{-0.04}^{+0.03}$ |

Table 10: Predictions of ratios between branching fractions in different
schemes, in which the definition of $\mathcal{R}_{i}$ are given in Table 4.

Channel | Fit-I | Fit-I’ | Fit-II | Fit-II’ | Fit-III | Fit-III’ | Expt.
---|---|---|---|---|---|---|---
$\mathcal{R}_{1}$ | $0.24\pm{0.04}$ | $0.22\pm{0.04}$ | $0.22\pm{0.04}$ | $0.25_{-0.16}^{+0.06}$ | $0.23_{-0.06}^{+0.05}$ | $0.24\pm 0.04$ |  $0.25\pm 0.03$ Belle:2022bsi
$\mathcal{R}_{2}$ | $0.31\pm{0.04}$ | $0.05\pm{0.03}$ | $0.33_{-0.06}^{+0.05}$ | $0.20_{-0.43}^{+0.04}$ | $0.30_{-0.07}^{+0.06}$ | $0.19\pm 0.04$ |  $0.33\pm 0.06$ Belle:2022bsi
$\mathcal{R}_{3}$ | $1.31_{-0.26}^{+0.24}$ | $0.20_{-0.24}^{+0.22}$ | $1.49_{-0.36}^{+0.34}$ | $0.81_{-1.37}^{+0.14}$ | $1.30_{-0.39}^{+0.38}$ | $0.77_{-0.21}^{+0.20}$ |  $1.34\pm 0.29$ Belle:2022bsi
$10^{2}\mathcal{R}_{4}$ | $4.98_{-0.34}^{+0.30}$ | $5.04_{-0.34}^{+0.30}$ | $5.13_{-0.38}^{+0.37}$ | $5.22_{-0.40}^{+0.38}$ | $5.03_{-0.65}^{+0.51}$ | $5.22_{-0.41}^{+0.39}$ |  $4.78\pm 0.39$ BESIII:2022tnm
|  |  |  |  |  |  |  $5.05\pm 0.16$ Belle:2022uod
$10^{2}\mathcal{R}_{5}$ | $2.85_{-0.26}^{+0.21}$ | $2.81_{-0.17}^{+0.14}$ | $2.98_{-0.18}^{+0.19}$ | $2.74\pm{0.13}$ | $2.93_{-0.35}^{+0.31}$ | $2.74_{-0.14}^{+0.13}$ |  $3.61\pm 0.73$ BESIII:2022wxj
|  |  |  |  |  |  |  $2.78\pm 0.16$ Belle:2022uod
$\mathcal{R}_{6}$ | $0.97\pm{0.07}$ | $0.99\pm{0.07}$ | $0.97\pm{0.08}$ | $1.02\pm{0.08}$ | $0.97_{-0.13}^{+0.11}$ | $1.02\pm 0.09$ | $0.98\pm 0.05$
$\mathcal{R}_{7}$ | $0.46\pm{0.05}$ | $0.49\pm{0.05}$ | $0.79\pm{0.11}$ | $0.93_{-0.18}^{+0.19}$ | $0.27_{-0.09}^{+0.05}$ | $1.09\pm 0.33$ | $0.225\pm 0.013$
$10^{2}\mathcal{R}_{8}$ | $7.55_{-3.58}^{+3.69}$ | $8.85_{-2.30}^{+2.90}$ | $15.31_{-7.35}^{+7.54}$ | $16.97_{-5.92}^{+6.31}$ | $6.04_{-2.64}^{+2.38}$ | $20.58_{-9.13}^{+9.67}$ | $3.8\pm 0.7$
$10^{2}\mathcal{R}_{9}$ | $12.23_{-6.49}^{+5.66}$ | $11.89_{-2.34}^{+1.92}$ | $17.28_{-10.08}^{+10.22}$ | $18.64_{-6.78}^{+6.13}$ | $7.64_{-3.05}^{+2.49}$ | $24.88_{-10.93}^{+10.98}$ | $12.3\pm 1.22$
$10^{2}\mathcal{R}_{10}$ | $2.61_{-0.14}^{+0.15}$ | $4.38\pm{0.03}$ | $3.90_{-0.18}^{+0.19}$ | $4.39\pm{0.02}$ | $2.46_{-2.76}^{+4.26}$ | $4.40\pm{0.02}$ | $2.75\pm 0.57$

third group is defined based on Fit-II/II’ by subtracting
$\alpha(\Lambda_{c}^{+}\to pK_{S})$,
$\mathcal{B}(\Xi_{c}^{0}\to\Xi^{-}\pi^{+})$ and
$\mathcal{B}(\Xi_{c}^{+}\to\Xi^{0}\pi^{+})$.

In Table 9 we show predictions of branching fractions and decay asymmetries in
various schemes by comparing them with experimental values. To catch a glimpse
of goodness of fit, explicit values of $\chi^{2}_{\rm{min}}/{\rm{d.o.f.}}$ in
each scheme are also listed therein. We also theoretically calculate the
ratios between particular modes which have been measured as an auxiliary
illustration in Table 10.

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