# Review of the low-lying excited baryons $\Sigma^{*}(1/2^{-})$

En Wang<EMAIL_ADDRESS>School of Physics, Zhengzhou University, Zhengzhou
450001, China  Guangxi Key Laboratory of Nuclear Physics and Nuclear
Technology, Guangxi Normal University, Guilin 541004, China  Li-Sheng Geng
<EMAIL_ADDRESS>School of Physics, Beihang University, Beijing
102206, China  Peng Huanwu Collaborative Center for Research and Education,
Beihang University, Beijing 100191, China  Beijing Key Laboratory of Advanced
Nuclear Materials and Physics, Beihang University, Beijing 102206, China
Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern
Physics, Chinese Academy of Sciences, Huizhou 516000, China  Jia-Jun Wu
<EMAIL_ADDRESS>School of Physics, University of Chinese Academy of
Sciences, Beijing 100049, China  Southern Center for Nuclear-Science Theory
(SCNT), Institute of Modern Physics, Chinese Academy of Sciences, Huizhou
516000, China  Ju-Jun Xie<EMAIL_ADDRESS>Southern Center for Nuclear-
Science Theory (SCNT), Institute of Modern Physics, Chinese Academy of
Sciences, Huizhou 516000, China  Institute of Modern Physics, Chinese Academy
of Sciences, Lanzhou 730000, China  School of Nuclear Sciences and Technology,
University of Chinese Academy of Sciences, Beijing 101408, China  Bing-Song
Zou<EMAIL_ADDRESS>Department of Physics, Tsinghua University,
Beijing 100084, China  CAS Key Laboratory of Theoretical Physics, Institute of
Theoretical Physics, Chinese Academy of Sciences, Beijing 100190,China  School
of Physics, University of Chinese Academy of Sciences, Beijing 100049, China
Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern
Physics, Chinese Academy of Sciences, Huizhou 516000, China

###### Abstract

Strong empirical and phenomenological indications exist for large sea-quark
admixtures in the low-lying excited baryons. Investigating the low-lying
excited baryon $\Sigma^{*}(1/2^{-})$ is important to determine the nature of
the low-lying excited baryons. We review the experimental and theoretical
progress on the studies of the $\Sigma^{*}(1/2^{-})$. Although several
candidates have received intensive discussions, such as $\Sigma(1620)$ and
$\Sigma(1480)$, their existence needs further confirmation. Following the
prediction of the unquenched quark models for the $\Sigma^{*}(1/2^{-})$, many
theoretical works suggested the existence of these states in various
processes. Future experimental measurements could shed light on the existence
of the low-lying excited $\Sigma^{*}(1/2^{-})$ state.

##  I Introduction

As composite subatomic particles made of quarks and gluons held together by
the strong interaction, hadrons are categorized into $q\bar{q}$ mesons and
$qqq$ baryons within the conventional quark model proposed by Gell-Mann and
Zweig Gell-Mann:1964ewy ; Zweig:1964ruk . So far, all the ground-state mesons
and baryons composed of light quarks ($u,d,s$) have been well established, and
a huge number of excited light mesons and baryons have been observed
experimentally ParticleDataGroup:2022pth , some of which can not be well
described in the conventional quark models. For instance, recently BESIII has
observed one isoscalar state $\eta_{1}(1855)$ with exotic quantum numbers
$J^{PC}=1^{-+}$ in the process $J/\psi\to\gamma\eta\eta^{\prime}$, which could
be a candidate for hybrid hadrons BESIII:2022riz ; Chen:2022qpd . Those exotic
light hadrons challenge our understanding of the quark model and provide
insights into the strong interaction dynamics within Quantum Chromodynamics
(QCD) Chen:2016spr ; Chen:2022asf ; Guo:2017jvc ; Oset:2016lyh ; Liu:2024uxn ;
Gao:1999ar ; Hall:2014uca .

There are some puzzles for the low-lying excited baryons with the quantum
numbers of spin-parity $J^{P}=1/2^{-}$. One puzzle is the “mass reverse
problem”. In the naive quark model, the mass of the $N(1535)$ with spin-parity
quantum numbers $J^{P}=1/2^{-}$ should be lower than the one of the radial
excitation $N(1440)$ with $J^{P}=1/2^{+}$, and the one of the $\Lambda(1405)$
($J^{P}=1/2^{-}$) with strangeness $S=-1$ ParticleDataGroup:2022pth ;
Liu:2005pm ; Geng:2008cv ; Capstick:1986ter , as shown in Fig. 1. Meanwhile,
the $N(1535)$ strongly couples to the channels with strangeness, such as the
$\eta N$, $K\Lambda$, and $K\Sigma$, which implies that this state has
significant sea-quark admixtures. The low-lying excited baryon $\Lambda(1670)$
exhibits itself as an enhancement in the $\eta\Lambda$ invariant mass
distribution of the processes $K^{-}p\to\eta\Lambda$ CrystalBall:2001uhc  and
$\Lambda_{c}^{+}\to\eta\Lambda\pi^{+}$ BESIII:2018qyg ; Belle:2020xku , and as
a dip structure around the $\eta\Lambda$ threshold in the $\bar{K}^{0}n$
invariant mass distribution of the process $K^{-}p\to\bar{K}^{0}n$
Gopal:1976gs . Recently, the Belle Collaboration observed a cusp structure
around the $\eta\Lambda$ threshold in the $pK^{-}$ invariant mass distribution
of the process $\Lambda_{c}^{+}\to pK^{-}\pi^{+}$, which could be associated
with the $\Lambda(1670)$ Belle:2022cbs ; Zhang:2024jby . Thus, these exotic
properties of the low-lying excited baryons with the quantum numbers of spin-
parity $J^{P}=1/2^{-}$ are difficult to explain in the simple quenched quark
model and should have a much richer structure than the three valence quark
composition Zou:2010tc .

Figure 1: Masses of the low-lying excited baryons with $J^{P}=1/2^{-}$. The
results of ‘quark model’ are taken from the predictions of the quark model
Capstick:1986ter , the results of ‘pentaquark’ are taken from the predictions
of the pentaquark model Zhang:2004xt , and the results of ‘RPP’ are the
experimental data ParticleDataGroup:2022pth . In addition, the
$N^{*}_{1/2^{-}}$, $\Lambda^{*}_{1/2^{-}}$/$\Sigma^{*}_{1/2^{-}}$, and
$\Xi^{*}_{1/2^{-}}$ masses of ‘molecule’ are taken from the predictions of
Ref. Garzon:2014ida , Ref. Lu:2022hwm , and Ref. Ramos:2002xh , respectively.

In the classical quenched quark model, which ignores the creation of the
quark-antiquark pairs, the quarks cannot be separated from a hadron due to an
infinitely large confinement potential. However, as shown in Fig. 2, the
spatial excitation energy of a quark in a baryon is comparable to pulling a
$q\bar{q}$ pair from the gluon field Zou:2016bxw , which implies that the
creation of quark-antiquark pairs from the vacuum plays a crucial role in
understanding quark confinement and hadron spectroscopy. The 5-quark
components in baryons can be either in the form of molecule, such as
$\bar{K}N-\Sigma\pi$ for $\Lambda(1405)$, or in other forms of quark
correlations, such as pentaquark configurations $[ud][us]\bar{s}$ for
$N(1535)$, as depicted in Fig. 2.

Figure 2: The $qqq$ and $qqqq\bar{q}$ configurations for the low-lying excited
baryons with $J^{P}=1/2^{-}$.

The low-lying baryon $\Lambda(1405)$ with $J^{P}=1/2^{-}$ is difficult to
understand in the naive quark model as a $P$-wave excited baryon Isgur:1978xj
. It has been proposed that this state can be generated by the $\bar{K}N$
coupled-channel interaction within the unitary chiral approach Kaiser:1995eg ;
Oset:1997it ; Oller:2000fj ; Jido:2003cb , or contains both a $qqq$ component
and a $qqqq\bar{q}$ component that is strongly coupled to the $\bar{K}N$ and
$\pi\Sigma$ channels Helminen:2000jb . In Refs. Helminen:2000jb ; Zou:2007mk ;
Zhang:2004xt , it was proposed that the $N(1535)$ resonance could be the
lowest $L=1$ orbital excited $uud$ state with a large admixing of the
$[ud][us]\bar{s}$ pentaquark component, which makes $N(1535)$ heavier than
$N(1440)$, and provides a natural explanation for the large couplings of
$N(1535)$ to the strangeness channels $\eta N$, $K\Lambda$, and $K\Sigma$. In
addition, $N(1535)$ could be dynamically generated from the $S$-wave
pseudoscalar meson-octet baryon interactions within the chiral unitary
approach and is predicted to strongly couple to the channels $\eta N$,
$K\Lambda$, and $K\Sigma$ Kaiser:1995eg ; Kaiser:1996js ; Inoue:2001ip ;
Bruns:2010sv ; Nieves:2011gb ; Gamermann:2011mq ; Khemchandani:2013nma . The
physical picture remains unchanged, even after the pseudoscalar meson-baryon
mixing with the vector-baryon sates was considered Garzon:2014ida . The $\phi$
production in the $\pi p$ and $pp$ collisions suggest that the $N(1535)$
strongly couples to the $K\Sigma$, $K\Lambda$, and $\phi N$ channels
Xie:2007qt ; Doring:2008sv , consistent with the results of the chiral unitary
approach. We have proposed to study the $N(1535)$ within the chiral unitary
approach through the three-body decays of $\Lambda_{b}$ Lyu:2023aqn ;
Lu:2016roh ; Wang:2015pcn . Recently, it was suggested to test the molecular
nature of $N(1535)$ by measuring its correlation functions Molina:2023jov , or
the scattering length and effective range of the $K\Sigma$, $K\Lambda$, and
$\eta p$ channels Li:2023pjx . The $N(1535)$ was also interpreted as the
three-quark core dressed by meson-nucleon scattering contributions within the
Hamiltonian Effective Field Theory Abell:2023nex ; Guo:2022hud ; Liu:2015ktc .

However, the low-lying excited $\Sigma^{*}(1/2^{-})$ has not yet been well
established experimentally and theoretically ParticleDataGroup:2022pth .
Although the mass of the $\Sigma(1620)$ with $J^{P}=1/2^{-}$ listed in the
Review of Particle Physics (RPP) ParticleDataGroup:2022pth  is consistent with
the quenched quark model’s prediction of the $\Sigma^{*}(1/2^{-})$ mass
Capstick:1986ter ; Capstick:2000qj ; Glozman:1995fu , as shown in Fig. 1, its
status is only 1-star (meaning that the evidence of its existence is poor). It
is omitted from the summary tables, which implies that it still needs to be
confirmed.

It should be pointed out that the unquenched quark models have also made some
interesting predictions for the isovector partner $\Sigma^{*}(1/2^{-})$ of the
$\Lambda(1405)$ and $N(1535)$. In Ref. Helminen:2000jb , the authors suggested
that most of the low-lying baryon resonances in all flavor sectors of the
baryon spectrum have strong sea-quark components of the form $qqqq\bar{q}$,
and predicted a $\Sigma^{*}(1/2^{-})$ resonance with a mass about 1509 MeV.
Within Jaffe and Wilczek’s diquark model framework, Ref. Zhang:2004xt
predicted the mass of $\Sigma^{*}(1/2^{-})$ to be about 1360 MeV, as shown in
Fig. 1.

On the other hand, the $S$-wave meson-baryon interaction in the $S=-1$ sector
was studied using the chiral unitary approach, and one $\Sigma^{*}(1/2^{-})$
with a mass around the $\bar{K}N$ threshold was predicted in Refs. Oset:2001cn
; Oller:2000fj ; Oset:1997it ; Khemchandani:2018amu ; Kamiya:2016jqc ;
Jido:2003cb ; Oller:2006jw ; Garcia-Recio:2002yxy ; Lutz:2001yb ; Guo:2012vv ,
which is supported by the analysis of the CLAS data on the process $\gamma
p\to K\Sigma\pi$ CLAS:2013rjt ; Roca:2013cca .

In addition, as shown in Fig. 1, two $\Sigma^{*}(1/2^{-})$ states were
predicted in the covariant baryon chiral perturbation theory, including the
next-to-next-to-leading order contributions, with the narrower pole located at
$(1432,-i18)$ MeV showing up as a cusp, and the broader one located at
$(1364,-i110)$ MeV becoming a broad enhancement in the real axis of the
amplitude squared Lu:2022hwm . Indeed, Ref. Jido:2003cb  also found two poles
of $(1401,40i)$ MeV and $(1488,114)$ Mev within the chiral unitary approach by
considering the results obtained from the model of Ref. Oller:2000fj .

In a word, the $\Sigma^{*}(1/2^{-})$ with a mass around 1.4 GeV as a partner
of $\Lambda(1405)$ has not been found yet. Since searching for the state
$\Sigma^{*}(1/2^{-})$ is crucial to deepening our understanding of low-lying
excited baryons Crede:2013kia ; Klempt:2009pi ; Oset:2016lyh ; Chen:2022zgm ;
Ramos:2002xh  and there are many theoretical works have been devoted to this
issue, we would like to review the experimental and theoretical progress on
the studies of the low-lying excited baryon $\Sigma^{*}(1/2^{-})$.

In this review, we give the experimental progress and theoretical studies
about the $\Sigma^{*}(1/2^{-})$ in Section II and Section III, respectively.
Finally, we give a summary and outlook in Section IV.

##  II Experimental information on $\Sigma^{*}(1/2^{-})$

###  II.1 $\Sigma(1620)$

The $\Sigma(1620)$ with $J^{P}=1/2^{-}$ was listed as a 2-star resonance in
the previous version of PRR and downgraded to 1-star in the present version
ParticleDataGroup:2022pth . Although its mass $(1600\sim 1650)$ MeV is
consistent with the predicted mass around 1650 NeV of the quenched quark
models Capstick:1986ter , the evidence for its existence is very weak
Zou:2016bxw . Based on the multi-channel analysis of the $\bar{K}N$ reactions,
both Ref. Kim:1971zxa  and Ref. Langbein:1972uhb  claimed the evidence for a
$\Sigma^{*}(1/2^{-})$ with a mass around 1620 MeV, but predicted different
branching ratios for this resonance. However, Ref. Baillon:1975sk  found no
signal for $\Sigma^{*}(1/2^{-})$ between 1600 MeV and 1650 MeV through the
analysis of the reaction $\bar{K}N\to\Lambda\pi$, and suggested the existence
of the $\Sigma(1660)$ with $J^{P}=1/2^{-}$, supported by the multi-channel
analysis of the $\bar{K}N$ reactions Kim:1971zxa ; Langbein:1972uhb . In Ref.
Carroll:1976gs , the analysis of the total cross sections for $K^{-}p$ and
$K^{-}n$ with all proper final states indicated some $\Sigma$ resonances near
1600 MeV with quantum numbers undetermined. In Ref. Morris:1978ia , the
authors analyzed the reaction $K^{-}n\to\pi^{-}\Lambda$ and obtained two
solutions, with one solution indicating a $\Sigma^{*}(1/2^{-})$ near 1600 MeV,
and the other showing no resonant structure below the $\Sigma(1670)$
($J^{P}=3/2^{-}$).

In order to clarify the status of the $\Sigma(1620)$ and the $\Sigma(1660)$,
Ref. Gao:2010ve  analyzed the differential cross sections and $\Lambda$
polarization for both $K^{-}p\to\pi^{0}\Lambda$ and $K^{-}n\to\pi^{-}\Lambda$
reactions with an effective Lagrangian approach, and claimed the evidence for
the $\Sigma(1620)$, and found that the Crystal Ball $\Lambda$ polarization
data demand the existence of a $\Sigma$ resonance with $J^{P}=1/2^{+}$ and
mass near 1635 MeV, compatible with $\Sigma(1660)$ listed in RPP
ParticleDataGroup:2022pth , while the data do not demand the $\Sigma(1620)$.

Recently, Ref. Sarantsev:2019xxm  performed the fit of nearly all published
data on medium-energy $K^{-}p$ elastic, charge exchange, and inelastic
scattering and did not find the evidence for the $\Sigma(1620)$. Thus, the
existence of the $\Sigma(1620)$ is still in debate and needs to be confirmed
with more experimental measurements.

###  II.2 $\Sigma(1480)$

In the 2019 version of the RPP ParticleDataGroup:2018ovx , there is one
$\Sigma(1480)$ with a mass of about 1480 MeV and a width of $30\sim 60$ MeV,
which has only 1-star, and is omitted from the summary tables. This state was
first observed in the $\Lambda\pi$ and $\Sigma\pi$ spectra in the reaction
$\pi^{+}p\to(Y\pi)K^{+}$ Pan:1970ez . Later, Ref. AMSTERDAM-CERN-NIJMEGEN-
OXFORD:1980gfp  performed a multichannel analysis of $K^{-}p\to
p\bar{K}^{0}\pi^{-}$ at $4.2$ GeV and observed a 3.5 standard deviation signal
at 1480 Mev in the $p\bar{K}^{0}$ which cannot be explained as a reflection of
any competing channel.

The ZEUS Collaboration performed a resonance search in the $K_{S}^{0}p$ and
$K_{S}^{0}\bar{p}$ invariant mass spectrum of the inclusive deep inelastic
scattering at an $ep$ center-of-mass energy of $300\sim 318$ GeV and found one
resonance with a mass of $1465.1\pm 2.9$ MeV and a width of $15.5\pm 3.4$ MeV
ZEUS:2004lje .

In addition, one hyperon state $Y^{0*}$ with mass of $1480\pm 15$ MeV and
width of $60\pm 15$ MeV was observed in the reaction $pp\to pK^{+}Y^{0*}$ with
at least 4.5 standard deviations by the ANKE spectrometer at COSY-J$\ddot{\rm
u}$lich at $p_{\rm bean}=3.65$ GeV Zychor:2005sj . Since the isospin of the
$Y^{0*}$ has not been determined, it could either be the $\Sigma(1480)$, or,
alternatively, a $\Lambda$ hyperon.

However, no more precise data supports the existence of the $\Sigma(1480)$,
and the present RPP has removed this state ParticleDataGroup:2022pth .

###  II.3 $\Sigma^{*}(1/2^{-})$-like structure

Very recently, the Belle Collaboration reported the $\Lambda\pi^{+}$ and
$\Lambda\pi^{-}$ invariant mass distributions of the process
$\Lambda_{c}^{+}\rightarrow\Lambda\pi^{+}\pi^{+}\pi^{-}$, which show a
significant cusp signal around the $\bar{K}N$ threshold Belle:2022ywa . When
interpreted as a resonance, they found for the $\Lambda\pi^{+}$ combination a
mass of $1434.3\pm 0.6$(stat.)${}^{+0.9}_{-0.0}$(syst.) MeV, and an intrinsic
width of $11.5\pm 2.8$(stat.)${}^{+0.1}_{-5.3}$(syst.) MeV with a significance
of 7.5$\sigma$, and for the $\Lambda\pi^{-}$ combination a mass of $1438.5\pm
0.9$(stat.)${}^{+0.2}_{-2.5}$(syst.) MeV, and an intrinsic width of $33.0\pm
7.5$(stat.)${}^{+0.1}_{-23.6}$(syst.) MeV with a significance of 6.2$\sigma$
Belle:2022ywa . This state is consistent with the theoretical predicted
$\Sigma^{*}(1/2^{-})$ with a mass around $\bar{K}N$ threshold, which is
dynamically generated from the $S$-wave meson-baryon interaction in the $S=-1$
sector within the chiral unitary approach Oset:2001cn ; Oller:2000fj ;
Oset:1997it ; Khemchandani:2018amu ; Kamiya:2016jqc ; Jido:2003cb ;
Oller:2006jw ; Garcia-Recio:2002yxy ; Lutz:2001yb .

##  III Theoretical studies of $\Sigma^{*}(1/2^{-})$

###  III.1 $K^{-}p\to\Lambda\pi^{+}\pi^{-}$, $\Lambda p\to\Lambda p\pi^{0}$

In Ref. Wu:2009tu , the authors re-examined some old data of the
$K^{-}p\to\Lambda\pi^{+}\pi^{-}$ reactions obtained from an exposure of the
Laboratory’s 72-in. hydrogen bubble chamber to a separated $K^{-}$ beam of the
Bevatron Huwe:1969te . They found that, besides the well-established
$\Sigma^{*}(1385)$ ($J^{P}=3/2^{+}$), there is indeed some evidence for the
possible existence of a new $\Sigma^{*}(1/2^{-})$ resonance with the same mass
but a broader width. Later, the authors re-studied the
$K^{-}p\to\Lambda\pi^{+}\pi^{-}$ reaction within the effective Lagrangian
approach Wu:2009nw . By comparing with the experimental data of the Lawrence
Berkeley Laboratory 25-inch hydrogen bubble chamber, they found strong
evidence of the resonance $\Sigma^{*}(1/2^{-})$ with a mass around 1380 MeV.

The experimental measurements of the process $\Lambda p\to\Lambda p\pi^{0}$
show a strong threshold enhancement in the $\pi^{0}\Lambda$ invariant mass
distribution Kadyk:1971tc . Since the $\Sigma(1385)$ with spin-parity
$J^{P}=3/2^{+}$ decays to $\pi\Lambda$ in $P$-wave, its contribution would be
suppressed at low energies. Thus, Ref. Xie:2014zga  studied this process by
considering the proposed $\Sigma^{*}(1/2^{-})$ state, which decays into
$\pi\Lambda$ in $S$-wave, and described the near-threshold enhancement fairly
well, which supports the existence of the $\Sigma^{*}(1/2^{-})$ state in this
process.

###  III.2 $\Sigma^{*}(1/2^{-})$ photo- and $\nu$-productions

In Ref. Chen:2013vxa , the authors studied the reactions $\gamma N\to
K^{+}\Sigma(1385)\to K^{+}\pi^{0}\Lambda$ near threshold within an effective
Lagrangian approach. They found that the $\Sigma^{*}(1/2^{-})$ and the
interference of $\Sigma(1385)$ and $\Sigma^{*}(1/2^{-})$ play significant
roles near threshold. The predicted ratios of the helicity cross sections
$\sigma_{3/2}/\sigma_{1/2}$ and the angular distribution of the $\pi$ are
distinctly different depending on whether the $\Sigma^{*}(1/2^{-})$ is
present.

In Ref. Roca:2013av , the authors analyzed the CLAS data for the
$\Lambda(1405)$ photoproduction in the process $\gamma p\to
K^{+}\pi^{0}\Sigma^{0}$ CLAS:2013rjt , which filters isospin $I=0$ for the
$\pi^{0}\Sigma^{0}$ system, and determined the mass and width of the two
$\Lambda(1405)$ resonances, which are dynamically generated from the meson-
baryon interactions within the chiral unitary approach. Following the same
strategy, they extended the method to analyze the experimental data on $\gamma
p\to K^{+}\pi^{\pm}\Sigma^{\mp}$ by incorporating the isospin $I=1$ component
within the chiral unitary approach CLAS:2013rjt . They suggested the existence
of an $I=1$ resonance around the $\bar{K}N$ threshold Roca:2013cca . Even
though there is no pole in the usual unphysical Riemann sheet connected to the
physical one, the $I=1$ amplitude has a resonance-like structure around the
$\bar{K}N$ threshold, with a clear cusp similar to the case of $a_{0}(980)$
Oller:1998hw ; BESIII:2016tqo ; Liang:2016hmr .

The LEPS Collaboration measured the $\gamma n\to K^{+}\Sigma(1385)^{0}$
reaction using a linearly polarized photon beam and found that the azimuthal
angle $\phi$-dependence of the asymmetry is negative at $E_{\gamma}=1.8-2.4$
GeV LEPS:2008azb , in contrast to the theoretical prediction Oh:2007jd . By
including the proposed $\Sigma^{*}(1/2^{-})$ with a mass around 1380 MeV, Ref.
Gao:2010hy  well reproduced the experimental data for both the $\gamma n$ and
$\gamma p$ experiments.

Based on the cross-section data of $K^{-}p$ scattering, Ref.
Khemchandani:2018amu  studied the hyperon resonances by including both
pseudoscalar-baryon and vector-baryon dynamics, and produced the properties of
the $\Lambda(1405)$, supporting the existence of an isospin $I=1$ state
$\Sigma^{*}(1/2)$ with a mass around 1400 MeV. Furthermore, they calculated
the total and differential cross-sections for $\gamma p\to
K^{+}\Lambda(1405)$, in agreement with the CLAS data CLAS:2013rxx , and
predicted the cross-section of the process $\gamma p\to
K^{+}\Sigma^{*}(1/2^{-})$ Kim:2021wov .

Recently, Ref. Lyu:2023oqn  studied the $\Sigma^{*}(1/2^{-})$ photoproduction
within the Regge-effective Lagrangian approach by considering that the
$\Sigma^{*}(1/2^{-})$ couples strongly to the $\bar{K}N$ channel, and
predicted the differential and total cross sections. The total cross section
of $\gamma n\to K^{+}\Sigma^{*}(1/2^{-})$ is approximately 4.2 $\mu$b around
$E_{\gamma}=2.4$ GeV.

Furthermore, similar to photoproduction, Ref. Ren:2015bsa  proposed to study
the strangeness baryon in the antineutrino induced reactions
$\bar{\nu}_{l}p\to l+\Phi B$ where $\Phi B$ could be various meson-baryon
systems with strangeness, for instance, $\bar{K}N$, $\pi\Sigma$, $\pi\Lambda$,
and $K\Xi$. They showed that a combined study of $\pi^{0}\Sigma^{0}$,
$\pi^{+}\Sigma^{-}$, and $\pi^{-}\Sigma^{+}$ production induced by
antineutrinos could provide useful information about the $\Sigma^{*}(1/2^{-})$
state.

###  III.3 Charmonium decays
$\chi_{c0}\to\bar{\Sigma}\Sigma\pi,\bar{\Lambda}\Sigma\pi$,
$J/\psi\to\Lambda\bar{\Lambda}\pi$

In Ref. Wang:2015qta , the authors proposed to search for the
$\Sigma^{*}(1/2^{-})$ in the process $\chi_{c0}(1P)\to\bar{\Sigma}\Sigma\pi$.
Since the quantum numbers of the $\chi_{c0}(1P)$ are
$I^{G}(J^{PC})=0^{+}(0^{++})$, and the $\chi_{c0}(1P)$, blind to SU(3),
behaves like an SU(3) singlet, this process is a good isospin filter that
guarantees the $\pi\Sigma$ system has isospin $I=1$. On the other hand, the
$\chi_{c0}(1P)$ is a SU(3) singlet, and $\bar{\Sigma}$ belongs to an SU(3)
octet. Thus the $\pi\Sigma$ system will also be in an octet state, which
suppresses the contribution from the decuplet state $\Sigma(1385)$.

Considering the $\pi\Sigma$ and $\pi\bar{\Sigma}$ final-state interactions
within the chiral unitary approach, Ref. Wang:2015qta  predicted one cusp
structure around 1430 MeV in the $\pi\Sigma$ invariant mass distribution,
which could be associated with the $\Sigma^{*}(1/2^{-})$ predicted in Refs.
Roca:2013cca ; Jido:2003cb ; Oller:2000fj .

Furthermore, Ref. Liu:2017hdx  performed a theoretical study on the process
$\chi_{c0}\to\bar{\Lambda}\Sigma\pi$, considering the contributions from the
$\pi\Sigma$ and $\pi\bar{\Lambda}$ final state interactions within the chiral
unitary approach, which could dynamically generate the resonance
$\Sigma^{*}(1/2^{-})$ with a mass about 1430 MeV and $\Lambda(1405)$.
Meanwhile, another $\Sigma^{*}(1/2^{-})$ with a mass of 1380 MeV and a width
of 120 MeV is considered. The predicted $\pi\bar{\Lambda}$ invariant mass
distribution of the process $\chi_{c0}\to\bar{\Lambda}\Sigma\pi$ has a broad
peak around 1380 MeV and a cusp structure around the $\bar{K}N$ threshold,
which could be associated with the resonance $\Sigma^{*}(1/2^{-})$ Liu:2017hdx
.

Meanwhile, the possible existence of such a $\Sigma^{*}(1/2^{-})$ structure in
$J/\psi$ decays was discussed in Ref. Zou:2006uh . Recently, Ref.
Huang:2024oai  investigated the isospin-breaking process
$J/\psi\to\Lambda\bar{\Lambda}\pi$, and found that the triangle singularity
plays a significant role in this process, resulting in the creation of a
resonance-like structure around 1.4 GeV in the $\Lambda\pi$
($\bar{\Lambda}\pi$) invariant mass spectrum. It is suggested that the process
$J/\psi\to\Lambda\bar{\Lambda}\pi$ could be used to offer direct evidence for
the predicted triangle singularity and the additional evidence of the
$\Sigma^{*}(1/2^{-})$.

###  III.4 Charmed decays $\Lambda_{c}^{+}\to\eta\Lambda\pi^{+}$

Based on a 980 fb-1 data sample, the Belle Collaboration measured the process
$\Lambda^{+}_{c}\to\eta\pi^{+}\Lambda$. Significant signals of the resonances
$\Lambda(1670)$ and $\Sigma(1385)$ appear in the $\eta\Lambda$ and
$\pi^{+}\Lambda$ invariant mass distributions Belle:2020xku . This process has
been measured by CLEO CLEO:1995cbq  and BESIII BESIII:2018qyg  earlier, and
was also suggested to search for the $\Sigma^{*}(1/2^{-})$ in Ref. Xie:2017xwx
. It is shown that the contribution of $\Sigma^{*}(1/2^{-})$ with a mass about
1380 MeV makes the peak of $\pi^{+}\Lambda$ invariant mass distribution
broader, and the shapes of the angle distributions are very different for
$\Sigma(1385)$ and $\Sigma^{*}(1/2^{-})$ Xie:2017xwx .

In Ref. Wang:2022nac , the authors analyzed this process by considering the
intermediate state $\Sigma^{*}(1385)$ ($J^{P}=3/2^{+}$), and the $S$-wave
$\eta\Lambda$ and $\eta\pi^{+}$ final-state interactions in the chiral unitary
approach, which dynamically generate the $\Lambda(1670)$ and $a_{0}(980)$
resonances. Their results could reproduce the Belle data of the
$\pi^{+}\Lambda$ and $\eta\Lambda$ invariant mass distributions. However, the
Dalitz plot of the process $\Lambda^{+}_{c}\to\eta\pi^{+}\Lambda$ has more
yields appearing in the region of $M^{2}_{\Lambda\pi^{+}}<1.85$ GeV2 and
$M^{2}_{\eta\Lambda}<4$ GeV2, which cannot be well described in Ref.
Wang:2022nac . These discrepancies may imply some contributions from the
resonance $\Sigma^{*}(1/2^{-})$ Xie:2017xwx .

Motivated by the Belle measurements of the process
$\Lambda_{c}^{+}\to\eta\pi^{+}\Lambda$ Kim:2021wov , Ref. Lyu:2024qgc
investigated this process by considering the contributions from the
$a_{0}(980)$, $\Lambda(1670)$, and the intermediate resonances $\Sigma(1385)$
and $\Sigma^{*}(1/2^{-})$. It was shown that the results involving the
$\Sigma^{*}(1/2^{-})$ are favored by the Belle data of the $\eta\Lambda$ and
$\pi^{+}\Lambda$ invariant mass distributions. Furthermore, the $\eta\pi^{+}$
invariant mass distribution and the angular distribution $d\Gamma/d{\rm
cos}\theta$ were predicted, which are significantly different depending on
whether the $\Sigma^{*}(1/2^{-})$ is present or not. Finally, it was shown
that with the contribution from the $\Sigma^{*}(1/2^{-})$, the calculated
Dalitz plot agrees with the Belle measurements.

###  III.5 Charmed decays $\Lambda_{c}^{+}\to p\bar{K}^{0}\eta$

Recently, the Belle Collaboration measured the branching fraction
$\mathcal{B}(\Lambda_{c}^{+}\to pK^{0}_{S}\eta)=(4.35\pm 0.10\pm 0.20\pm
0.22)\times 10^{-3}$, and reported the invariant mass squared distributions of
this process Belle:2022pwd . In the $\eta p$ invariant mass squared
distribution, one can find a narrow peak around $M^{2}_{\eta p}=2.3$ GeV2 and
a broad peak around $M^{2}_{\eta p}=2.75$ GeV2, which should correspond to the
$N(1535)$ and $N(1650)$ resonances, respectively Xie:2017erh ; Pavao:2018wdf .
In addition, a threshold enhancement around $M^{2}_{p{K}_{S}^{0}}=2.1$ GeV2 in
the $p{K}_{S}^{0}$ invariant mass squared distribution could be associated
with the predicted low-lying excited $\Sigma^{*}(1/2^{-})$ baryon state below
the $pK^{0}_{S}$ threshold.

Motivated by the Belle measurements on the process $\Lambda_{c}^{+}\to
p\bar{K}^{0}\eta$, Ref. Li:2024rqb  investigated this process by considering
the contributions from the $N(1535)$, $N(1650)$, and the predicted low-lying
baryon $\Sigma^{*}(1/2^{-})$.

It was demonstrated that with the contribution from $\Sigma^{*}(1/2^{-})$
considered, and the $\chi^{2}/d.o.f.$ of fit reduces from 5.67 to 1.55, and
the calculated invariant mass spectrum agrees with the Belle measurements. The
near-threshold enhancement in the $p\bar{K}^{0}$ invariant mass squared
distribution could also be well reproduced, which implies that the
$\Sigma^{*}(1/2^{-})$ plays an important role in this process Li:2024rqb .

###  III.6 Charmed decays $\Lambda_{c}^{+}\to\pi^{+}\pi^{0}\pi^{-}\Sigma^{+}$

Since triangle singularity plays an important role in many processes
Liu:2015taa ; Bayar:2016ftu ; Liang:2019jtr ; Dai:2018hqb ; Wang:2016dtb ;
Guo:2019twa , Ref. Xie:2018gbi  studied the $\Sigma^{*}(1/2^{-})$ production
in the decay of $\Lambda_{c}^{+}\to\pi^{+}\pi^{0}\pi^{-}\Sigma^{+}$ by
considering the triangle loop, which develops a singularity of the invariant
mass of $\pi^{0}\pi^{-}\Sigma^{+}$ system around 1880 MeV. Here, the
$\Sigma^{*}(1/2^{-})$ with a mass around 1430 MeV is a dynamically generated
resonance from the meson-baryon interaction and is a cusp structure
Roca:2013cca . It is found that a narrow peak, tied to the
$\Sigma^{*}(1/2^{-})$ state, appears in the final $\pi^{-}\Sigma^{+}$ mass
spectrum at the energy around the $\bar{K}N$ threshold. Meanwhile, the
branching ratio for the $\Sigma^{*}(1/2^{-})$ signal is predicted to be of the
order of $10^{-4}$ Xie:2018gbi . In the future, Belle II, BESIII, and the
future Super Tau-Charm Factory (STCF) experiments could test such predictions.

##  IV Summary and outlook

Although quenched quark models have successfully described the ground-state
baryons as three-quark states, there are strong empirical and phenomenological
indications for larger sea-quark admixtures in the low-lying excited baryons.
Investigating the low-lying excited baryon $\Sigma^{*}(1/2^{-})$ is crucial to
determining the nature of the low-lying baryons and understanding meson-baryon
interactions. As the isovector partner of the $\Lambda(1405)$ and $N(1535)$,
the $\Sigma^{*}(1/2^{-})$ has motivated many experimental and theoretical
works in the last decades.

This review presents the experimental information on the
$\Sigma^{*}(1/2^{-})$. The $\Sigma(1620)$ is a 1-star resonance in the present
RPP, and its existence is still in debate. The $\Sigma(1480)$ appeared in the
previous version of RPP before 2019, and its existence is still to be
confirmed. Recently, Belle has observed one cusp structure around the
$\bar{K}N$ threshold in the $\Lambda\pi^{+}\pi^{+}\pi^{-}$ reaction, which
could be associated with the predicted $\Sigma^{*}(1/2^{-})$ resonance.

Next, we have provided a brief introduction of the theoretical studies on the
$\Sigma^{*}(1/2^{-})$ productions in the processes
$K^{-}p\to\Lambda\pi^{+}\pi^{-}$, $\Lambda p\to\Lambda p\pi^{0}$, $\gamma p\to
K^{+}\pi^{\pm}\Sigma^{\mp}$, which show some indications for the existence of
$\Sigma^{*}(1/2^{-})$. Meanwhile, it is shown that the processes of
$\chi_{c0}\to\bar{\Sigma}\Sigma\pi,\bar{\Lambda}\Sigma\pi$,
$J/\psi\to\Lambda\bar{\Lambda}\pi$,
$\Lambda_{c}^{+}\to\eta\Lambda\pi^{+},p\bar{K}^{0}\eta$, and
$\Lambda_{c}^{+}\to\pi^{+}\pi^{0}\pi^{-}\Sigma^{+}$ could be used to search
for the $\Sigma^{*}(1/2^{-})$ in the future.

By now, the BESIII, Belle, LHCb, and J-PARC Collaborations have accumulated a
wealth of data about the low-lying excited baryons. In the future, BESIII
BESIII:2020nme , Belle II Jia:2023upb , LHCb LHCb:2022sck , and J-PARC
Agari:2012kid  experiments, also the future STCF Achasov:2023gey , are
expected to yield more data on the relevant processes, which could shed light
on the existence of the low-lying excited $\Sigma^{*}(1/2^{-})$ state.

Another possible ideal clean platform to study the low-lying excited
$\Sigma^{*}(1/2^{-})$ state is the $\bar{\nu}_{l}+N\to l^{+}+\Sigma^{*}\to
l^{+}+\Lambda\pi$ reactions Wu:2013kla  to avoid complicated strong final-
state interactions with other hadrons in electric or hadron production
processes. With increasing intensity of neutrino fluxes, there are many new
measurements of neutrino nuclear scattering from various groups at Fermilab
ArgoNeuT:2014rlj , KEK T2K:2014hih  and CERN FASER:2023zcr . The study of
$\bar{\nu}_{l}+N\to l^{+}+\Sigma^{*}\to l^{+}+\Lambda\pi$ reaction may be
considered and realized.

## Acknowledgements

We would like to thank Eulogio Oset for useful comments. LSG and JJX are
partly supported by the National Key R&D Program of China under Grant No.
2023YFA1606700. This work is supported by the Natural Science Foundation of
Henan under Grant No. 232300421140 and No. 222300420554, the National Natural
Science Foundation of China under Grant No. 12192263, 12205075, 12175239,
12221005, 12075288, 12361141819, and also by the Project of Youth Backbone
Teachers of Colleges and Universities of Henan Province (2020GGJS017), and
also by the Open Project of Guangxi Key Laboratory of Nuclear Physics and
Nuclear Technology, No. NLK2021-08, and also by the Central Government
Guidance Funds for Local Scientific and Technological Development, China (No.
Guike ZY22096024), and also by the National Key Research and Development
Program of China under Contracts 2020YFA0406400, and also by the Chinese
Academy of Sciences under Grant No. YSBR-101, and also by the Youth Innovation
Promotion Association CAS.

## References

  * (1) M. Gell-Mann, Phys. Lett. 8 (1964), 214-215 doi:10.1016/S0031-9163(64)92001-3
  * (2) G. Zweig, doi:10.17181/CERN-TH-401
  * (3) R. L. Workman et al. [Particle Data Group], PTEP 2022, 083C01 (2022) doi:10.1093/ptep/ptac097
  * (4) M. Ablikim et al. [BESIII], Phys. Rev. Lett. 129, no.19, 192002 (2022) [erratum: Phys. Rev. Lett. 130, no.15, 159901 (2023)] doi:10.1103/PhysRevLett.129.192002 [arXiv:2202.00621 [hep-ex]].
  * (5) H. X. Chen, N. Su and S. L. Zhu, Chin. Phys. Lett. 39 (2022) no.5, 051201 doi:10.1088/0256-307X/39/5/051201 [arXiv:2202.04918 [hep-ph]].
  * (6) H. X. Chen, W. Chen, X. Liu, Y. R. Liu and S. L. Zhu, Rept. Prog. Phys. 80, no.7, 076201 (2017) doi:10.1088/1361-6633/aa6420 [arXiv:1609.08928 [hep-ph]].
  * (7) H. X. Chen, W. Chen, X. Liu, Y. R. Liu and S. L. Zhu, Rept. Prog. Phys. 86 (2023) no.2, 026201 doi:10.1088/1361-6633/aca3b6 [arXiv:2204.02649 [hep-ph]].
  * (8) F. K. Guo, C. Hanhart, U. G. Meißner, Q. Wang, Q. Zhao and B. S. Zou, Rev. Mod. Phys. 90 (2018) no.1, 015004 [erratum: Rev. Mod. Phys. 94 (2022) no.2, 029901] doi:10.1103/RevModPhys.90.015004 [arXiv:1705.00141 [hep-ph]].
  * (9) E. Oset, W. H. Liang, M. Bayar, J. J. Xie, L. R. Dai, M. Albaladejo, M. Nielsen, T. Sekihara, F. Navarra and L. Roca, et al. Int. J. Mod. Phys. E 25 (2016), 1630001 doi:10.1142/S0218301316300010 [arXiv:1601.03972 [hep-ph]].
  * (10) M. Z. Liu, Y. W. Pan, Z. W. Liu, T. W. Wu, J. X. Lu and L. S. Geng, [arXiv:2404.06399 [hep-ph]].
  * (11) H. Y. Gao and B. Q. Ma, Mod. Phys. Lett. A 14 (1999), 2313-2319 doi:10.1142/S021773239900239X [arXiv:hep-ph/0305294 [hep-ph]].
  * (12) J. M. M. Hall, W. Kamleh, D. B. Leinweber, B. J. Menadue, B. J. Owen, A. W. Thomas and R. D. Young, Phys. Rev. Lett. 114 (2015) no.13, 132002 doi:10.1103/PhysRevLett.114.132002 [arXiv:1411.3402 [hep-lat]].
  * (13) B. C. Liu and B. S. Zou, Phys. Rev. Lett. 96 (2006) no.4, 042002 doi:10.1103/PhysRevLett.96.042002 [arXiv:nucl-th/0503069 [nucl-th]].
  * (14) L. S. Geng, E. Oset, B. S. Zou and M. Doring, Phys. Rev. C 79 (2009), 025203 doi:10.1103/PhysRevC.79.025203 [arXiv:0807.2913 [hep-ph]].
  * (15) S. Capstick and N. Isgur, Phys. Rev. D 34 (1986) no.9, 2809-2835 doi:10.1103/physrevd.34.2809
  * (16) A. Starostin et al. [Crystal Ball], Phys. Rev. C 64 (2001), 055205 doi:10.1103/PhysRevC.64.055205
  * (17) M. Ablikim et al. [BESIII], Phys. Rev. D 99 (2019) no.3, 032010 doi:10.1103/PhysRevD.99.032010 [arXiv:1812.10731 [hep-ex]].
  * (18) J. Y. Lee et al. [Belle], Phys. Rev. D 103, no.5, 052005 (2021) doi:10.1103/PhysRevD.103.052005 [arXiv:2008.11575 [hep-ex]].
  * (19) G. P. Gopal et al. [Rutherford-London], Nucl. Phys. B 119 (1977), 362-400 doi:10.1016/0550-3213(77)90002-5
  * (20) S. B. Yang et al. [Belle], Phys. Rev. D 108 (2023) no.3, L031104 doi:10.1103/PhysRevD.108.L031104 [arXiv:2209.00050 [hep-ex]].
  * (21) S. C. Zhang, M. Y. Duan, W. T. Lyu, G. Y. Wang, J. Y. Zhu and E. Wang, [arXiv:2405.14235 [hep-ph]].
  * (22) B. S. Zou, Nucl. Phys. A 835 (2010), 199-206 doi:10.1016/j.nuclphysa.2010.01.194 [arXiv:1001.1084 [nucl-th]].
  * (23) B. S. Zou, [arXiv:1603.03927 [hep-ph]].
  * (24) N. Isgur and G. Karl, Phys. Rev. D 18 (1978), 4187 doi:10.1103/PhysRevD.18.4187
  * (25) N. Kaiser, P. B. Siegel and W. Weise, Nucl. Phys. A 594 (1995), 325-345 doi:10.1016/0375-9474(95)00362-5 [arXiv:nucl-th/9505043 [nucl-th]].
  * (26) E. Oset and A. Ramos, Nucl. Phys. A 635 (1998), 99-120 doi:10.1016/S0375-9474(98)00170-5 [arXiv:nucl-th/9711022 [nucl-th]].
  * (27) J. A. Oller and U. G. Meissner, Phys. Lett. B 500, 263-272 (2001) doi:10.1016/S0370-2693(01)00078-8 [arXiv:hep-ph/0011146 [hep-ph]].
  * (28) D. Jido, J. A. Oller, E. Oset, A. Ramos and U. G. Meissner, Nucl. Phys. A 725 (2003), 181-200 doi:10.1016/S0375-9474(03)01598-7 [arXiv:nucl-th/0303062 [nucl-th]].
  * (29) C. Helminen and D. O. Riska, Nucl. Phys. A 699 (2002), 624-648 doi:10.1016/S0375-9474(01)01294-5 [arXiv:nucl-th/0011071 [nucl-th]].
  * (30) B. S. Zou, Eur. Phys. J. A 35 (2008), 325-328 doi:10.1140/epja/i2007-10561-8 [arXiv:0711.4860 [nucl-th]].
  * (31) A. Zhang, Y. R. Liu, P. Z. Huang, W. Z. Deng, X. L. Chen and S. L. Zhu, HEPNP 29 (2005), 250 [arXiv:hep-ph/0403210 [hep-ph]].
  * (32) N. Kaiser, T. Waas and W. Weise, Nucl. Phys. A 612 (1997), 297-320 doi:10.1016/S0375-9474(96)00321-1 [arXiv:hep-ph/9607459 [hep-ph]].
  * (33) T. Inoue, E. Oset and M. J. Vicente Vacas, Phys. Rev. C 65, 035204 (2002) doi:10.1103/PhysRevC.65.035204 [arXiv:hep-ph/0110333 [hep-ph]].
  * (34) P. C. Bruns, M. Mai and U. G. Meissner, Phys. Lett. B 697 (2011), 254-259 doi:10.1016/j.physletb.2011.02.008 [arXiv:1012.2233 [nucl-th]].
  * (35) K. P. Khemchandani, A. Martinez Torres, H. Nagahiro and A. Hosaka, Phys. Rev. D 88 (2013) no.11, 114016 doi:10.1103/PhysRevD.88.114016 [arXiv:1307.8420 [nucl-th]].
  * (36) J. Nieves, A. Pich and E. Ruiz Arriola, Phys. Rev. D 84 (2011), 096002 doi:10.1103/PhysRevD.84.096002 [arXiv:1107.3247 [hep-ph]].
  * (37) D. Gamermann, C. Garcia-Recio, J. Nieves and L. L. Salcedo, Phys. Rev. D 84 (2011), 056017 doi:10.1103/PhysRevD.84.056017 [arXiv:1104.2737 [hep-ph]].
  * (38) E. J. Garzon and E. Oset, Phys. Rev. C 91 (2015) no.2, 025201 doi:10.1103/PhysRevC.91.025201 [arXiv:1411.3547 [hep-ph]].
  * (39) J. J. Xie, B. S. Zou and H. C. Chiang, Phys. Rev. C 77 (2008), 015206 doi:10.1103/PhysRevC.77.015206 [arXiv:0705.3950 [nucl-th]].
  * (40) M. Doring, E. Oset and B. S. Zou, Phys. Rev. C 78 (2008), 025207 doi:10.1103/PhysRevC.78.025207 [arXiv:0805.1799 [nucl-th]].
  * (41) J. X. Lu, E. Wang, J. J. Xie, L. S. Geng and E. Oset, Phys. Rev. D 93 (2016), 094009 doi:10.1103/PhysRevD.93.094009 [arXiv:1601.00075 [hep-ph]].
  * (42) W. T. Lyu, Y. H. Lyu, M. Y. Duan, G. Y. Wang, D. Y. Chen and E. Wang, [arXiv:2310.11139 [hep-ph]].
  * (43) E. Wang, H. X. Chen, L. S. Geng, D. M. Li and E. Oset, Phys. Rev. D 93, no.9, 094001 (2016) doi:10.1103/PhysRevD.93.094001 [arXiv:1512.01959 [hep-ph]].
  * (44) R. Molina, C. W. Xiao, W. H. Liang and E. Oset, Phys. Rev. D 109 (2024) no.5, 054002 doi:10.1103/PhysRevD.109.054002 [arXiv:2310.12593 [hep-ph]].
  * (45) H. P. Li, J. Song, W. H. Liang, R. Molina and E. Oset, [arXiv:2311.14365 [hep-ph]].
  * (46) C. D. Abell, D. B. Leinweber, Z. W. Liu, A. W. Thomas and J. J. Wu, Phys. Rev. D 108 (2023) no.9, 094519 doi:10.1103/PhysRevD.108.094519 [arXiv:2306.00337 [hep-lat]].
  * (47) D. Guo and Z. W. Liu, Phys. Rev. D 105 (2022) no.11, 114039 doi:10.1103/PhysRevD.105.114039 [arXiv:2201.11555 [hep-ph]].
  * (48) Z. W. Liu, W. Kamleh, D. B. Leinweber, F. M. Stokes, A. W. Thomas and J. J. Wu, Phys. Rev. Lett. 116 (2016) no.8, 082004 doi:10.1103/PhysRevLett.116.082004 [arXiv:1512.00140 [hep-lat]].
  * (49) S. Capstick and W. Roberts, Prog. Part. Nucl. Phys. 45 (2000), S241-S331 doi:10.1016/S0146-6410(00)00109-5 [arXiv:nucl-th/0008028 [nucl-th]].
  * (50) L. Y. Glozman and D. O. Riska, Phys. Rept. 268 (1996), 263-303 doi:10.1016/0370-1573(95)00062-3 [arXiv:hep-ph/9505422 [hep-ph]].
  * (51) E. Oset, A. Ramos and C. Bennhold, Phys. Lett. B 527, 99-105 (2002) [erratum: Phys. Lett. B 530, 260-260 (2002)] doi:10.1016/S0370-2693(01)01523-4 [arXiv:nucl-th/0109006 [nucl-th]].
  * (52) K. P. Khemchandani, A. Martínez Torres and J. A. Oller, Phys. Rev. C 100, no.1, 015208 (2019) doi:10.1103/PhysRevC.100.015208 [arXiv:1810.09990 [hep-ph]].
  * (53) Y. Kamiya, K. Miyahara, S. Ohnishi, Y. Ikeda, T. Hyodo, E. Oset and W. Weise, Nucl. Phys. A 954, 41-57 (2016) doi:10.1016/j.nuclphysa.2016.04.013 [arXiv:1602.08852 [hep-ph]].
  * (54) J. A. Oller, Eur. Phys. J. A 28, 63-82 (2006) doi:10.1140/epja/i2006-10011-3 [arXiv:hep-ph/0603134 [hep-ph]].
  * (55) C. Garcia-Recio, J. Nieves, E. Ruiz Arriola and M. J. Vicente Vacas, Phys. Rev. D 67, 076009 (2003) doi:10.1103/PhysRevD.67.076009 [arXiv:hep-ph/0210311 [hep-ph]].
  * (56) M. F. M. Lutz and E. E. Kolomeitsev, Nucl. Phys. A 700, 193-308 (2002) doi:10.1016/S0375-9474(01)01312-4 [arXiv:nucl-th/0105042 [nucl-th]].
  * (57) Z. H. Guo and J. A. Oller, Phys. Rev. C 87 (2013) no.3, 035202 doi:10.1103/PhysRevC.87.035202 [arXiv:1210.3485 [hep-ph]].
  * (58) L. Roca and E. Oset, Phys. Rev. C 88, no.5, 055206 (2013) doi:10.1103/PhysRevC.88.055206 [arXiv:1307.5752 [nucl-th]].
  * (59) K. Moriya et al. [CLAS], Phys. Rev. C 87, no.3, 035206 (2013) doi:10.1103/PhysRevC.87.035206 [arXiv:1301.5000 [nucl-ex]].
  * (60) J. X. Lu, L. S. Geng, M. Doering and M. Mai, Phys. Rev. Lett. 130 (2023) no.7, 071902 doi:10.1103/PhysRevLett.130.071902 [arXiv:2209.02471 [hep-ph]].
  * (61) V. Crede and W. Roberts, Rept. Prog. Phys. 76, 076301 (2013) doi:10.1088/0034-4885/76/7/076301 [arXiv:1302.7299 [nucl-ex]].
  * (62) E. Klempt and J. M. Richard, Rev. Mod. Phys. 82, 1095-1153 (2010) doi:10.1103/RevModPhys.82.1095 [arXiv:0901.2055 [hep-ph]].
  * (63) C. Chen, W. Q. Niu and H. Q. Zheng, Chin. Phys. C 46 (2022) no.8, 081001 doi:10.1088/1674-1137/ac68d7 [arXiv:2203.03747 [hep-ph]].
  * (64) A. Ramos, E. Oset and C. Bennhold, Phys. Rev. Lett. 89 (2002), 252001 doi:10.1103/PhysRevLett.89.252001 [arXiv:nucl-th/0204044 [nucl-th]].
  * (65) J. K. Kim, Phys. Rev. Lett. 27 (1971), 356-360 doi:10.1103/PhysRevLett.27.356
  * (66) W. Langbein and F. Wagner, Nucl. Phys. B 47 (1972), 477-525 doi:10.1016/0550-3213(72)90447-6
  * (67) P. Baillon and P. J. Litchfield, Nucl. Phys. B 94 (1975), 39-66 doi:10.1016/0550-3213(75)90085-1
  * (68) A. S. Carroll, I. H. Chiang, T. F. Kycia, K. K. Li, P. O. Mazur, D. N. Michael, P. M. Mockett, D. C. Rahm and R. Rubinstein, Phys. Rev. Lett. 37 (1976), 806 doi:10.1103/PhysRevLett.37.806
  * (69) W. A. Morris, J. R. Albright, A. P. Colleraine, J. D. Kimel and J. E. Lannutti, Phys. Rev. D 17 (1978), 55-61 doi:10.1103/PhysRevD.17.55
  * (70) P. Gao, B. S. Zou and A. Sibirtsev, Nucl. Phys. A 867 (2011), 41-51 doi:10.1016/j.nuclphysa.2011.07.008 [arXiv:1011.2387 [nucl-th]].
  * (71) A. V. Sarantsev, M. Matveev, V. A. Nikonov, A. V. Anisovich, U. Thoma and E. Klempt, Eur. Phys. J. A 55 (2019) no.10, 180 doi:10.1140/epja/i2019-12880-5 [arXiv:1907.13387 [nucl-ex]].
  * (72) M. Tanabashi et al. [Particle Data Group], Phys. Rev. D 98 (2018) no.3, 030001 doi:10.1103/PhysRevD.98.030001
  * (73) Y. L. Pan, F. L. Forman, W. Ko, V. Hagopian and W. Selove, Phys. Rev. D 2 (1970), 449-462 doi:10.1103/PhysRevD.2.449
  * (74) J. J. Engelen et al. [AMSTERDAM-CERN-NIJMEGEN-OXFORD], Nucl. Phys. B 167 (1980), 61-97 doi:10.1016/0550-3213(80)90119-4
  * (75) S. Chekanov et al. [ZEUS], Phys. Lett. B 591 (2004), 7-22 doi:10.1016/j.physletb.2004.04.024 [arXiv:hep-ex/0403051 [hep-ex]].
  * (76) I. Zychor, V. Koptev, M. Buscher, A. Dzyuba, I. Keshelashvili, V. Kleber, H. R. Koch, S. Krewald, Y. Maeda and S. Mikirtichyants, et al. Phys. Rev. Lett. 96 (2006), 012002 doi:10.1103/PhysRevLett.96.012002 [arXiv:nucl-ex/0506014 [nucl-ex]].
  * (77) Y. Ma et al. [Belle], Phys. Rev. Lett. 130 (2023) no.15, 151903 doi:10.1103/PhysRevLett.130.151903 [arXiv:2211.11151 [hep-ex]].
  * (78) J. J. Wu, S. Dulat and B. S. Zou, Phys. Rev. D 80, 017503 (2009) doi:10.1103/PhysRevD.80.017503 [arXiv:0906.3950 [hep-ph]].
  * (79) D. O. Huwe, Phys. Rev. 181 (1969), 1824-1853 doi:10.1103/PhysRev.181.1824
  * (80) J. J. Wu, S. Dulat and B. S. Zou, Phys. Rev. C 81, 045210 (2010) doi:10.1103/PhysRevC.81.045210 [arXiv:0909.1380 [hep-ph]].
  * (81) J. A. Kadyk, G. Alexander, J. H. Chan, P. Gaposchkin and G. H. Trilling, Nucl. Phys. B 27 (1971), 13-22 doi:10.1016/0550-3213(71)90076-9
  * (82) J. J. Xie, J. J. Wu and B. S. Zou, Phys. Rev. C 90 (2014) no.5, 055204 doi:10.1103/PhysRevC.90.055204 [arXiv:1407.7984 [nucl-th]].
  * (83) Y. H. Chen and B. S. Zou, Phys. Rev. C 88 (2013) no.2, 024304 doi:10.1103/PhysRevC.88.024304 [arXiv:1302.5466 [nucl-th]].
  * (84) L. Roca and E. Oset, Phys. Rev. C 87 (2013) no.5, 055201 doi:10.1103/PhysRevC.87.055201 [arXiv:1301.5741 [nucl-th]].
  * (85) J. A. Oller, E. Oset and J. R. Pelaez, Phys. Rev. D 59 (1999), 074001 [erratum: Phys. Rev. D 60 (1999), 099906; erratum: Phys. Rev. D 75 (2007), 099903] doi:10.1103/PhysRevD.59.074001 [arXiv:hep-ph/9804209 [hep-ph]].
  * (86) M. Ablikim et al. [BESIII], Phys. Rev. D 95 (2017) no.3, 032002 doi:10.1103/PhysRevD.95.032002 [arXiv:1610.02479 [hep-ex]].
  * (87) W. H. Liang, J. J. Xie and E. Oset, Eur. Phys. J. C 76 (2016) no.12, 700 doi:10.1140/epjc/s10052-016-4563-1 [arXiv:1609.03864 [hep-ph]].
  * (88) K. Hicks et al. [LEPS], Phys. Rev. Lett. 102 (2009), 012501 doi:10.1103/PhysRevLett.102.012501 [arXiv:0812.0771 [nucl-ex]].
  * (89) Y. Oh, C. M. Ko and K. Nakayama, Phys. Rev. C 77 (2008), 045204 doi:10.1103/PhysRevC.77.045204 [arXiv:0712.4285 [nucl-th]].
  * (90) P. Gao, J. J. Wu and B. S. Zou, Phys. Rev. C 81, 055203 (2010) doi:10.1103/PhysRevC.81.055203 [arXiv:1001.0805 [nucl-th]].
  * (91) K. Moriya et al. [CLAS], Phys. Rev. C 88 (2013), 045201 doi:10.1103/PhysRevC.88.045201 [arXiv:1305.6776 [nucl-ex]].
  * (92) S. H. Kim, K. P. Khemchandani, A. Martinez Torres, S. i. Nam and A. Hosaka, Phys. Rev. D 103 (2021) no.11, 114017 doi:10.1103/PhysRevD.103.114017 [arXiv:2101.08668 [hep-ph]].
  * (93) Y. H. Lyu, H. Zhang, N. C. Wei, B. C. Ke, E. Wang and J. J. Xie, Chin. Phys. C 47 (2023) no.5, 053108 [arXiv:2301.08615 [hep-ph]].
  * (94) X. L. Ren, E. Oset, L. Alvarez-Ruso and M. J. Vicente Vacas, Phys. Rev. C 91 (2015) no.4, 045201 doi:10.1103/PhysRevC.91.045201 [arXiv:1501.04073 [hep-ph]].
  * (95) E. Wang, J. J. Xie and E. Oset, Phys. Lett. B 753, 526-532 (2016) doi:10.1016/j.physletb.2015.12.060 [arXiv:1509.03367 [hep-ph]].
  * (96) L. J. Liu, E. Wang, J. J. Xie, K. L. Song and J. Y. Zhu, Phys. Rev. D 98, no.11, 114017 (2018) doi:10.1103/PhysRevD.98.114017 [arXiv:1712.07469 [hep-ph]].
  * (97) B. S. Zou, Int. J. Mod. Phys. A 21 (2006), 5552-5557 doi:10.1142/S0217751X06034732
  * (98) Q. Huang, Z. X. Ma, J. J. Wu, R. G. Ping, J. He and H. X. Huang, [arXiv:2405.11127 [hep-ph]].
  * (99) R. Ammar et al. [CLEO], Phys. Rev. Lett. 74 (1995), 3534-3537 doi:10.1103/PhysRevLett.74.3534
  * (100) J. J. Xie and L. S. Geng, Phys. Rev. D 95, no.7, 074024 (2017) doi:10.1103/PhysRevD.95.074024 [arXiv:1703.09502 [hep-ph]].
  * (101) G. Y. Wang, N. C. Wei, H. M. Yang, E. Wang, L. S. Geng and J. J. Xie, Phys. Rev. D 106, no.5, 056001 (2022) doi:10.1103/PhysRevD.106.056001 [arXiv:2206.01425 [hep-ph]].
  * (102) W. T. Lyu, S. C. Zhang, G. Y. Wang, J. J. Wu, E. Wang, L. S. Geng and J. J. Xie, [arXiv:2405.09226 [hep-ph]].
  * (103) L. K. Li et al. [Belle], Phys. Rev. D 107 (2023) no.3, 032004 doi:10.1103/PhysRevD.107.032004 [arXiv:2210.01995 [hep-ex]].
  * (104) J. J. Xie and L. S. Geng, Phys. Rev. D 96 (2017) no.5, 054009 doi:10.1103/PhysRevD.96.054009 [arXiv:1704.05714 [hep-ph]].
  * (105) R. Pavao, S. Sakai and E. Oset, Phys. Rev. C 98 (2018) no.1, 015201 doi:10.1103/PhysRevC.98.015201 [arXiv:1802.07882 [nucl-th]].
  * (106) Y. Li, S. W. Liu, E. Wang, D. M. Li, L. S. Geng and J. J. Xie, [arXiv:2406.01209 [hep-ph]].
  * (107) X. H. Liu, M. Oka and Q. Zhao, Phys. Lett. B 753 (2016), 297-302 doi:10.1016/j.physletb.2015.12.027 [arXiv:1507.01674 [hep-ph]].
  * (108) M. Bayar, F. Aceti, F. K. Guo and E. Oset, Phys. Rev. D 94 (2016) no.7, 074039 doi:10.1103/PhysRevD.94.074039 [arXiv:1609.04133 [hep-ph]].
  * (109) W. H. Liang, H. X. Chen, E. Oset and E. Wang, Eur. Phys. J. C 79 (2019) no.5, 411 doi:10.1140/epjc/s10052-019-6928-8 [arXiv:1903.01252 [hep-ph]].
  * (110) L. R. Dai, R. Pavao, S. Sakai and E. Oset, Phys. Rev. D 97 (2018) no.11, 116004 doi:10.1103/PhysRevD.97.116004 [arXiv:1804.01136 [hep-ph]].
  * (111) E. Wang, J. J. Xie, W. H. Liang, F. K. Guo and E. Oset, Phys. Rev. C 95 (2017) no.1, 015205 doi:10.1103/PhysRevC.95.015205 [arXiv:1610.07117 [hep-ph]].
  * (112) F. K. Guo, X. H. Liu and S. Sakai, Prog. Part. Nucl. Phys. 112 (2020), 103757 doi:10.1016/j.ppnp.2020.103757 [arXiv:1912.07030 [hep-ph]].
  * (113) J. J. Xie and E. Oset, Phys. Lett. B 792 (2019), 450-453 doi:10.1016/j.physletb.2019.04.011 [arXiv:1811.07247 [hep-ph]].
  * (114) M. Ablikim et al. [BESIII], Chin. Phys. C 44 (2020) no.4, 040001 doi:10.1088/1674-1137/44/4/040001 [arXiv:1912.05983 [hep-ex]].
  * (115) S. Jia, W. Xiong and C. Shen, Chin. Phys. Lett. 40 (2023) no.12, 121301 doi:10.1088/0256-307X/40/12/121301 [arXiv:2312.00403 [hep-ex]].
  * (116) R. Aaij et al. [LHCb], Phys. Rev. D 108 (2023) no.1, 012023 doi:10.1103/PhysRevD.108.012023 [arXiv:2208.03262 [hep-ex]].
  * (117) K. Agari, S. Enomoto, H. Fujioka, Y. Fujiwara, T. Hashimoto, R. S. Hayano, T. Hiraiwa, E. Hirose, M. Ieiri and Y. Igarashi, et al. PTEP 2012 (2012), 02B009 doi:10.1093/ptep/pts038
  * (118) M. Achasov, X. C. Ai, R. Aliberti, L. P. An, Q. An, X. Z. Bai, Y. Bai, O. Bakina, A. Barnyakov and V. Blinov, et al. Front. Phys. (Beijing) 19 (2024) no.1, 14701 doi:10.1007/s11467-023-1333-z [arXiv:2303.15790 [hep-ex]].
  * (119) J. J. Wu and B. S. Zou, Few Body Syst. 56 (2015) no.4-5, 165-183 doi:10.1007/s00601-015-0973-0 [arXiv:1307.0574 [hep-ph]].
  * (120) R. Acciarri et al. [ArgoNeuT], Phys. Rev. D 89 (2014) no.11, 112003 doi:10.1103/PhysRevD.89.112003 [arXiv:1404.4809 [hep-ex]].
  * (121) K. Abe et al. [T2K], Phys. Rev. D 92 (2015) no.11, 112003 doi:10.1103/PhysRevD.92.112003 [arXiv:1411.6264 [hep-ex]].
  * (122) H. Abreu et al. [FASER], Phys. Rev. Lett. 131 (2023) no.3, 3 doi:10.1103/PhysRevLett.131.031801 [arXiv:2303.14185 [hep-ex]].