<EMAIL_ADDRESS>
# Measurements of $K_{S}^{0}$-$K_{L}^{0}$ asymmetries in the decays
$\Lambda_{c}^{+}\to pK_{L,S}^{0}$, $pK_{L,S}^{0}\pi^{+}\pi^{-}$ and
$pK_{L,S}^{0}\pi^{0}$

###### Abstract

Using $e^{+}e^{-}$ annihilation data sets corresponding to an integrated
luminosity of 4.5 $\text{fb}^{-1}$, collected with the BESIII detector at
center-of-mass energies between 4.600 and 4.699 GeV, we report the first
measurements of the absolute branching fractions
$\mathcal{B}(\Lambda_{c}^{+}\to pK_{L}^{0})=(1.67\pm 0.06\pm 0.04)\%$,
$\mathcal{B}(\Lambda_{c}^{+}\to pK_{L}^{0}\pi^{+}\pi^{-})=(1.69\pm 0.10\pm
0.05)\%$, and $\mathcal{B}(\Lambda_{c}^{+}\to pK_{L}^{0}\pi^{0})=(2.02\pm
0.13\pm 0.05)\%$, where the first uncertainties are statistical and the second
systematic. Combining with the known branching fractions of
$\Lambda_{c}^{+}\to pK_{S}^{0}$, $\Lambda_{c}^{+}\to
pK_{S}^{0}\pi^{+}\pi^{-}$, and $\Lambda_{c}^{+}\to pK_{S}^{0}\pi^{0}$, we
present the first measurements of the $K_{S}^{0}$-$K_{L}^{0}$ asymmetries
$R(\Lambda_{c}^{+},K_{S,L}^{0}X)=\frac{\mathcal{B}(\Lambda_{c}^{+}\to
K_{S}^{0}X)-\mathcal{B}(\Lambda_{c}^{+}\to
K_{L}^{0}X)}{\mathcal{B}(\Lambda_{c}^{+}\to
K_{S}^{0}X)+\mathcal{B}(\Lambda_{c}^{+}\to K_{L}^{0}X)}$ in charmed baryon
decays: $R(\Lambda_{c}^{+},pK_{S,L}^{0})=-0.025\pm 0.031$,
$R(\Lambda_{c}^{+},pK_{S,L}^{0}\pi^{+}\pi^{-})=-0.027\pm 0.048$, and
$R(\Lambda_{c}^{+},pK_{S,L}^{0}\pi^{0})=-0.015\pm 0.046$. No significant
asymmetries within the uncertainties are observed.

###### Keywords:

$\Lambda_{c}^{+}$ baryon, $K^{0}_{S}$-$K^{0}_{L}$ asymmetry, the BESIII
detector

††arxiv: 1234.56789

##  1 Introduction

The lightest charmed baryon, $\Lambda_{c}^{+}$, provides a unique environment
for studying the behavior of light di-quarks in the presence of a heavy quark
Cheng:2021qpd . Its hadronic decays occur only through the weak interaction,
and various theoretical models have been proposed. These include the covariant
confined quark model Korner:1992wi ; Ivanov:1997ra , the pole model
Cheng:1991sn ; Xu:1992vc ; Cheng:1993gf ; Xu:1992sw ; Zenczykowski:1993jm ;
Sharma:1998rd , current algebra Korner:1978ec ; Uppal:1994pt , and SU(3)
flavor symmetry approaches Lu:2016ogy ; Savage:1989qr ; Kohara:1991ug ;
Verma:1995dk ; Chau:1995gk . Its decay falls into three categories: Cabibbo-
favored (CF) decays, singly Cabibbo-suppressed decays, and doubly Cabibbo-
suppressed (DCS) decays. The decay amplitudes of the CF and DCS modes are
expected to be proportional to the products of the Cabibbo-Kobayashi-Maskawa
elements $|V_{ud}^{*}V_{cs}|$ and $|V_{us}^{*}V_{cd}|$, respectively. The
ratio of their decays is approximately of the order of $\mathcal{O}(10^{-3})$,
resulting in a small branching fraction (BF) for the DCS decay and making it
challenging to observe directly in experiments.

In addition to direct measurements of DCS decays, the amplitudes of DCS modes
can be probed using the $K^{0}_{S}$-$K^{0}_{L}$ asymmetry in the decays into
neutral kaons, which arises from the interference between CF and DCS
amplitudes Bigi:1994aw ; Wang:2017gxe . The $K^{0}_{S}$-$K^{0}_{L}$ asymmetry
has been studied in the decays of charmed $D$ mesons, where the asymmetry is
defined by

$R(D,K_{S,L}^{0}X)=\frac{\mathcal{B}(D\to K^{0}_{S}X)-\mathcal{B}(D\to
K^{0}_{L}X)}{\mathcal{B}(D\to K^{0}_{S}X)+\mathcal{B}(D\to K^{0}_{L}X)},$ (1)

and $X$ can be $\pi^{0}$, $\eta$, $\eta^{\prime}$, $\omega$, $\rho^{0}$ or
$\phi$. A large asymmetry for $R(D^{0},K_{S,L}^{0}\pi^{0})$ was reported in a
previous measurement by the CLEO experiment as
$R(D^{0},K_{S,L}^{0}\pi^{0})=0.108\pm 0.025\pm 0.024$ CLEO:2007rhw , where the
first uncertainty is statistical and the second is systematic. The BESIII
experiment reported measurements of the $K^{0}_{S}$-$K^{0}_{L}$ asymmetries
$R(D^{0},K_{S,L}^{0}X)$, where $X=\phi$, $\eta$, $\omega$ BESIII:2022xhe .
Significant asymmetries were observed in $D^{0}\to K^{0}_{L}\eta$ and
$D^{0}\to K^{0}_{L}\eta^{\prime}$ decays with
$R(D^{0},K_{S,L}^{0}\eta)=0.080\pm 0.022$ and
$R(D^{0},K_{S,L}^{0}\eta^{\prime})=0.108\pm 0.035$, respectively. In addition,
this asymmetry has been investigated for the lightest charmed strange meson,
and $R(D_{s}^{+},K_{S,L}^{0}K^{+})$ was determined to be $(-2.1\pm 1.9\pm
1.6)\%$ BESIII:2019kfh . However, such measurements have not been made for the
decays of charmed baryons.

Using flavor SU(3) asymmetry Lu:2016ogy ; Savage:1989qr ; Kohara:1991ug ;
Verma:1995dk ; Chau:1995gk , theoretical predictions Wang:2017gxe  for
$K^{0}_{S}$-$K^{0}_{L}$ asymmetries have been made for charmed baryon two-body
decays into a light baryon and a neutral kaon. Similar to Equation 1, the
asymmetry of $\mathcal{B}(\Lambda_{c}^{+}\to K^{0}_{S}X)$ and
$\mathcal{B}(\Lambda_{c}^{+}\to K^{0}_{L}X)$ in charmed baryon decays is
defined as

$\displaystyle
R(\Lambda_{c}^{+},K_{S,L}^{0}X)=\frac{\mathcal{B}(\Lambda_{c}^{+}\to
K^{0}_{S}X)-\mathcal{B}(\Lambda_{c}^{+}\to
K^{0}_{L}X)}{\mathcal{B}(\Lambda_{c}^{+}\to
K^{0}_{S}X)+\mathcal{B}(\Lambda_{c}^{+}\to K^{0}_{L}X)},$ (2)

where $X$ is $p$, $p\pi^{+}\pi^{-}$ or $p\pi^{0}$. Equation 2 can be further
reduced as $R(\Lambda_{c}^{+}\to K^{0}_{S,L}X)\simeq-2r_{f}\cos{\delta_{f}}$,
where $r_{f}$ and $\delta_{f}$ are the relative strength and phase between the
DCS ($\Lambda_{c}^{+}\to K^{0}X$) and CF ($\Lambda_{c}^{+}\to\bar{K}^{0}X$)
amplitudes, respectively. The parameter $r_{f}$ is expected to be proportional
to the ratio $|V_{cd}^{*}V_{us}/V_{cs}^{*}V_{ud}|\sim\lambda^{2}$
ParticleDataGroup:2022pth . A non-zero asymmetry value indicates the presence
of DCS processes. The asymmetry of $\Lambda_{c}^{+}\to pK_{S,L}^{0}$ is
predicted to be in the range of ($-0.010$, $0.087$) in Ref. Wang:2017gxe . The
$K^{0}_{S}$-$K^{0}_{L}$ asymmetry is a promising observable with which to
search for the two-body DCS processes of charmed baryons.

In this paper, we report the first measurements of the absolute BFs of
$\Lambda_{c}^{+}\to pK^{0}_{L}$, $\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{+}\pi^{-}$
and $\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{0}$ based on $e^{+}e^{-}$ annihilation
data samples corresponding to a total integrated luminosity of 4.5 fb-1
collected at the center-of-mass (c.m.) energies $\sqrt{s}$ between 4.600 and
4.699 GeV. The luminosities are listed in Table 1 BESIII:2022dxl ;
BESIII:2022ulv . Using the results of $\mathcal{B}(\Lambda_{c}^{+}\to
pK^{0}_{S})$, $\mathcal{B}(\Lambda_{c}^{+}\to pK^{0}_{S}\pi^{+}\pi^{-})$, and
$\mathcal{B}(\Lambda_{c}^{+}\to pK^{0}_{S}\pi^{0})$ from the Particle Data
Group (PDG) ParticleDataGroup:2022pth , we present the $K^{0}_{S}$-$K^{0}_{L}$
asymmetries $R(\Lambda_{c}^{+},K_{S,L}^{0}X)$, where $X=p$, $p\pi^{+}\pi^{-}$
or $p\pi^{0}$. Charge conjugate channels are implied throughout this paper,
unless explicitly stated.

Table 1: The integrated luminosities at each c.m. energy BESIII:2022dxl ; BESIII:2022ulv . $\sqrt{s}$ (GeV) | Integrated luminosity $(\mbox{pb${}^{-1}$})$
---|---
4.600 | 586.9 $\pm$ 0.1 $\pm$ 3.9
4.612 | 103.7 $\pm$ 0.1 $\pm$ 0.6
4.628 | 521.5 $\pm$ 0.1 $\pm$ 2.8
4.641 | 551.7 $\pm$ 0.1 $\pm$ 2.9
4.661 | 529.4 $\pm$ 0.1 $\pm$ 2.8
4.682 | 1667.4 $\pm$ 0.2 $\pm$ 8.8
4.699 | 535.5 $\pm$ 0.1 $\pm$ 2.8

##  2 BESIII experiment and Monte Carlo simulation

The BESIII detector BESIII:2009fln  records symmetric $e^{+}e^{-}$ collisions
provided by the BEPCII storage ring Yu:2016cof , which operates at c.m.
energies ranging from 1.85 to 4.95 GeV, with a peak luminosity of $1.1\times
10^{33}\;\mathrm{cm}^{-2}\mathrm{s}^{-1}$ achieved at
$\sqrt{s}=3.773\;\mathrm{GeV}$. The BESIII detector has collected large data
samples in this energy region BESIII:2020nme . The cylindrical core of the
BESIII detector covers 93% of the full solid angle and consists of a helium-
based multilayer drift chamber (MDC), a plastic scintillator time-of-flight
system (TOF), and a CsI(Tl) electromagnetic calorimeter (EMC), which are all
enclosed in a superconducting solenoidal magnet providing a 1.0 T magnetic
field Huang:2022wuo . The solenoid is supported by an octagonal flux-return
yoke with resistive plate counter muon identification modules interleaved with
steel. The charged-particle momentum resolution at $1\,\mbox{GeV/$c$}$ is
$0.5\%$, and the $\mathrm{d}E/\mathrm{d}x$ resolution is $6\%$ for electrons
from Bhabha scattering. The EMC measures photon energies with a resolution of
$2.5\%$ ($5\%$) at $1\,\mbox{GeV}$ in the barrel (end cap) region. The time
resolution in the TOF barrel region is 68 ps, while that in the end cap region
was initially 110 ps. The end cap TOF system was upgraded in 2015 using multi-
gap resistive plate chamber technology, providing a time resolution of 60 ps
Li:2017jpg ; Guo:2017sjt ; Lange:2001uf . Of the data used in this analysis,
87% was with the upgraded end cap TOF.

Simulated samples generated with geant4-based GEANT4:2002zbu  Monte Carlo (MC)
software, which includes the geometric description of the BESIII detector and
the detector response performance YouZhengYun2008 ; Liang:2009zzb ;
Huang:2022wuo , are used to determine detection efficiencies and to estimate
potential background contributions. The simulation describes the beam energy
spread and the initial state radiation (ISR) in the $e^{+}e^{-}$ annihilations
with the generator kkmc Jadach:2000ir ; Jadach:1999vf . The inclusive MC
samples, corresponding to about 40 times the number of events of the data
samples, include the production of $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$
pairs, open charm processes, the ISR production of vector charmonium(-like)
states, and the continuum processes incorporated in kkmc Jadach:2000ir ;
Jadach:1999vf . The known decay modes are modeled with evtgen Lange:2001uf ;
Ping:2008zz  using BFs taken from the PDG ParticleDataGroup:2022pth , and the
remaining unknown charmonium decays are modeled with lundcharm Chen:2000tv ;
Lange:2001uf . Final state radiation from charged final state particles is
incorporated using photos Richter-Was:1992hxq . For the production of
$e^{+}e^{-}\to\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ events, the Born cross-
section line shape from BESIII measurements is used BESIII:2017kqg ;
BESIII:2023rwv . Exclusive $e^{+}e^{-}\to\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$
signal MC samples are generated with $\bar{\Lambda}_{c}^{-}$ decaying to
twelve specific tag modes (as described in Section 3) and $\Lambda_{c}^{+}$
decaying to $pK^{0}_{L}$, $pK^{0}_{L}\pi^{+}\pi^{-}$ and $pK^{0}_{L}\pi^{0}$.
The angular distribution of the decay $\Lambda_{c}^{+}\to pK^{0}_{L}$ is
modeled with decay asymmetry parameters obtained from Ref. BESIII:2019odb .
For processes from $\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{+}\pi^{-}$ and
$\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{0}$ channels, signal models are tunned
based on the data. Additional MC samples are generated to estimate
contributions from peaking background processes, where $\bar{\Lambda}_{c}^{-}$
decays into tag modes and $\Lambda_{c}^{+}$ decays into $pK^{0}_{S}$, $p\eta$,
$pK^{0}_{S}\pi^{0}$, and $pK^{0}_{S}\pi^{+}\pi^{-}$, with $K^{0}_{S}$ and
$\eta$ decaying inclusively. Each tag mode of the exclusive MC samples is
generated with the same number of events.

##  3 Data analysis

Taking advantage of the threshold production of the
$\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ pair, the double-tag (DT) method MARK-
III:1985hbd ; MARK-III:1987jsm ; Ke:2023qzc ; Li:2021iwf  is employed to study
$\Lambda_{c}^{+}\to pK^{0}_{L}$, $\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{+}\pi^{-}$
and $\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{0}$, where $K^{0}_{L}$ is reconstructed
by the missing-mass technique. A single-tag (ST) event is selected by tagging
a $\bar{\Lambda}_{c}^{-}$ baryon with one of the following twelve tag modes:
$\bar{p}K_{S}^{0}$, $\bar{p}K^{+}\pi^{-}$, $\bar{p}K_{S}^{0}\pi^{0}$,
$\bar{p}K_{S}^{0}\pi^{-}\pi^{+}$, $\bar{p}K^{+}\pi^{-}\pi^{0}$,
$\bar{p}\pi^{-}\pi^{+}$, $\bar{\Lambda}\pi^{-}$,
$\bar{\Lambda}\pi^{-}\pi^{0}$, $\bar{\Lambda}\pi^{-}\pi^{+}\pi^{-}$,
$\bar{\Sigma}^{0}\pi^{-}$, $\bar{\Sigma}^{-}\pi^{0}$, and
$\bar{\Sigma}^{-}\pi^{-}\pi^{+}$. The ST event selection criteria,
efficiencies, and yields are described in Ref. BESIII:2022xne . The signal
decays $\Lambda_{c}^{+}\to pK^{0}_{L}$, $pK^{0}_{L}\pi^{+}\pi^{-}$, and
$pK^{0}_{L}\pi^{0}$ are reconstructed using the remaining charged tracks and
photons recoiling against the ST $\bar{\Lambda}_{c}^{-}$ candidates, and
referred to as DT events.

Charged tracks are required to be within $|\cos\theta|<0.93$, where $\theta$
is the polar angle defined with respect to the $z$-axis, which is the symmetry
axis of the MDC. The distance of closest approach to the interaction point
(IP) must be less than 10 cm along the $z$ axis and less than 1 cm in the
perpendicular plane. Particle identification (PID) for charged tracks combines
measurements of the energy deposited in the MDC (d$E/$d$x$) and the flight
time in the TOF to form a likelihood value $\mathcal{L}(h)$ for each hadron
($h$) hypothesis, where $h=p,K$, or $\pi$. Charged tracks are identified as
protons if the proton hypothesis has the highest likelihood
($\mathcal{L}(p)>\mathcal{L}(K)$ and $\mathcal{L}(p)>\mathcal{L}(\pi)$), or as
pions if $\mathcal{L}(\pi)>\mathcal{L}(K)$ is satisfied.

Photon candidates are reconstructed from showers that are not associated with
any charged tracks in the EMC BESIII:2009fln . The deposited energy of each
shower in the EMC is required to be greater than 25 MeV in the barrel region
($|\cos\theta|<0.80$), and greater than 50 MeV in the end cap region
($0.86<|\cos\theta|<0.92$). The EMC time difference from the event start time
is required to be less than 700 ns, to exclude electronic noise and showers
unrelated to the events. The opening angle between each shower and $\bar{p}$
must be greater than $20^{\circ}$, to suppress the background from
annihilation of $\bar{p}$ with the detector material. The $\pi^{0}$ candidates
are reconstructed from photon pairs with invariant mass $M(\gamma\gamma)$ in
the range 0.115 GeV/$c^{2}$ $<M(\gamma\gamma)<$ 0.150 GeV/$c^{2}$. To improve
momentum resolution and exclude background, a kinematic fit is performed to
constrain $M(\gamma\gamma)$ to the known $\pi^{0}$ mass
ParticleDataGroup:2022pth , and candidates with fit quality $\chi^{2}<20$ are
retained for further analysis.

The signal candidates of $\Lambda_{c}^{+}\to pK^{0}_{L}$ and
$\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{0}$ are required to have only one charged
track with opposite charge to the tagged $\bar{\Lambda}_{c}^{-}$ satisfying
the proton PID criteria. For $\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{0}$ decay, the
$\pi^{0}$ candidate with the highest energy is selected. In the reconstruction
of $\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{+}\pi^{-}$, events must have only three
remaining charged tracks with correct charges and PID. Candidates with
additional charged tracks, whose distances of closest approaches to the IP are
within $\pm$20 cm along the beam direction, are excluded. The presence of the
$K^{0}_{L}$ is inferred by the kinematic variable $M^{2}_{\mathrm{miss}}$,
defined as

$M^{2}_{\mathrm{miss}}\equiv\left(E_{\mathrm{beam}}-E_{\mathrm{selected}}\right)^{2}/c^{4}-\left|\vec{p}_{\Lambda_{c}^{+}}-\vec{p}_{\mathrm{selected}}\right|^{2}/c^{2},$
(3)

where $E_{\mathrm{beam}}$ is the beam energy and $E_{\mathrm{selected}}$
($\vec{p}_{\mathrm{selected}}$) is the total measured energy (momentum) of the
selected particles in the DT signal side, boosted into the c.m. system of
$e^{+}e^{-}$. To improve the momentum resolution, the momentum of
$\Lambda_{c}^{+}$ is determined by

$\vec{p}_{\Lambda_{c}^{+}}\equiv-\hat{p}_{\mathrm{\bar{\Lambda}_{c}^{-}}}\sqrt{E_{\mathrm{beam}}^{2}/c^{2}-m^{2}_{\Lambda_{c}^{+}}c^{2}},$
(4)

where $\hat{p}_{\mathrm{\bar{\Lambda}_{c}^{-}}}$ is the direction of the
tagged $\bar{\Lambda}_{c}^{-}$ and $m_{\Lambda_{c}^{+}}$ is the known
$\Lambda_{c}^{+}$ baryon mass taken from the PDG ParticleDataGroup:2022pth .
For all three decays, the $M^{2}_{\mathrm{miss}}$ distributions are expected
to have a peak around the known mass squared of $K^{0}_{L}$
ParticleDataGroup:2022pth .

Based on studies of inclusive MC samples, the dominant background events for
the signal mode $\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{+}\pi^{-}$ are from
processes with $\Lambda\to p\pi^{-}$ and $K^{0}_{S}\to\pi^{+}\pi^{-}$. They
are rejected by vetoing events with $M(p\pi^{-})$ ($M(\pi^{+}\pi^{-})$)
invariant masses in the interval of 1.11 $\mbox{GeV/$c^{2}$}<M(p\pi^{-})<$
1.12 GeV/$c^{2}$ (0.48 GeV/$c^{2}$$<M(\pi^{+}\pi^{-})<$ 0.52 GeV/$c^{2}$). The
combinatorial backgrounds are suppressed by requiring the recoil mass of the
proton
$M_{\mathrm{recoil}}(p)\equiv\sqrt{E_{\mathrm{beam}}^{2}-|\vec{p}_{\Lambda_{c}^{+}}-\vec{p}_{p}|^{2}}>1.0$
GeV/$c^{2}$, which removes only about 3% of the signal. Here $\vec{p}_{p}$ is
the momentum of the proton. For the $\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{0}$
signal mode, background events of $\Lambda_{c}^{+}\to
pK^{0}_{S}(\to\pi^{0}\pi^{0})$ and $pK^{0}_{L}$ are excluded by requiring
$M_{\mathrm{recoil}}(p)>0.65\mbox{GeV/$c^{2}$}$, which removes less than 1% of
signal. Events within the range 1.17 $\mbox{GeV/$c^{2}$}<M(p\pi^{0})<1.20$
GeV/$c^{2}$ are discarded to suppress the background of the $\Sigma^{+}\to
p\pi^{0}$ decay.

To improve the momentum resolution, a six constraint (6C) kinematic fit is
performed requiring total four-momentum conservation with respect to that of
the initial $e^{+}e^{-}$ collision and constraining both masses of the tagged
$\bar{\Lambda}_{c}^{-}$ and the signal $\Lambda_{c}^{+}$ to
$m_{\Lambda_{c}^{+}}$. The $K^{0}_{L}$ is treated as a missing particle, and
its four-momentum and mass are free in the kinematic fit. The $\chi^{2}$ of
the kinematic fit for each signal mode is required to be less than the
optimized value that maximizes the figure of merit $S/\sqrt{S+B}$, where $S$
and $B$ are the numbers of signal and background events from MC simulations,
scaled to the data luminosity. The optimized requirements are $\chi^{2}<60$
for $\Lambda_{c}^{+}\to pK^{0}_{L}$, $\chi^{2}<25$ for $\Lambda_{c}^{+}\to
pK^{0}_{L}\pi^{+}\pi^{-}$, and $\chi^{2}<20$ for $\Lambda_{c}^{+}\to
pK^{0}_{L}\pi^{0}$. The resulting $M^{2}_{\mathrm{miss}}$ distributions of the
DT events are shown in Figure 1, which combine all data samples at the seven
c.m. energies. Signal events are indicated by the significant peaks around the
$K^{0}_{L}$ mass squared.

Figure 1: The $M^{2}_{\mathrm{miss}}$ distributions of the selected DT events
for (a) $\Lambda_{c}^{+}\to pK^{0}_{L}$, (b) $\Lambda_{c}^{+}\to
pK^{0}_{L}\pi^{+}\pi^{-}$, and (c) $\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{0}$
decays. The points with error bars are data combined from seven c.m. energies,
the red histograms indicate the signal processes, and the green and orange
histograms are the peaking backgrounds. The cyan and violet histograms
represent non-signal $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ and inclusive
hadronic background processes, respectively.

There are peaking backgrounds remaining from $\Lambda_{c}^{+}\to
pK^{0}_{S}(\to\pi^{0}\pi^{0})$ and $\Lambda_{c}^{+}\to p\eta(\to\gamma\gamma$
or $3\pi^{0})$, $\Lambda_{c}^{+}\to
pK^{0}_{S}(\to\pi^{0}\pi^{0})\pi^{+}\pi^{-}$, and $\Lambda_{c}^{+}\to
pK^{0}_{S}(\to\pi^{0}\pi^{0})\pi^{0}$ in the corresponding signal modes. The
peaking background events from $\Lambda_{c}^{+}\to K^{0}_{S}X$ decays
$N^{\mathrm{Bkg}}_{K^{0}_{S}X}$ are determined by

$\displaystyle
N^{\mathrm{Bkg}}_{K^{0}_{S}X}=N^{\mathrm{Data}}_{\mathrm{DT}\,K^{0}_{S}X}\cdot
w_{K^{0}_{S}X},\
w_{K^{0}_{S}X}=\frac{\sum_{i}s_{i}\cdot\frac{N^{\mathrm{ST}}_{i}}{\varepsilon^{\mathrm{ST}}_{i}}\cdot
N^{\mathrm{MC},i}_{\mathrm{DT}\,K^{0}_{L}X}}{\sum_{i}\frac{N^{\mathrm{ST}}_{i}}{\varepsilon^{\mathrm{ST}}_{i}}\cdot
N^{\mathrm{MC},i}_{\mathrm{DT}\,K^{0}_{S}X}},$ (5)

where $i$ represents the tag mode, and
$N^{\mathrm{Data}}_{\mathrm{DT}\,K^{0}_{S}X}$ denotes the data yields passing
the DT selection criteria of $\Lambda_{c}^{+}\to K^{0}_{S}X$. Here, the DT
selection criteria of $\Lambda_{c}^{+}\to K^{0}_{S}X$ require a fully
reconstructed $K^{0}_{S}$ from $\pi^{+}\pi^{-}$ combinations, as described in
Ref. BESIII:2015bjk . $N^{\mathrm{Data}}_{\mathrm{DT}\,K^{0}_{S}X}$ is
corrected by the factor $w_{K^{0}_{S}X}$, which is derived from the exclusive
MC simulation samples of $\Lambda_{c}^{+}\to K^{0}_{S}X$.
$N_{\mathrm{DT}\,K^{0}_{L}X}^{\mathrm{MC},i}$ and
$N_{\mathrm{DT}\,K^{0}_{S}X}^{\mathrm{MC},i}$ are the numbers of the
$K^{0}_{S}X$ MC events that satisfy the DT selection criteria of
$\Lambda_{c}^{+}\to K^{0}_{L}X$ and $\Lambda_{c}^{+}\to K^{0}_{S}X$,
respectively. $N^{\mathrm{ST}}_{i}$ and $\varepsilon^{\mathrm{ST}}_{i}$ are
the ST yields and ST efficiencies from Ref. BESIII:2022xne . A scale factor
$s_{i}$ is specified for each tag mode, and $s_{i}$ is set to 2 if both the
tag and signal modes are $K^{0}_{S}X$. Otherwise, it is set to 1. For peaking
background events from $\Lambda_{c}^{+}\to p\eta$, the contribution is
evaluated based on the corresponding exclusive MC samples using

$\displaystyle N^{\mathrm{Bkg}}_{p\eta}=\mathcal{B}(\Lambda_{c}^{+}\to
p\eta)\cdot w_{p\eta},\
w_{p\eta}=\sum_{i}\left(\frac{N^{\mathrm{ST}}_{i}}{\varepsilon^{\mathrm{ST}}_{i}}\cdot\frac{N^{\prime\mathrm{MC},i}_{pK^{0}_{L}}}{N^{\prime\mathrm{MC},i}_{\mathrm{tot}}}\right),$
(6)

with $\mathcal{B}(\Lambda_{c}^{+}\to p\eta)=(1.41\pm 0.11)\times 10^{-3}$
ParticleDataGroup:2022pth . $N^{\prime\mathrm{MC},i}_{pK^{0}_{L}}$ is the
number of surviving DT events for the $i$-th tag mode, that satisfy the DT
selection criteria of $\Lambda_{c}^{+}\to pK^{0}_{L}$, and
$N^{\prime\mathrm{MC},i}_{\mathrm{tot}}$ is the total number of MC events
generated for the $i$-th tag mode. Table 2 summarizes the contributions
arising from each peaking background process.

Table 2: Estimated yields of peaking backgrounds at each c.m. energy. The uncertainties are statistical only. $\sqrt{s}$ (GeV) | $\Lambda_{c}^{+}\to pK^{0}_{S}$ | $\Lambda_{c}^{+}\to p\eta$ | $\Lambda_{c}^{+}\to pK^{0}_{S}\pi^{+}\pi^{-}$ | $\Lambda_{c}^{+}\to pK^{0}_{S}\pi^{0}$
---|---|---|---|---
4.600 | 59 $\pm$ 6 | 13 $\pm$ 1 | 25 $\pm$ 5 | 44 $\pm$ 6
4.612 | 14 $\pm$ 3 | 2.3$\pm$ 0.2 | 4 $\pm$ 1 | 8 $\pm$ 3
4.628 | 70 $\pm$ 7 | 11 $\pm$ 1 | 31 $\pm$ 4 | 38 $\pm$ 6
4.641 | 68 $\pm$ 7 | 12 $\pm$ 1 | 30 $\pm$ 5 | 38 $\pm$ 7
4.661 | 60 $\pm$ 6 | 12 $\pm$ 1 | 39 $\pm$ 5 | 51 $\pm$ 8
4.682 | 198 $\pm$ 12 | 35 $\pm$ 3 | 97 $\pm$ 9 | 150 $\pm$ 13
4.699 | 54 $\pm$ 6 | 10 $\pm$ 1 | 24 $\pm$ 5 | 37 $\pm$ 6

A simultaneous unbinned maximum-likelihood fit is performed on the
$M^{2}_{\mathrm{miss}}$ distributions of the seven c.m. energies. The signal
and peaking backgrounds are modeled by individual MC-simulated shapes
convolved with Gaussian functions to account for differences between the data
and MC simulations. The Gaussian means and widths are free parameters in the
fit. The yields of the peaking background events are free with their mean and
standard deviation values set to the results listed in Table 2. For the signal
mode $\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{0}$, a truth-matching method is
employed to obtain the pure signal shape by comparing the two photons from the
$\pi^{0}$ with their corresponding MC truth information. The opening angle
$\theta_{\mathrm{truth}}$ between the truth and the reconstructed photons is
required to be less than $10^{\circ}$. The combinatorial background shape is
taken from the inclusive MC samples, including non-signal
$\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ and continuum hadron production events.

The BFs of the decays $\Lambda_{c}^{+}\to pK^{0}_{L}$, $\Lambda_{c}^{+}\to
pK^{0}_{L}\pi^{+}\pi^{-}$, and $\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{0}$ are
shared variables for the seven c.m. energies in the simultaneous fit,
determined by

$\displaystyle\mathcal{B}_{\mathrm{sig}}=\frac{N^{\mathrm{DT}}}{N^{\mathrm{ST}}\cdot\varepsilon_{\mathrm{avg}}\cdot\mathcal{B}_{\mathrm{int}}},$
(7)

where
$\varepsilon_{\mathrm{avg}}=\left(\sum_{i}N^{\mathrm{ST}}_{i}\cdot\varepsilon^{\mathrm{DT}}_{i}/\varepsilon^{\mathrm{ST}}_{i}\right)/N^{\mathrm{ST}}$
is the average detection efficiency for detecting signal modes in ST events
and $i$ represents the $i$-th ST tag mode. Table 3 lists the ST events and the
average detection efficiencies for each c.m. energy. $N^{\mathrm{DT}}$ and
$\varepsilon^{\mathrm{DT}}_{i}$ are the DT yields and corresponding
efficiencies, respectively. $\mathcal{B}_{\mathrm{int}}$ is the intermediate
BF of $\pi^{0}$, $\mathcal{B}(\pi^{0}\to\gamma\gamma)=(98.823\pm 0.034)\%$
ParticleDataGroup:2022pth  for $\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{0}$ decay.
Figure 2 shows the results of fits to the $M^{2}_{\mathrm{miss}}$
distributions, combining all data samples. From these fits, the BFs are
$\mathcal{B}(\Lambda_{c}^{+}\to pK^{0}_{L})=(1.67\pm 0.06)\%$,
$\mathcal{B}(\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{+}\pi^{-})=(1.69\pm 0.10)\%$,
and $\mathcal{B}(\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{0})=(2.02\pm 0.13)\%$,
where the uncertainties are statistical only. The total DT signal yields from
all c.m. energies are $N^{\mathrm{DT}}_{pK^{0}_{L}}=1627\pm 56$,
$N^{\mathrm{DT}}_{pK^{0}_{L}\pi^{+}\pi^{-}}=648\pm 39$, and
$N^{\mathrm{DT}}_{pK^{0}_{L}\pi^{0}}=652\pm 41$, for $\Lambda_{c}^{+}\to
pK^{0}_{L}$, $\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{+}\pi^{-}$, and
$\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{0}$, respectively.

Table 3: ST events ($N^{\mathrm{ST}}$), average detection efficiencies
($\varepsilon_{\mathrm{avg}}$) and DT signal yields ($N^{\mathrm{DT}}$) for
$\Lambda_{c}^{+}\to pK^{0}_{L}$, $\Lambda_{c}^{+}\to
pK^{0}_{L}\pi^{+}\pi^{-}$, and $\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{0}$ decays
at each c.m. energy. The errors are statistical only.

| $4.600~{}\mbox{GeV}$ | $4.612~{}\mbox{GeV}$ | $4.628~{}\mbox{GeV}$ | $4.641~{}\mbox{GeV}$ | $4.661~{}\mbox{GeV}$ | $4.682~{}\mbox{GeV}$ | $4.698~{}\mbox{GeV}$
---|---|---|---|---|---|---|---
$N^{\mathrm{ST}}$ | 17391 $\pm$ 171 | 3114 $\pm$ 75 | 14558 $\pm$ 135 | 15545 $\pm$ 165 | 15235 $\pm$ 164 | 44704 $\pm$ 284 | 12971 $\pm$ 158
modes | $\varepsilon_{\mathrm{avg}}(\%)$
$\Lambda_{c}^{+}\to pK^{0}_{L}$ | 76.32 $\pm$ 0.08 | 76.13 $\pm$ 0.08 | 78.12 $\pm$ 0.08 | 78.83 $\pm$ 0.08 | 78.78 $\pm$ 0.08 | 79.66 $\pm$ 0.09 | 79.89 $\pm$ 0.09
$\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{+}\pi^{-}$ | 28.61 $\pm$ 0.16 | 28.86 $\pm$ 0.17 | 29.87 $\pm$ 0.18 | 30.96 $\pm$ 0.17 | 31.12 $\pm$ 0.17 | 31.93 $\pm$ 0.17 | 32.83 $\pm$ 0.18
$\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{0}$ | 23.83 $\pm$ 0.15 | 24.28 $\pm$ 0.15 | 25.79 $\pm$ 0.16 | 26.40 $\pm$ 0.16 | 26.68 $\pm$ 0.16 | 27.43 $\pm$ 0.17 | 27.27 $\pm$ 0.17
modes | $N^{\mathrm{DT}}$
$\Lambda_{c}^{+}\to pK^{0}_{L}$ | 222 $\pm$ 8 | 40 $\pm$ 1 | 190 $\pm$ 7 | 205 $\pm$ 7 | 201 $\pm$ 7 | 596 $\pm$ 21 | 173 $\pm$ 6
$\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{+}\pi^{-}$ | 84 $\pm$ 5 | 15 $\pm$ 1 | 74 $\pm$ 4 | 81 $\pm$ 5 | 80 $\pm$ 5 | 242 $\pm$ 14 | 72 $\pm$ 4
$\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{0}$ | 83 $\pm$ 5 | 15 $\pm$ 1 | 75 $\pm$ 5 | 82 $\pm$ 5 | 81 $\pm$ 5 | 245 $\pm$ 16 | 71 $\pm$ 4

Figure 2: Combined fit results of the $M^{2}_{\mathrm{miss}}$ distributions
from all data samples for (a) $\Lambda_{c}^{+}\to pK^{0}_{L}$, (b)
$\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{+}\pi^{-}$ and (c) $\Lambda_{c}^{+}\to
pK^{0}_{L}\pi^{0}$ decays. The black dots with error bars are data, while the
black solid curves are the fit results. The red dashed curves are the signal
shapes, and the blue and green dashed curves represent the peaking backgrounds
$\Lambda_{c}^{+}\to pK^{0}_{S}$ and $\Lambda_{c}^{+}\to p\eta$, respectively.
The violet dashed curves are the combinatorial background shapes.

##  4 Systematic uncertainties

In the DT method, most of the systematic uncertainties associated with the ST
selections cancel. The major sources of systematic uncertainties in the BFs
measurements are described below and are reported relative to the measured
BFs.

  * •

Tracking and PID efficiencies. The tracking and PID efficiencies of the
charged protons and pions are studied using a control sample of $J/\psi\to
p\bar{p}\pi^{+}\pi^{-}$ BESIII:2022bkj . The MC simulation samples are
weighted by the efficiency ratio between data and MC as function of charged
particle momentum and $\cos{\theta}$. The systematic uncertainties of tracking
and PID are 0.5% and 0.1% for $\Lambda_{c}^{+}\to pK^{0}_{L}$, 1.6% and 0.8%
for $\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{+}\pi^{-}$, and 0.7% and 0.4% for
$\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{0}$, respectively.

  * •

No extra charged track requirement. The number of good charged tracks is
required to be exactly one (three) for $pK^{0}_{L}$ and $pK^{0}_{L}\pi^{0}$
($pK^{0}_{L}\pi^{+}\pi^{-}$) DT candidates in the recoil system of the tagged
$\bar{\Lambda}_{c}^{-}$. The difference between data and MC simulation from
this selection is studied using a control sample of $\Lambda_{c}^{+}\to
pK^{+}\pi^{-}$. The systematic uncertainty is 1.9%.

  * •

MC statistics. The exclusive MC simulation samples are used to obtain the ST
and DT detection efficiencies and to estimate the peaking background events.
The systematic uncertainties associated with the limited MC sample sizes are
estimated to be 0.1%, 0.5%, and 0.5% for $\Lambda_{c}^{+}\to pK^{0}_{L}$,
$\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{+}\pi^{-}$, and $\Lambda_{c}^{+}\to
pK^{0}_{L}\pi^{0}$, respectively.

  * •

ST yield. The systematic uncertainty arising from the total ST yield is
assigned to be 0.2% BESIII:2022xne .

  * •

Kinematic fit. The model of the MC simulation is much simpler than the real
detector performance, resulting in a difference between the data and MC
simulation in the track parameters of the charged tracks BESIII:2012mpj . The
helix parameters of the charged tracks are corrected, and the BFs are re-
evaluated with the updated MC simulation samples. The differences from the
measured BFs are taken as the systematic uncertainties associated with the
kinematic fit, which are 0.5%, 1.0%, and 0.5% for $\Lambda_{c}^{+}\to
pK^{0}_{L}$, $\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{+}\pi^{-}$ and
$\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{0}$, respectively.

  * •

Angle($\gamma$,$\bar{p}$) requirement. To estimate the systematic uncertainty
of the Angle($\gamma,\bar{p}$) requirement, the difference between the data
and MC simulation samples of this requirement is investigated from a control
sample of $\psi(3686)\to\pi^{+}\pi^{-}J/\psi,J/\psi\to p\bar{p}\pi^{0}$. The
systematic uncertainty is 0.2% for $\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{0}$.

  * •

$\pi^{0}$ reconstruction. The systematic uncertainty due to the $\pi^{0}$
reconstruction is determined using the control sample of $J/\psi\to
p\bar{p}\pi^{0}$ BESIII:2022ahw . The MC simulation samples are corrected
depending on the $\pi^{0}$ momentum. The systematic uncertainty is determined
to be 0.5%.

  * •

Truth-match method. The systematic uncertainty from the truth-match method is
determined comparing the measured BFs with and without the truth-match
requirements. The resulting systematic uncertainty is taken as 0.2%.

  * •

Signal model. For $\Lambda_{c}^{+}\to pK^{0}_{L}$, the systematic uncertainty
from the signal model is determined varying the decay asymmetry parameters
within $\pm 1\sigma$. The deviation from the measured BF is found to be
negligible. For $\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{+}\pi^{-}$ and
$\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{0}$, the signal models in the nominal
analysis is tunned based on the data. The possible intermediate resonances are
considered in the amplitude analysis, composed of $\Sigma^{*}$, $\Delta^{*}$,
$N^{*}$, $\bar{K}^{*}$ and $\rho$. The nominal amplitude models are then
replaced by alternative ones with equivalent descriptions of the data. The
alternative model of $\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{0}$ is selected by
removing the insignificant intermediate resonances. For $\Lambda_{c}^{+}\to
pK^{0}_{L}\pi^{+}\pi^{-}$, the amplitude fit is not stable due to the limited
statistics of data. An alternative model is chosen with a similar fit quality
to the nominal. The systematic uncertainties are determined to be 1.1% and
0.1% for $\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{+}\pi^{-}$ and $\Lambda_{c}^{+}\to
pK^{0}_{L}\pi^{0}$, respectively.

  * •

Background shape. To investigate the systematic uncertainty from the
background shape, the nominal background shape is replaced with a second-order
Chebychev polynomial function in the simultaneous fit. The systematic
uncertainties are 0.9%, 0.6%, and 0.4% for $\Lambda_{c}^{+}\to pK^{0}_{L}$,
$\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{+}\pi^{-}$ and $\Lambda_{c}^{+}\to
pK^{0}_{L}\pi^{0}$, respectively.

  * •

Fit bias. The systematic uncertainty from the simultaneous fit is studied with
5000 sets of toy MC samples, which are simulated with all parameters from the
fit model fixed. The BFs obtained from the toy samples are fitted with a
Gaussian function. The deviations between the Gaussian mean value and nominal
BFs are assigned as systematic uncertainties. For the decay
$\Lambda_{c}^{+}\to pK_{L}\pi^{0}$, the fit bias is found to be 0.27%, while
for the other two signal modes it is negligible.

Other sources of systematic uncertainties, such as the BF of
$\pi^{0}\to\gamma\gamma$, are neglected due to their negligible effects.
Assuming that all sources of systematic uncertainties in the BFs measurements
are uncorrelated, the quadratic sums of the different sources are considered
as the total systematic uncertainties, which are 2.2%, 3.1%, and 2.3% for
$\Lambda_{c}^{+}\to pK^{0}_{L}$, $\Lambda_{c}^{+}\to
pK^{0}_{L}\pi^{+}\pi^{-}$, and $\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{0}$,
respectively. Table 4 lists all the systematic uncertainties discussed above.

Table 4: Relative systematic uncertainties in the BF measurements. Source | $\Lambda_{c}^{+}\to pK^{0}_{L}\ (\%)$ | $\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{+}\pi^{-}\ (\%)$ | $\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{0}\ (\%)$
---|---|---|---
Tracking | 0.5 | 1.6 | 0.7
PID | 0.1 | 0.8 | 0.4
No extra charged track | 1.9 | 1.9 | 1.9
MC statistics | 0.2 | 0.5 | 0.5
ST yields | 0.2 | 0.2 | 0.2
Kinematic fit | 0.5 | 1.0 | 0.5
Angle($\gamma$,$\bar{p}$) requirement | - | - | 0.2
$\pi^{0}$ reconstruction | - | - | 0.5
Truth-match method | - | - | 0.2
Signal model | - | 1.1 | 0.1
Background shape | 0.9 | 0.6 | 0.4
Fit bias | - | - | 0.3
Total | 2.2 | 3.1 | 2.3

##  5 Summary

In summary, we report the BFs of $\Lambda_{c}^{+}\to pK^{0}_{L}$,
$\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{+}\pi^{-}$ and $\Lambda_{c}^{+}\to
pK^{0}_{L}\pi^{0}$ for the first time, by analyzing $e^{+}e^{-}$ annihilation
data samples corresponding to an integrated luminosity of 4.5 fb-1 collected
at c.m. energies between 4.600 and 4.699 GeV. The measured BFs of these decays
are $\mathcal{B}(\Lambda_{c}^{+}\to pK^{0}_{L})=(1.67\pm 0.06\pm 0.04)\%$,
$\mathcal{B}(\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{+}\pi^{-})=(1.69\pm 0.10\pm
0.05)\%$, and $\mathcal{B}(\Lambda_{c}^{+}\to pK^{0}_{L}\pi^{0})=(2.02\pm
0.13\pm 0.05)\%$. Combining the BFs measurements in this work with the values
of $\mathcal{B}(\Lambda_{c}^{+}\to K^{0}_{S}X)$ ParticleDataGroup:2022pth ,
the $K^{0}_{S}$-$K^{0}_{L}$ asymmetries are determined, as summarized in Table
5. The uncertainties are derived through the standard error propagation
procedure, assuming that the uncertainties of the estimated
$\mathcal{B}(\Lambda_{c}^{+}\to K^{0}_{L}X)$ and the quoted
$\mathcal{B}(\Lambda_{c}^{+}\to K^{0}_{S}X)$ are uncorrelated. Taking into
account the uncertainties, no obvious asymmetry is observed in any of the
three decays. The $K^{0}_{S}$-$K^{0}_{L}$ asymmetry of $\Lambda_{c}^{+}\to
pK_{S,L}^{0}$ $R(\Lambda_{c}^{+},pK_{S,L}^{0})=-0.025\pm 0.031$ is compatible
with the prediction of ($-$0.010, 0.087) based on SU(3) flavor symmetry
Wang:2017gxe . Our measurements of the $K^{0}_{S}$-$K^{0}_{L}$ asymmetries in
charmed baryon decays offer the possibility to access the DCS processes
involving neutral kaons and provide further constraints on their amplitudes.

Table 5: The BFs $\mathcal{B}(\Lambda_{c}^{+}\to K^{0}_{L}X)$, the known BFs
$\mathcal{B}(\Lambda_{c}^{+}\to K^{0}_{S}X)$, and $K^{0}_{S}$-$K^{0}_{L}$
asymmetries.

Mode |  $\mathcal{B}(\Lambda_{c}^{+}\to K^{0}_{L}X)$ (%) |  $\mathcal{B}(\Lambda_{c}^{+}\to K^{0}_{S}X)$ (%) ParticleDataGroup:2022pth  |  $R(\Lambda_{c}^{+}$, $K_{L,S}^{0}X)$
---|---|---|---
$\Lambda_{c}^{+}\to pK_{L,S}^{0}$ | 1.67 $\pm$ 0.06 $\pm$ 0.04 | 1.59 $\pm$ 0.07 | -0.025 $\pm$ 0.031
$\Lambda_{c}^{+}\to pK_{L,S}^{0}\pi^{+}\pi^{-}$ | 1.69 $\pm$ 0.10 $\pm$ 0.05 | 1.60 $\pm$ 0.11 | -0.027 $\pm$ 0.048
$\Lambda_{c}^{+}\to pK_{L,S}^{0}\pi^{0}$ | 2.02 $\pm$ 0.13 $\pm$ 0.05 | 1.96 $\pm$ 0.12 | -0.015 $\pm$ 0.046

###### Acknowledgements.

The BESIII Collaboration thanks the staff of BEPCII and the IHEP computing
center for their strong support. This work is supported in part by National
Key R&D Program of China under Contracts Nos. 2020YFA0406400, 2020YFA0406300,
2023YFA1606000; National Natural Science Foundation of China (NSFC) under
Contracts Nos. 11635010, 11735014, 11935015, 11935016, 11935018, 12025502,
12035009, 12035013, 12061131003, 12192260, 12192261, 12192262, 12192263,
12192264, 12192265, 12221005, 12225509, 12235017, 12361141819; the Chinese
Academy of Sciences (CAS) Large-Scale Scientific Facility Program; the CAS
Center for Excellence in Particle Physics (CCEPP); Joint Large-Scale
Scientific Facility Funds of the NSFC and CAS under Contract No. U1832207; 100
Talents Program of CAS; The Institute of Nuclear and Particle Physics (INPAC)
and Shanghai Key Laboratory for Particle Physics and Cosmology; German
Research Foundation DFG under Contracts Nos. 455635585, FOR5327, GRK 2149;
Istituto Nazionale di Fisica Nucleare, Italy; Ministry of Development of
Turkey under Contract No. DPT2006K-120470; National Research Foundation of
Korea under Contract No. NRF-2022R1A2C1092335; National Science and Technology
fund of Mongolia; National Science Research and Innovation Fund (NSRF) via the
Program Management Unit for Human Resources & Institutional Development,
Research and Innovation of Thailand under Contract No. B16F640076; Polish
National Science Centre under Contract No. 2019/35/O/ST2/02907; The Swedish
Research Council; U. S. Department of Energy under Contract No. DE-
FG02-05ER41374

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## The BESIII collaboration

M. Ablikim1, M. N. Achasov4,c, P. Adlarson76, O. Afedulidis3, X. C. Ai81, R.
Aliberti35, A. Amoroso75A,75C, Q. An72,58,a, Y. Bai57, O. Bakina36, I.
Balossino29A, Y. Ban46,h, H.-R. Bao64, V. Batozskaya1,44, K. Begzsuren32, N.
Berger35, M. Berlowski44, M. Bertani28A, D. Bettoni29A, F. Bianchi75A,75C, E.
Bianco75A,75C, A. Bortone75A,75C, I. Boyko36, R. A. Briere5, A. Brueggemann69,
H. Cai77, X. Cai1,58, A. Calcaterra28A, G. F. Cao1,64, N. Cao1,64, S. A.
Cetin62A, J. F. Chang1,58, G. R. Che43, G. Chelkov36,b, C. Chen43, C. H.
Chen9, Chao Chen55, G. Chen1, H. S. Chen1,64, H. Y. Chen20, M. L. Chen1,58,64,
S. J. Chen42, S. L. Chen45, S. M. Chen61, T. Chen1,64, X. R. Chen31,64, X. T.
Chen1,64, Y. B. Chen1,58, Y. Q. Chen34, Z. J. Chen25,i, Z. Y. Chen1,64, S. K.
Choi10, G. Cibinetto29A, F. Cossio75C, J. J. Cui50, H. L. Dai1,58, J. P.
Dai79, A. Dbeyssi18, R. E. de Boer3, D. Dedovich36, C. Q. Deng73, Z. Y. Deng1,
A. Denig35, I. Denysenko36, M. Destefanis75A,75C, F. De Mori75A,75C, B.
Ding67,1, X. X. Ding46,h, Y. Ding40, Y. Ding34, J. Dong1,58, L. Y. Dong1,64,
M. Y. Dong1,58,64, X. Dong77, M. C. Du1, S. X. Du81, Y. Y. Duan55, Z. H.
Duan42, P. Egorov36,b, Y. H. Fan45, J. Fang59, J. Fang1,58, S. S. Fang1,64, W.
X. Fang1, Y. Fang1, Y. Q. Fang1,58, R. Farinelli29A, L. Fava75B,75C, F.
Feldbauer3, G. Felici28A, C. Q. Feng72,58, J. H. Feng59, Y. T. Feng72,58, M.
Fritsch3, C. D. Fu1, J. L. Fu64, Y. W. Fu1,64, H. Gao64, X. B. Gao41, Y. N.
Gao46,h, Yang Gao72,58, S. Garbolino75C, I. Garzia29A,29B, L. Ge81, P. T.
Ge19, Z. W. Ge42, C. Geng59, E. M. Gersabeck68, A. Gilman70, K. Goetzen13, L.
Gong40, W. X. Gong1,58, W. Gradl35, S. Gramigna29A,29B, M. Greco75A,75C, M. H.
Gu1,58, Y. T. Gu15, C. Y. Guan1,64, A. Q. Guo31,64, L. B. Guo41, M. J. Guo50,
R. P. Guo49, Y. P. Guo12,g, A. Guskov36,b, J. Gutierrez27, K. L. Han64, T. T.
Han1, F. Hanisch3, X. Q. Hao19, F. A. Harris66, K. K. He55, K. L. He1,64, F.
H. Heinsius3, C. H. Heinz35, Y. K. Heng1,58,64, C. Herold60, T. Holtmann3, P.
C. Hong34, G. Y. Hou1,64, X. T. Hou1,64, Y. R. Hou64, Z. L. Hou1, B. Y. Hu59,
H. M. Hu1,64, J. F. Hu56,j, S. L. Hu12,g, T. Hu1,58,64, Y. Hu1, G. S.
Huang72,58, K. X. Huang59, L. Q. Huang31,64, X. T. Huang50, Y. P. Huang1, Y.
S. Huang59, T. Hussain74, F. Hölzken3, N. Hüsken35, N. in der Wiesche69, J.
Jackson27, S. Janchiv32, J. H. Jeong10, Q. Ji1, Q. P. Ji19, W. Ji1,64, X. B.
Ji1,64, X. L. Ji1,58, Y. Y. Ji50, X. Q. Jia50, Z. K. Jia72,58, D. Jiang1,64,
H. B. Jiang77, P. C. Jiang46,h, S. S. Jiang39, T. J. Jiang16, X. S.
Jiang1,58,64, Y. Jiang64, J. B. Jiao50, J. K. Jiao34, Z. Jiao23, S. Jin42, Y.
Jin67, M. Q. Jing1,64, X. M. Jing64, T. Johansson76, S. Kabana33, N. Kalantar-
Nayestanaki65, X. L. Kang9, X. S. Kang40, M. Kavatsyuk65, B. C. Ke81, V.
Khachatryan27, A. Khoukaz69, R. Kiuchi1, O. B. Kolcu62A, B. Kopf3, M.
Kuessner3, X. Kui1,64, N. Kumar26, A. Kupsc44,76, W. Kühn37, J. J. Lane68, L.
Lavezzi75A,75C, T. T. Lei72,58, Z. H. Lei72,58, M. Lellmann35, T. Lenz35, C.
Li47, C. Li43, C. H. Li39, Cheng Li72,58, D. M. Li81, F. Li1,58, G. Li1, H. B.
Li1,64, H. J. Li19, H. N. Li56,j, Hui Li43, J. R. Li61, J. S. Li59, K. Li1, K.
L. Li19, L. J. Li1,64, L. K. Li1, Lei Li48, M. H. Li43, P. R. Li38,k,l, Q. M.
Li1,64, Q. X. Li50, R. Li17,31, S. X. Li12, T. Li50, W. D. Li1,64, W. G.
Li1,a, X. Li1,64, X. H. Li72,58, X. L. Li50, X. Y. Li1,64, X. Z. Li59, Y. G.
Li46,h, Z. J. Li59, Z. Y. Li79, C. Liang42, H. Liang1,64, H. Liang72,58, Y. F.
Liang54, Y. T. Liang31,64, G. R. Liao14, Y. P. Liao1,64, J. Libby26, A.
Limphirat60, C. C. Lin55, D. X. Lin31,64, T. Lin1, B. J. Liu1, B. X. Liu77, C.
Liu34, C. X. Liu1, F. Liu1, F. H. Liu53, Feng Liu6, G. M. Liu56,j, H.
Liu38,k,l, H. B. Liu15, H. H. Liu1, H. M. Liu1,64, Huihui Liu21, J. B.
Liu72,58, J. Y. Liu1,64, K. Liu38,k,l, K. Y. Liu40, Ke Liu22, L. Liu72,58, L.
C. Liu43, Lu Liu43, M. H. Liu12,g, P. L. Liu1, Q. Liu64, S. B. Liu72,58, T.
Liu12,g, W. K. Liu43, W. M. Liu72,58, X. Liu38,k,l, X. Liu39, Y. Liu81, Y.
Liu38,k,l, Y. B. Liu43, Z. A. Liu1,58,64, Z. D. Liu9, Z. Q. Liu50, X. C.
Lou1,58,64, F. X. Lu59, H. J. Lu23, J. G. Lu1,58, X. L. Lu1, Y. Lu7, Y. P.
Lu1,58, Z. H. Lu1,64, C. L. Luo41, J. R. Luo59, M. X. Luo80, T. Luo12,g, X. L.
Luo1,58, X. R. Lyu64, Y. F. Lyu43, F. C. Ma40, H. Ma79, H. L. Ma1, J. L.
Ma1,64, L. L. Ma50, L. R. Ma67, M. M. Ma1,64, Q. M. Ma1, R. Q. Ma1,64, T.
Ma72,58, X. T. Ma1,64, X. Y. Ma1,58, Y. Ma46,h, Y. M. Ma31, F. E. Maas18, M.
Maggiora75A,75C, S. Malde70, Y. J. Mao46,h, Z. P. Mao1, S. Marcello75A,75C, Z.
X. Meng67, J. G. Messchendorp13,65, G. Mezzadri29A, H. Miao1,64, T. J. Min42,
R. E. Mitchell27, X. H. Mo1,58,64, B. Moses27, N. Yu. Muchnoi4,c, J.
Muskalla35, Y. Nefedov36, F. Nerling18,e, L. S. Nie20, I. B. Nikolaev4,c, Z.
Ning1,58, S. Nisar11,m, Q. L. Niu38,k,l, W. D. Niu55, Y. Niu 50, S. L.
Olsen64, Q. Ouyang1,58,64, S. Pacetti28B,28C, X. Pan55, Y. Pan57, A. Pathak34,
Y. P. Pei72,58, M. Pelizaeus3, H. P. Peng72,58, Y. Y. Peng38,k,l, K.
Peters13,e, J. L. Ping41, R. G. Ping1,64, S. Plura35, V. Prasad33, F. Z. Qi1,
H. Qi72,58, H. R. Qi61, M. Qi42, T. Y. Qi12,g, S. Qian1,58, W. B. Qian64, C.
F. Qiao64, X. K. Qiao81, J. J. Qin73, L. Q. Qin14, L. Y. Qin72,58, X. P.
Qin12,g, X. S. Qin50, Z. H. Qin1,58, J. F. Qiu1, Z. H. Qu73, C. F. Redmer35,
K. J. Ren39, A. Rivetti75C, M. Rolo75C, G. Rong1,64, Ch. Rosner18, S. N.
Ruan43, N. Salone44, A. Sarantsev36,d, Y. Schelhaas35, K. Schoenning76, M.
Scodeggio29A, K. Y. Shan12,g, W. Shan24, X. Y. Shan72,58, Z. J. Shang38,k,l,
J. F. Shangguan16, L. G. Shao1,64, M. Shao72,58, C. P. Shen12,g, H. F.
Shen1,8, W. H. Shen64, X. Y. Shen1,64, B. A. Shi64, H. Shi72,58, H. C.
Shi72,58, J. L. Shi12,g, J. Y. Shi1, Q. Q. Shi55, S. Y. Shi73, X. Shi1,58, J.
J. Song19, T. Z. Song59, W. M. Song34,1, Y. J. Song12,g, Y. X. Song46,h,n, S.
Sosio75A,75C, S. Spataro75A,75C, F. Stieler35, S. S Su40, Y. J. Su64, G. B.
Sun77, G. X. Sun1, H. Sun64, H. K. Sun1, J. F. Sun19, K. Sun61, L. Sun77, S.
S. Sun1,64, T. Sun51,f, W. Y. Sun34, Y. Sun9, Y. J. Sun72,58, Y. Z. Sun1, Z.
Q. Sun1,64, Z. T. Sun50, C. J. Tang54, G. Y. Tang1, J. Tang59, M. Tang72,58,
Y. A. Tang77, L. Y. Tao73, Q. T. Tao25,i, M. Tat70, J. X. Teng72,58, V.
Thoren76, W. H. Tian59, Y. Tian31,64, Z. F. Tian77, I. Uman62B, Y. Wan55, S.
J. Wang 50, B. Wang1, B. L. Wang64, Bo Wang72,58, D. Y. Wang46,h, F. Wang73,
H. J. Wang38,k,l, J. J. Wang77, J. P. Wang 50, K. Wang1,58, L. L. Wang1, M.
Wang50, N. Y. Wang64, S. Wang12,g, S. Wang38,k,l, T. Wang12,g, T. J. Wang43,
W. Wang73, W. Wang59, W. P. Wang35,58,72,o, X. Wang46,h, X. F. Wang38,k,l, X.
J. Wang39, X. L. Wang12,g, X. N. Wang1, Y. Wang61, Y. D. Wang45, Y. F.
Wang1,58,64, Y. L. Wang19, Y. N. Wang45, Y. Q. Wang1, Yaqian Wang17, Yi
Wang61, Z. Wang1,58, Z. L. Wang73, Z. Y. Wang1,64, Ziyi Wang64, D. H. Wei14,
F. Weidner69, S. P. Wen1, Y. R. Wen39, U. Wiedner3, G. Wilkinson70, M.
Wolke76, L. Wollenberg3, C. Wu39, J. F. Wu1,8, L. H. Wu1, L. J. Wu1,64, X.
Wu12,g, X. H. Wu34, Y. Wu72,58, Y. H. Wu55, Y. J. Wu31, Z. Wu1,58, L.
Xia72,58, X. M. Xian39, B. H. Xiang1,64, T. Xiang46,h, D. Xiao38,k,l, G. Y.
Xiao42, S. Y. Xiao1, Y. L. Xiao12,g, Z. J. Xiao41, C. Xie42, X. H. Xie46,h, Y.
Xie50, Y. G. Xie1,58, Y. H. Xie6, Z. P. Xie72,58, T. Y. Xing1,64, C. F.
Xu1,64, C. J. Xu59, G. F. Xu1, H. Y. Xu67,2,p, M. Xu72,58, Q. J. Xu16, Q. N.
Xu30, W. Xu1, W. L. Xu67, X. P. Xu55, Y. Xu40, Y. C. Xu78, Z. S. Xu64, F.
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Yang51,f, H. L. Yang34, H. X. Yang1, T. Yang1, Y. Yang12,g, Y. F. Yang43, Y.
F. Yang1,64, Y. X. Yang1,64, Z. W. Yang38,k,l, Z. P. Yao50, M. Ye1,58, M. H.
Ye8, J. H. Yin1, Junhao Yin43, Z. Y. You59, B. X. Yu1,58,64, C. X. Yu43, G.
Yu1,64, J. S. Yu25,i, M. C. Yu40, T. Yu73, X. D. Yu46,h, Y. C. Yu81, C. Z.
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Yuan59, C. X. Yue39, A. A. Zafar74, F. R. Zeng50, S. H. Zeng63A,63B,63C,63D,
X. Zeng12,g, Y. Zeng25,i, Y. J. Zeng59, Y. J. Zeng1,64, X. Y. Zhai34, Y. C.
Zhai50, Y. H. Zhan59, A. Q. Zhang1,64, B. L. Zhang1,64, B. X. Zhang1, D. H.
Zhang43, G. Y. Zhang19, H. Zhang72,58, H. Zhang81, H. C. Zhang1,58,64, H. H.
Zhang59, H. H. Zhang34, H. Q. Zhang1,58,64, H. R. Zhang72,58, H. Y. Zhang1,58,
J. Zhang81, J. Zhang59, J. J. Zhang52, J. L. Zhang20, J. Q. Zhang41, J. S.
Zhang12,g, J. W. Zhang1,58,64, J. X. Zhang38,k,l, J. Y. Zhang1, J. Z.
Zhang1,64, Jianyu Zhang64, L. M. Zhang61, Lei Zhang42, P. Zhang1,64, Q. Y.
Zhang34, R. Y. Zhang38,k,l, S. H. Zhang1,64, Shulei Zhang25,i, X. D. Zhang45,
X. M. Zhang1, X. Y Zhang40, X. Y. Zhang50, Y. Zhang73, Y. Zhang1, Y. T.
Zhang81, Y. H. Zhang1,58, Y. M. Zhang39, Yan Zhang72,58, Z. D. Zhang1, Z. H.
Zhang1, Z. L. Zhang34, Z. Y. Zhang77, Z. Y. Zhang43, Z. Z. Zhang45, G. Zhao1,
J. Y. Zhao1,64, J. Z. Zhao1,58, L. Zhao1, Lei Zhao72,58, M. G. Zhao43, N.
Zhao79, R. P. Zhao64, S. J. Zhao81, Y. B. Zhao1,58, Y. X. Zhao31,64, Z. G.
Zhao72,58, A. Zhemchugov36,b, B. Zheng73, B. M. Zheng34, J. P. Zheng1,58, W.
J. Zheng1,64, Y. H. Zheng64, B. Zhong41, X. Zhong59, H. Zhou50, J. Y. Zhou34,
L. P. Zhou1,64, S. Zhou6, X. Zhou77, X. K. Zhou6, X. R. Zhou72,58, X. Y.
Zhou39, Y. Z. Zhou12,g, Z. C. Zhou20, A. N. Zhu64, J. Zhu43, K. Zhu1, K. J.
Zhu1,58,64, K. S. Zhu12,g, L. Zhu34, L. X. Zhu64, S. H. Zhu71, T. J. Zhu12,g,
W. D. Zhu41, Y. C. Zhu72,58, Z. A. Zhu1,64, J. H. Zou1, J. Zu72,58

(BESIII Collaboration)

1 Institute of High Energy Physics, Beijing 100049, People’s Republic of China
2 Beihang University, Beijing 100191, People’s Republic of China
3 Bochum Ruhr-University, D-44780 Bochum, Germany
4 Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
5 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
6 Central China Normal University, Wuhan 430079, People’s Republic of China
7 Central South University, Changsha 410083, People’s Republic of China
8 China Center of Advanced Science and Technology, Beijing 100190, People’s
Republic of China
9 China University of Geosciences, Wuhan 430074, People’s Republic of China
10 Chung-Ang University, Seoul, 06974, Republic of Korea
11 COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind
Road, 54000 Lahore, Pakistan
12 Fudan University, Shanghai 200433, People’s Republic of China
13 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
14 Guangxi Normal University, Guilin 541004, People’s Republic of China
15 Guangxi University, Nanning 530004, People’s Republic of China
16 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
17 Hebei University, Baoding 071002, People’s Republic of China
18 Helmholtz Institute Mainz, Staudinger Weg 18, D-55099 Mainz, Germany
19 Henan Normal University, Xinxiang 453007, People’s Republic of China
20 Henan University, Kaifeng 475004, People’s Republic of China
21 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
22 Henan University of Technology, Zhengzhou 450001, People’s Republic of
China
23 Huangshan College, Huangshan 245000, People’s Republic of China
24 Hunan Normal University, Changsha 410081, People’s Republic of China
25 Hunan University, Changsha 410082, People’s Republic of China
26 Indian Institute of Technology Madras, Chennai 600036, India
27 Indiana University, Bloomington, Indiana 47405, USA
28 INFN Laboratori Nazionali di Frascati , (A)INFN Laboratori Nazionali di
Frascati, I-00044, Frascati, Italy; (B)INFN Sezione di Perugia, I-06100,
Perugia, Italy; (C)University of Perugia, I-06100, Perugia, Italy
29 INFN Sezione di Ferrara, (A)INFN Sezione di Ferrara, I-44122, Ferrara,
Italy; (B)University of Ferrara, I-44122, Ferrara, Italy
30 Inner Mongolia University, Hohhot 010021, People’s Republic of China
31 Institute of Modern Physics, Lanzhou 730000, People’s Republic of China
32 Institute of Physics and Technology, Peace Avenue 54B, Ulaanbaatar 13330,
Mongolia
33 Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica
1000000, Chile
34 Jilin University, Changchun 130012, People’s Republic of China
35 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
36 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
37 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
38 Lanzhou University, Lanzhou 730000, People’s Republic of China
39 Liaoning Normal University, Dalian 116029, People’s Republic of China
40 Liaoning University, Shenyang 110036, People’s Republic of China
41 Nanjing Normal University, Nanjing 210023, People’s Republic of China
42 Nanjing University, Nanjing 210093, People’s Republic of China
43 Nankai University, Tianjin 300071, People’s Republic of China
44 National Centre for Nuclear Research, Warsaw 02-093, Poland
45 North China Electric Power University, Beijing 102206, People’s Republic of
China
46 Peking University, Beijing 100871, People’s Republic of China
47 Qufu Normal University, Qufu 273165, People’s Republic of China
48 Renmin University of China, Beijing 100872, People’s Republic of China
49 Shandong Normal University, Jinan 250014, People’s Republic of China
50 Shandong University, Jinan 250100, People’s Republic of China
51 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
52 Shanxi Normal University, Linfen 041004, People’s Republic of China
53 Shanxi University, Taiyuan 030006, People’s Republic of China
54 Sichuan University, Chengdu 610064, People’s Republic of China
55 Soochow University, Suzhou 215006, People’s Republic of China
56 South China Normal University, Guangzhou 510006, People’s Republic of China
57 Southeast University, Nanjing 211100, People’s Republic of China
58 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
59 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
60 Suranaree University of Technology, University Avenue 111, Nakhon
Ratchasima 30000, Thailand
61 Tsinghua University, Beijing 100084, People’s Republic of China
62 Turkish Accelerator Center Particle Factory Group, (A)Istinye University,
34010, Istanbul, Turkey; (B)Near East University, Nicosia, North Cyprus,
99138, Mersin 10, Turkey
63 University of Bristol, (A)H H Wills Physics Laboratory; (B)Tyndall Avenue;
(C)Bristol; (D)BS8 1TL
64 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
65 University of Groningen, NL-9747 AA Groningen, The Netherlands
66 University of Hawaii, Honolulu, Hawaii 96822, USA
67 University of Jinan, Jinan 250022, People’s Republic of China
68 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
69 University of Muenster, Wilhelm-Klemm-Strasse 9, 48149 Muenster, Germany
70 University of Oxford, Keble Road, Oxford OX13RH, United Kingdom
71 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
72 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
73 University of South China, Hengyang 421001, People’s Republic of China
74 University of the Punjab, Lahore-54590, Pakistan
75 University of Turin and INFN, (A)University of Turin, I-10125, Turin,
Italy; (B)University of Eastern Piedmont, I-15121, Alessandria, Italy;
(C)INFN, I-10125, Turin, Italy
76 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
77 Wuhan University, Wuhan 430072, People’s Republic of China
78 Yantai University, Yantai 264005, People’s Republic of China
79 Yunnan University, Kunming 650500, People’s Republic of China
80 Zhejiang University, Hangzhou 310027, People’s Republic of China
81 Zhengzhou University, Zhengzhou 450001, People’s Republic of China

a Deceased
b Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia
c Also at the Novosibirsk State University, Novosibirsk, 630090, Russia
d Also at the NRC "Kurchatov Institute", PNPI, 188300, Gatchina, Russia
e Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany
f Also at Key Laboratory for Particle Physics, Astrophysics and Cosmology,
Ministry of Education; Shanghai Key Laboratory for Particle Physics and
Cosmology; Institute of Nuclear and Particle Physics, Shanghai 200240,
People’s Republic of China
g Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and
Institute of Modern Physics, Fudan University, Shanghai 200443, People’s
Republic of China
h Also at State Key Laboratory of Nuclear Physics and Technology, Peking
University, Beijing 100871, People’s Republic of China
i Also at School of Physics and Electronics, Hunan University, Changsha
410082, China
j Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of
Quantum Matter, South China Normal University, Guangzhou 510006, China
k Also at MOE Frontiers Science Center for Rare Isotopes, Lanzhou University,
Lanzhou 730000, People’s Republic of China
l Also at Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou
730000, People’s Republic of China
m Also at the Department of Mathematical Sciences, IBA, Karachi 75270,
Pakistan
n Also at Ecole Polytechnique Federale de Lausanne (EPFL), CH-1015 Lausanne,
Switzerland
o Also at Helmholtz Institute Mainz, Staudinger Weg 18, D-55099 Mainz, Germany
p Also at School of Physics, Beihang University, Beijing 100191 , China


# Learning Subclass Representations for Visually-varied Image Classification

Xinchao Li  Peng Xu  Yue Shi  Martha Larson  Alan Hanjalic

Multimedia Information Retrieval Lab, Delft University of Technology  Delft,
The Netherlands  {x.li-3, p.xu, y.shi<EMAIL_ADDRESS>
(2013)

###### Abstract

In this paper, we present a subclass-representation approach that predicts the
probability of a social image belonging to one particular class. We explore
the co-occurrence of user-contributed tags to find subclasses with a strong
connection to the top level class. We then project each image on to the
resulting subclass space to generate a subclass representation for the image.
The novelty of the approach is that subclass representations make use of not
only the content of the photos themselves, but also information on the co-
occurrence of their tags, which determines membership in both subclasses and
top-level classes. The novelty is also that the images are classified into
smaller classes, which have a chance of being more visually stable and easier
to model. These subclasses are used as a latent space and images are
represented in this space by their probability of relatedness to all of the
subclasses. In contrast to approaches directly modeling each top-level class
based on the image content, the proposed method can exploit more information
for visually diverse classes. The approach is evaluated on a set of $2$
million photos with 10 classes, released by the Multimedia 2013 Yahoo! Large-
scale Flickr-tag Image Classification Grand Challenge. Experiments show that
the proposed system delivers sound performance for visually diverse classes
compared with methods that directly model top classes.

###### category:

H.3 Information Storage and Retrieval Content

###### keywords:

Large scale image classification, Subclass representation

††conference: Multimedia

##  1 Introduction

This paper describes the approach that we developed to address the Yahoo!
Large-scale Flickr-tag Image Classification Grand Challenge. This challenge is
formulated:

Figure 1: Example illustration of the subclass representation

_Given a training set of images together with their metadata, and a class
label corresponding to each training image, to build a ranking model that for
each class label ranks images in a test set (without metadata) as accurate as
possible._

We emphasize three particular characteristics of this challenge. First, the
ground truth for both the training and testing set is user generate tags. As
with any other tagging problem, these tags are expected to be noisy,
incomplete and subjective. The ranking models should have the ability to learn
useful information with difficult labels. Second, for this particular task,
the 10 target tags are selected from tags most frequently used by Flickr
users. The images from these classes (which we refer to as ‘top-level
classes’) show a broad variation in terms of visual representations. Thus, the
ranking model should be based on a representation that can model classes with
high intra-class variations. Third, the level of visual diversity varies
between the given 10 target tags. For example, images related to tag “sky” or
“beach” are expected to be more visually consistent than images related to
“2012” or “nature”. In view of this, the ranking model should have the ability
to deal with different abstraction levels adaptively.

In view of the above, we propose in this paper a new method for content-based
image classification/tagging based on subclass representation. As mentioned
above, the image classes in this challenge are user-generated tags. Since
social images are usually annotated by more than one tag, the co-occurrence of
different tags serves as a valuable information source that can be exploited
for elaborating the tags (classes) which are by themselves highly visually-
varied. As shown in the example in Fig. 1, the image class “nature” can cover
an extremely large range of visual representations. However, the subclasses of
it, such as “flower”, “bird”, “forest”, are much more homogeneous in terms of
the visual information. For this reason, we first discover the subclasses,
which are tags frequently and exclusively co-occurring with the top-level
class labels, for representing each image. Then, we train a binary classifier
for each selected subclass. For a given image, the confidence scores of all
subclass classifiers are concatenated to produce a high level representation.
Thus, the representation is developed to learn models of the top-level
classes. This is illustrated by Fig. 2. The final results are predicted by the
ranking model based on the the learned subclass representation.

The contribution of this paper lies in the following aspects:

  * •

This method uses a co-occurrence based method to discover subclasses. Compared
with semantic ontology based subclass generation methods [2] or using
predefined concepts as subclasses [9], the proposed subclass representation is
expected to be more discriminative in terms of predicting the target classes.

  * •

The proposed method uses confidence score instead of binary decision of the
subclass classifiers as the high level features. This strategy can develop
useful representations even if the performance of subclass classifier is not
reliable.

The remainder of the paper is organized as follows: in Section 2, we present
the details of the proposed method. The experimental framework and results are
presented in Section 3 and Section 4. Then, in Section 5 we discuss previous
research contributions that are related to our approach proposed in this
paper. Finally, Section 6 summarizes our contributions and discusses future
work.

##  2 Learning Subclass Representation

Figure 2: Overview of the propose approach. First subclasses are discovered,
then top-level classes are predicted using subclass representations.

###  2.1 Mining Subclasses

As discussed above, subclasses of one class (the target class) are expected to
be strongly connected with the target class and, moreover, relatively more
stably reflected in visual features than the target class. To define such
subclasses, we exploited the tags annotating the images. We first generate a
co-occurrence matrix between photos’ tags and their top-level classes, and
measure each tag’s connection to one class by its distinctive score, defined
as:

$S_{ij}=\frac{{C_{ij}}}{{\sum\limits_{j}{C_{ij}}}}$ (1)

where $C_{ij}$ is the number of co-occurrence of the $i$-th tag and the $j$-th
top level class. Note that in the setting of Yahoo! Challenge, the predefined
to-level classes are also chosen from the user-contributed tags.

The selected subclasses for class-$j$ can be then defined as the tags that
have their distinctive scores above a pre-defined threshold, as shown below:

$Tsel_{j}=\\{t_{i}|S_{ij}>Thr\\_distin,t_{i}\in T\\}$ (2)

where $t_{i}$ denotes the $i$-th tag, and $T$ denotes the set of all tags.
Note that some tags may be only assigned to a very small number of images in
one class. For those tags, the limited number of training images prevents them
from being effective subclasses. Taking this into account, we further rank all
selected tags, $t_{i}$, in $Tsel_{j}$ by the number of photos in class-$j$
that are tagged with $t_{i}$.

###  2.2 Subclass Representation

To generate a subclass-based representation for an image, we first use the
images tagged with the subclasses to train models, i.e., Support Vector
Machines (SVMs), for classifying subclasses, and then, use the confidence
scores for predicting each subclass as the new representation for the image.
In this sense, as illustrated in Fig. 2, the image features can be treated as
the fist level representation for an image, while the confidence scores of
subclasses are the high level representation. Based on the subclass
representations, we further learn the model that characterize the connection
between these representations and the top-level class.

##  3 Experimental Framework

###  3.1 Dataset

To verify the performance of the proposed approach, we carry out our
experiments on a dataset of photos released by the Multimedia 2013 Yahoo!
Large-scale Flickr-tag Image Classification Grand Challenge. The dataset
contains 2 million Flickr photos with 10 classes, i.e., 150K training and 50K
test images per class. The class labels are amongst the top tags annotated by
the Flickr users. Since the release does not includes the tags associated with
the photos, we re-crawled the photos’ tags using the photo ID provided in the
metadata.

To develop our system, the training dataset is randomly divided into three
parts with the ratio 4:3:1. The first is for training models for predicting
subclasses based on image features, the second is for training models for
target classes based on confidence scores from the learned subclass models,
and the last is for validation and parameter selection. The test data from the
grand challenge is used to evaluate the proposed system.

###  3.2 Multi-class Classification

To model the connection between image feature and subclass, and the connection
between subclass representation and top class, we choose an SVM-based
approach. For the purpose of classification for multiple classes, we apply
one-against-one training strategy, which is reported to have better training
time efficiency and prediction accuracy compared with other multi-class
support vector machines, e.g., one-against-all [3]. To generate probability
estimation from the SVM model, we apply the algorithm proposed in [10] and
modified its original implementation in LibSVM [1] to make it suitable for
distributed computing on a Hadoop-based distributed server.

###  3.3 Baselines

We compare our subclass-representation-based approach, denoted in the
following as $SVM\\_SubClassProb$, to two other approaches that closely relate
to our approach, as listed below.

  * •

$SVM\\_VisFeat$: Directly model target the 10 top-level classes based on image
features.

  * •

$SVM\\_ClassProb$: Project image features on to top-level 10-classes space.

##  4 Results

###  4.1 Learning Subclass Models

To learn the subclasses using the co-occurrence matrix between photos’ tags
and their classes, the $Thr\\_distin$ in Eq. (2) is set to $0.6$, and then we
manually select the subclasses in the top $10$ tags of each class, which
results in a total of 54 subclasses. As some classes may contains few
distinctive tags compared with other classes, they may have few subclasses,
i.e., class “travel” only contain 1 subclass. In contrast, some classes may
contain more distinctive tags, i.e., 14 subclasses for class “nature”. To
train these subclass models for projecting images on to the subclass space, we
use a maximum 10k images per subclass as training data. Note that some
subclasses may contain less than 10k images. The performance on the validation
set, Average Precision (AP) for these subclasses models are illustrated in
Fig. 3.

Figure 3: AP for 54 subclasses, labels on the horizontal axis indicate
subclass association with the top-level classes.

###  4.2 Classification Results

Fig.4 illustrates the performance in terms of mean average precision (MAP),
across all 10 top-level classes, for the proposed approach and the baselines
with respect to different training data scales. Overall, $SVM\\_SubClassProb$
performs better than $SVM\\_VisFeat$ and $SVM\\_ClassProb$ in different
training data scale, with an averaged gain $8\%$ for $SVM\\_VisFeat$ and $7\%$
for $SVM\\_ClassProb$. In addition, the $SVM\\_SubClassProb$ already achieves
good performance compared to others in case of a small training data scale,
e.g., 1k per class. Along with the increasing amounts of training data, the
performance for $SVM\\_SubClassProb$ levels out. This is due to the fact that
many subclasses do not contain enough training photos, e.g., $83\%$ of them
contains photos which are less than 10k.

Figure 4: MAP with different amounts of training data.

Fig. 5 further breaks down the performance in terms of AP over different
classes with respect to different training data scales. As can be seen, for
classes “food”, “people”,“sky”,“nature”, $SVM\\_SubClassProb$ gains more
improvements compared to other classes. This is due to the fact that, for
these classes, they own more subclasses compare to other classes, i.e., there
are more distinctive tags in these classes. Also, some classes,
“sky”,“people”, contains visually highly consistent subclasses, as illustrated
in Fig. 3, which give a strong support for top classes. This is especially
obvious for class “sky”, in which there is a subclass that has a very strong
visual consistency, providing a good support for the top class. Interestingly,
for the class “food”, which owns many subclasses, and each of them has a
relative low AP, however, this class still has reliable performance. We
conjecture that these subclass classifiers are providing useful discriminative
information in form of probabilities that they yield with respect to non-
relevant subclasses.

(a) music

(b) london

(c) wedding

(d) food

(e) travel

(f) beach

(g) people

(h) sky

(i) nature

(j) 2012

Figure 5: AP for 10 classes on different training data scale.

##  5 Related Work

Our work is closely related to method used for predicting tags with high
intra-class variation, in particular, sub-category based methods or methods
based on learning high level representations are often explored. We discuss
these methods here in turn.

Generating sub-categories has been considered as an effective method to deal
with classification problems where intra-class variation is high. The ImageNet
[2] organizes image dataset with labels corresponding to a semantic hierarchy.
This method is able to build comprehensive ontology for large scale dataset.
However, for a particular dataset, the sub-categories generated by data driven
strategies are expected to be more discriminative. [11] exploits co-watch
information to learn latent sub-tags for video tag prediction. [7] proposes to
discover the image hierarchy by using both visual and tag information. Our
method generates category-specific subclasses by exploring image/tag co-
occurrence, and trains classifiers for each subclass-tag. These subclasses
based models are expected to be discriminative in terms of estimating the
target tags, corresponding to top-level classes.

Learning higher level representation is adopted when the low-level image
features are not discriminative enough for the purpose of classification [8,
9]. For the supervised representation learning methods, a predefined set of
models are trained based on image features. The output of these models is
considered as the high level representations for predicting image categories.
Recently, deep neural networks have been used on unsupervised learning image
representations with large scale image dataset [6, 5]. This representation has
achieved promising results on different classification and tagging tasks [4].
Our methods used the output of trained subclass classifiers as the high level
features. This structure can be easily extend to deeper levels by finding
discriminative tags for the subclasses.

##  6 Conclusion

We have presented a subclass-representation approach to the task of
retrieving/ranking large scale social images to one particular class solely
based on visual content. The main contribution of the approach is that by
projecting the image feature representation on to a subclass space generated
by exploiting the co-occurrence information of user-contributed tags, it makes
use not only of the content of the photos themselves, but also of information
concerning the co-occurrence of the photo’s tags with tags corresponding to
top-level classes.

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