# Cluster structures of the $0^{+}$ states of ${}^{13}_{\Lambda}{\rm C}$ with
the 3$\alpha+\Lambda$ four-body orthogonality condition model

Qian Wu<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy of
Science, Lanzhou 730000, China  Yasuro Funaki<EMAIL_ADDRESS>College of Science and Engineering, Kanto Gakuin University, Yokohama
236-8501, Japan  RIKEN Nishina Center,RIKEN,2-1 Hirosawa,351-0115
Saitama,Japan  Xurong Chen  Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  School of Nuclear Science and Technology,
University of Chinese Academy of Sciences, Beijing 100049, China  Guangdong
Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter,
South China Normal University, Guangzhou 510006, China

###### Abstract

We study structures of the $0^{+}$ states of ${}^{13}_{\Lambda}$C within the
framework of $\alpha+\alpha+\alpha+\Lambda$ four-body orthogonality condition
model with the Gaussian expansion method. We obtain five $0^{+}$ states of
${}^{13}_{\Lambda}$C and compare them with the low-lying four $0^{+}$ states
of 12C. We find that the $0_{1}^{+}$, $0_{2}^{+}$ and $0_{4}^{+}$ states
correspond to the $0_{1}^{+}$, $0_{2}^{+}$ and $0_{4}^{+}$ states of 12C,
respectively. We also find that both the $0_{3}^{+}$ and $0_{5}^{+}$ states
have analogous nature to the $0_{3}^{+}$ state of 12C, where the additional
$\Lambda$ particle couples with 8Be cluster and one $\alpha$ cluster, in the
$0_{3}^{+}$ state of 12C, respectively. The former is then shown to have an
elongated ${}^{9}_{\Lambda}$Be$(0_{1}^{+})+\alpha$ cluster structure with a
higher nodal behaviour for their relative wave function, and the latter have a
dilute three-body gas consisting of $2\alpha+^{5}_{\Lambda}$He clusters.

OCM, hypernuclei, Microscopic cluster model

##  I Introduction

Nuclear clustering is extremely important in understanding the structures of
nuclear excitation wildermuth . According to the threshold rule, the so-called
Ikeda diagram ikedagram  tells us that the cluster structure appears around
the corresponding clusters breaking up threshold. The $0^{+}_{2}$ at 7.65 MeV
in 12C, known as the Hoyle state, is a typical example of the cluster states.
It is predicted by Fred Holye Hoyle1954  at 1954 which is a gateway state for
massive 12C production in stellar evolution and subsequently observed by Cook
et al Cook1957  at 1957. There is a long history in studying the structure of
the Hoyle state. In the early stage, Morinaga proposed a linear chain
structure of the $3\alpha$ clusters for the Hoyle state Morinag1956 . However
in 1975, with the semi-microscopic cluster model, known as the Orthogonality
Condition Model (OCM), Horiuchi proposed that the Hoyle state has a weakly
coupled $3\alpha$-cluster structure with relative $S$-waves HoriuchiOCM . That
was soon confirmed by Kamimura kamimura1981RGM  and Uegaki Uegaki1977  with
the fully microscopic cluster model calculations, Resonating Group Method
(RGM) and the Generator Coordinate Method (GCM), respectively. After a few
decades, an interesting conjecture was proposed that the Hoyle state has a
Bose-condensate-like structure, where all $3\alpha$ particles weakly interact
with each other and occupy an identical $S$-orbit THSR2001 . This new
interpretation was well established by the fact that the so-called Tohsaki-
Horiuchi-Schuck-Röpke (THSR) wave function was proved to be almost equal to
the RGM or GCM wave function for the Holyle state Funaki2003 ; Funaki2002 .

Moreover, the observed $0^{+}$ state at 10.3 MeV has been known for a long
time as a resonance with a broad width of about 3 MeV. Recently, Itoh et
al.Itoh_2013  reported that this broad $0^{+}$ state is decomposed into two
states, the $0^{+}_{3}$ and $0^{+}_{4}$ states at 9.04 MeV and 10.56 MeV, with
the width of 1.45 MeV and 1.42 MeV, respectively. These resonance parameters
are well reproduced by the OCM $+$ Complex Scaling Method (CSM) calculations
OCM2007 ; kami2013 . However, in several calculations without imposing a
resonance boundary condition, such like the Antisymmetrized Molecular Dynamics
(AMD) AMD2007 , the Fermionic Molecular Dynamics (FMD) FMD2007  and the GCM
GCM2015 , it seems that the observed new $0_{3}^{+}$ state is missing and only
the new $0^{+}_{4}$ state is reproduced. As for the structures of the
$0^{+}_{3}$ state, the OCM calculation in Ref.OCM2007  claimed that it has an
$S$-wave higher nodal structure for the relative wave function between the 8Be
and $\alpha$ clusters. Although the observed $0^{+}_{3}$ state is missing in
the AMD and FMD calculations AMD2007 ; FMD2007 , they give a single $0^{+}$
state above the Hoyle state, which has a linear-chain-like structure of the
$3\alpha$ clusters and may correspond to the observed $0^{+}_{4}$ state. Later
this linear-chain-like structure for the $0_{4}^{+}$ state was confirmed with
the THSR wave function, together with the simultaneous description of the
$0_{3}^{+}$ state, which behaves like an excited state of the Hoyle state
funaki2016 .

While the Hoyle state, which has a well-developed $3\alpha$ cluster structure,
locates slightly (0.38 MeV) above the $3\alpha$ breakup threshold, in
${}^{13}_{\Lambda}$C, the other new thresholds decaying into the
${}^{9}_{\Lambda}{\rm Be}+\alpha$, ${}^{5}_{\Lambda}{\rm He}+^{8}{\rm Be}$ and
${}^{12}{\rm C}+\Lambda$ clusters appear below the $3\alpha+\Lambda$ four-body
breakup threshold (see Fig. 2). Since the cluster states are likely to appear
around the corresponding thresholds, according to the threshold rule, it seems
that more variety of cluster structures appear and hence the clustering
becomes more important, when a $\Lambda$ particle is added into a core
nucleus, i.e. like in ${}^{13}_{\Lambda}$C.

Concerning the single $\Lambda$ hypernucleus, the simple $2\alpha+\Lambda$
system, ${}^{9}_{\Lambda}$Be, has already been well established by many
cluster model calculations qian2020 ; Lee2019few ; Hiyama1997ptep ; AMDbe9l ;
Bando1983ptp ; Motoba1985ptp ; Yamada1988prc . ${}^{13}_{\Lambda}$C has also
been investigated by using the microscopic and semi-microscopic cluster
models, RGM, GCM, and OCM. In Ref.Hiyama1997ptep , Hiyama et al. fully solved
the $3\alpha+\Lambda$ four-body problem in ${}^{13}_{\Lambda}$C with the OCM
and the Gaussian expansion method Hiyama2003gem . They obtained the
$(1/2)_{1}^{+}$ and $(1/2)_{2}^{+}$ states and concluded that both the states
correspond to the $0^{+}_{1}$ and the Hoyle state ($0_{2}^{+}$) in 12C,
respectively, where the former state has almost no change of the spatial size
of the ${}^{12}{\rm C}$ core and the latter has a drastic spatial shrinkage
from the 12C core, by adding the $\Lambda$ particle. Yamada et al. studied the
structures of the $(1/2)_{1}^{+}$, $(1/2)_{2}^{+}$, and $(1/2)_{3}^{+}$ states
by solving the $\Lambda+{{}^{12}{\rm C}}$ coupled channel equation
Yamada1985ptep , where ${}^{12}{\rm C}$ is described by the $3\alpha$ RGM wave
function kamimura1981RGM . They found that the $(1/2)_{1}^{+}$,
$(1/2)_{2}^{+}$ and $(1/2)_{3}^{+}$ states are dominated by the components of
$\Lambda(0s)+^{12}{\rm C}(0_{1}^{+})$, $\Lambda(0s)+^{12}{\rm C}(0_{2}^{+})$,
and $\Lambda(1s)+^{12}{\rm C}(0_{1}^{+})$ configurations, respectively.
However, all these calculations of ${}^{13}_{\Lambda}{\rm C}$ were done within
the bound state approximation, and also higher excitation energy region up to
around the $3\alpha+\Lambda$ threshold has never been investigated.

The study of exploring the higher excitation energy region in
${}^{13}_{\Lambda}$C was, however, done by using the so-called Hyper-THSR wave
function (H-THSR), where the THSR wave function is adapted to apply to the
$\Lambda$ hypernucleus Funaki2014ptep ; Funaki2018plb , though any correct
boundary condition of resonances is not imposed. The authors obtained the four
$0^{+}$ states, neglecting the spin-orbit coupling between the $\Lambda$
particle and core, and compared them with the $0_{1}^{+}$-$0_{4}^{+}$ states
of ${}^{12}{\rm C}$. While the $0_{1}^{+}$ and $0_{2}^{+}$ states of
${}^{13}_{\Lambda}$C are considered to correspond to the states, in which the
$\Lambda$ particle couples to the ground state and the Hoyle state in
${}^{12}{\rm C}$, respectively, the $0_{3}^{+}$ and $0_{4}^{+}$ states are
considered to correspond to the $0_{4}^{+}$ and $0_{3}^{+}$ states in
${}^{12}{\rm C}$, respectively. It was shown that the former has the linear-
chain-like structure of the $3\alpha$ clusters with the $\Lambda$ particle and
the latter has the more dilute gaslike structure of ${}^{5}_{\Lambda}{\rm
He}+2\alpha$ clusters, as being the analog to the Hoyle state. It was also
argued that the $\Lambda$ particle only couples to one $\alpha$ cluster to
form ${}^{5}_{\Lambda}{\rm He}$ in the $0_{4}^{+}$ state, and therefore the
state is less bound than the $0_{3}^{+}$ state, in which the additional
$\Lambda$ paticle couples all the $\alpha$ clusters, keeping the $3\alpha$
linear-chain-like structure.

In this work, we study the structures of the $0^{+}$ states of
${}^{13}_{\Lambda}$C up to around the $3\alpha+\Lambda$ breakup threshold
within the $3\alpha+\Lambda$ four-body OCM, where the relative motions between
the clusters are fully solved by using the Gaussian basis functions. Relative
angular momentum channels are taken to cover all the possible configurations
including the 12C+$\Lambda$, 8Be+${}^{5}_{\Lambda}$He and
${}^{9}_{\Lambda}$Be+$\alpha$ clustering in ${}^{13}_{\Lambda}$C. We use the
$\alpha$-$\alpha$ and $\Lambda$N interaction, which reproduce the binding
energies of 8Be, ${}^{5}_{\Lambda}$He, the ground and the first excited states
of ${}^{9}_{\Lambda}$Be, and the ground state of ${}^{13}_{\Lambda}$C. We
obtain five $0^{+}$ states in ${}^{13}_{\Lambda}$C. With calculating the S2
factors and mass radius, we find that the $0_{1}^{+}$, $0_{2}^{+}$ and
$0_{4}^{+}$ states in ${}^{13}_{\Lambda}$C correspond to the $0_{1}^{+}$,
$0_{2}^{+}$ and $0_{4}^{+}$ states of 12C, respectively. Both the $0_{3}^{+}$
and $0_{5}^{+}$ states have an analogous structure to the $0_{3}^{+}$ state of
12C, where the additional $\Lambda$ particle couples with 8Be cluster and one
$\alpha$ cluster, in the $0_{3}^{+}$ state of 12C, respectively.

The paper is organized as follows. In Sec. II, we introduce the realistic NN
and $\Lambda$N interaction and the unique adjustments of the potential
parameters. The energy levels of the $0_{1}^{+}$-$0_{5}^{+}$ states and their
rms radii are discussed in Sec. III.1. In Sec. III.2 and Sec. III.3, the
Reduced Width Amplitudes (RWAs) and the $S^{2}$-factors of the $0_{1}^{+}$,
$0_{2}^{+}$ and $0_{4}^{+}$ states and the $0_{3}^{+}$ and $0_{5}^{+}$ states
are discussed, respectively, and their structures are compared with those of
the corresponding $0^{+}$ state of ${}^{12}{\rm C}$. Sec. IV is devoted to
Summary.

##  II Hamiltonian and Method

We solve a four-body Schrödinger equation in ${}^{13}_{\Lambda}{\rm C}$
composed of the $3\alpha+\Lambda$ clusters with the semi-microscopic cluster
model, the OCM, where the hamiltonian is then written as follows:

$H=T+\sum_{i<j=1}^{3}V_{\alpha_{i}\alpha_{j}}+\sum_{i=1}^{3}V_{\alpha_{i}\Lambda}+V_{\alpha_{1}\alpha_{2}\alpha_{3}}+\sum_{i<j=1}^{3}V_{\alpha_{i}\alpha_{j}}^{\text{Pauli
}},$ (1)

where $T$ is the kinetic energy operator for the relative motions between the
clusters, and the $V_{\alpha_{i}\alpha_{j}}$ and $V_{\alpha_{i}\Lambda}$ are
the $\alpha-\alpha$ and $\alpha-\Lambda$ interaction operators, respectively.
The $V_{\alpha_{i}\alpha_{j}}^{\text{Pauli}}$ and
$V_{\alpha_{1}\alpha_{2}\alpha_{3}}$ represent the Pauli exclusion operator
between the $\alpha$ clusters and the phenomenological 3-body interaction
operator between the $3\alpha$ clusters, respectively, which will be explained
later in detail.

Figure 1:  Jacobi coordinates of ${}^{13}_{\Lambda}$C with $3\alpha+\Lambda$
model.

The total wave function with the $J^{\pi}=0^{+}$ of the four-body
$3\alpha+\Lambda$ cluster model can be described as follows:

$\displaystyle\Psi\left({}_{\Lambda}^{13}\mathrm{C}\right)$
$\displaystyle=\sum_{c=1}^{4}\sum_{L}\sum_{n_{1},l_{1}}\sum_{n_{2},l_{2}}\sum_{n_{3},l_{3}}C_{n_{1}l_{1}n_{2}l_{2}n_{3}l_{3}L}^{(c)}$
(2)
$\displaystyle\times\mathcal{A}_{\alpha}\Big{[}\Phi^{\left(\alpha_{1}\right)}\Phi^{\left(\alpha_{2}\right)}\Phi^{\left(\alpha_{3}\right)}$
$\displaystyle\times\left[\left[\phi_{n_{1}l_{1}}^{(c)}\left(\bm{r}_{c}\right)\psi_{n_{2}l_{2}}^{(\mathrm{c})}\left(\bm{R}_{c}\right)\right]_{L}\varphi_{n_{3}l_{3}}^{(c)}\left(\bm{\rho}_{c}\right)\right]_{J=0}\Big{]},$

where the four sets of Jacobian coordinates ($c=1-4$), shown in Fig. 1, are
used, the $\mathcal{A}_{\alpha}$ is the $\alpha-\alpha$ symmetrization
operator, and $\Phi^{(\alpha_{i})}$ with $i=1,2,3$ are the intrinsic wave
functions of the $\alpha$ cluster with the $(0s)^{4}$ configuration. The
relative wave functions between the clusters, corresponding to the three
Jacobi coordinates, $\phi_{n_{1}l_{1}}(\bm{r})$, $\psi_{n_{2}l_{2}}(\bm{R})$
and $\varphi_{n_{3}l_{3}}(\bm{\rho})$, are expanded by using the following
Gaussian basis functions, applying the Gaussian Expansion Method Hiyama2003gem
; Hiyama2009ppnp ,

$\displaystyle\phi_{n_{1}l_{1}}(\bm{r})=r^{\ell_{1}}e^{-(r/r_{n_{1}})^{2}}Y_{\ell_{1}m_{1}}(\hat{\mbox{\boldmath$r$}}),$
$\displaystyle\psi_{n_{2}l_{2}}(\bm{R})=R^{\ell_{2}}e^{-(R/R_{n_{2}})^{2}}Y_{\ell_{2}m_{2}}(\hat{\mbox{\boldmath$R$}}),$
$\displaystyle\varphi_{n_{3}l_{3}}(\bm{\rho})=\rho^{\ell_{3}}e^{-(\rho/\rho_{n_{3}})^{2}}Y_{\ell_{3}m_{3}}(\hat{\mbox{\boldmath$\rho$}}).$
(3)

The Gaussian variational parameters are chosen to have geometric progression
below,

$\displaystyle r_{n_{1}}=r_{\rm min}A_{1}^{n_{1}-1},\quad(n_{1}=1\sim
n_{1}^{\rm max}),$ $\displaystyle R_{n_{2}}=R_{\rm
min}A_{2}^{n_{2}-1},\quad(n_{2}=1\sim n_{2}^{\rm max}),$
$\displaystyle\rho_{n_{3}}=\rho_{\rm min}A_{3}^{n_{3}-1},\quad(n_{3}=1\sim
n_{3}^{\rm max}).$ (4)

Since the energy splitting of the $3/2^{+}-5/2^{+}$ is small enough only to be
measured by the high resolution $\gamma$-ray experimentsHiyama2009ppnp ;
Akikawa2002be9lgamma ; Tamura2004gammaray , we neglect the spin orbit coupling
between the $\Lambda$ and $\alpha$ clusters. The eigen energies and the
coefficients $C_{n_{1}l_{1}n_{2}l_{2}n_{3}l_{3}L}^{(c)}$ are obtained with
applying the Rayleigh-Ritz variational method.

The $\alpha$-$\alpha$ interaction is constructed by the folding procedure from
the modified Hasegawa-Nagata (MHN) effective nucleon-nucleon potential
hasegawa1971aa  and proton-proton coulomb potential, so as to reproduce the
observed $\alpha$-$\alpha$ scattering phase shift and the binding energy of
the ground state of 8Be within the $\alpha$-$\alpha$ OCM. The Pauli principle
between the two $\alpha$ clusters in the OCM is taken into account by
introducing the following Pauli exclusion operator into the Hamiltonian in Eq.
(1),

$V_{\alpha_{i}\alpha_{j}}^{\text{Pauli
}}=\lim\limits_{\lambda\rightarrow\infty}\lambda\sum_{f=0s,1s,0d}|\phi_{f}({\mbox{\boldmath$r$}}_{\alpha_{i}\alpha_{j}})\rangle\langle\phi_{f}({\mbox{\boldmath$r$}}^{\prime}_{\alpha_{i}\alpha_{j}})|.$
(5)

The Pauli forbidden states($0s,1s,0d$) are ruled out when $\lambda$ is an
infinity and practically the $\lambda$ is given around $\sim 10^{5}$MeV, which
is high enough to push the nonphysical states into a large energy region
without affecting the physical states. The harmonic oscillator size parameter
of the $0s$ wave function of the $\alpha$ cluster, $\Phi^{(\alpha_{i})}$ with
$i=1,2,3$ in Eq. (2), is taken as $1.358$ fm.

It is, however, well known that the folding $\alpha$-$\alpha$ potential given
by the MHN force significantly overbinds 12C, as discussed in
Ref.Hiyama1997ptep . We therefore introduce the following phenomenological
three-body repulsive force between the $3\alpha$ clusters,

$V_{\alpha_{1}\alpha_{2}\alpha_{3}}=V_{0}\exp\left[-\mu\left(r_{\alpha_{1}\alpha_{2}}^{2}+r_{\alpha_{2}\alpha_{3}}^{2}+r_{\alpha_{3}\alpha_{1}}^{2}\right)\right],$
(6)

where $V_{0}=154$ MeV and $\mu=0.195\ {\rm fm}^{-2}$ are adopted, to reproduce
the observed binding energy of the ground state of 12C, $E=-7.27$ MeV,
relative to the $3\alpha$ threshold. Then the Hoyle state is located at
$E=0.86$ MeV above the $3\alpha$ threshold, while the experimental value is
$0.38$ MeV.

The $\Lambda$-$\alpha$ interaction is also obtained by the folding procedure
of the $\Lambda$N interaction. We adopt the so-called YNG interaction
Yamamoto1994yng  as the $\Lambda$N interaction, which simulates the G-matrix
$\Lambda$N interaction derived from the Nijmegen model f (NF), of the three-
range Gaussian form as a function of fermi momentum $k_{F}$, given by,

$\displaystyle V_{\Lambda
N}(r,k_{F})=\sum\limits_{i=1}^{3}[(v_{0,even}^{i}+v_{\sigma\sigma,even}^{i}\sigma_{\Lambda}\cdot\sigma_{N})\frac{1+P_{r}}{2}$
$\displaystyle+(v_{0,odd}^{i}+v_{\sigma\sigma,odd}^{i}\sigma_{\Lambda}\cdot\sigma_{N})\frac{1-P_{r}}{2}]e^{-(r/\beta_{i})^{2}},$
(7)

where $P_{r}$ is the space exchange operator. However, it should be noted that
with this NF interaction, the ground state of ${}^{9}_{\Lambda}$Be is
overbound, due to the strong attraction of odd-state component of spin
independent part of the $\Lambda N$ interaction Hiyama1997ptep . We thus tune
the odd state part of the $\Lambda N$ interaction to reproduce the observed
binding energy of ${}^{5}_{\Lambda}$He and ${}^{9}_{\Lambda}$Be. The
parameters of this modified NF interaction are listed in Table 1.

Table 1: Parameters of the modified YNG-NF $\Lambda N$ interaction, where $k_{F}=0.963$ fm-1 is taken.  $\beta_{i}$ (fm) |  1.50 |  0.90 |  0.50
---|---|---|---
$v_{0,{\rm even}}^{i}$ |  $-9.22$ |  $-187.63$ |  795.43
$v_{0,{\rm odd}}^{i}$ |  $-5.67$ |  $-35.26$ |  2141.79(1172.21)

Then, with the $\alpha-\Lambda$ and $\alpha-\alpha$ interaction mentioned
above, the subsystems of ${}^{13}_{\Lambda}$C, including 8Be($0^{+}$),
8Be($2^{+}$), ${}^{5}_{\Lambda}$He, 12C($0_{1}^{+}$), 12C($0_{2}^{+}$),
${}^{9}_{\Lambda}$Be($0^{+}$) and ${}^{9}_{\Lambda}$Be($2^{+}$) are well
reproduced, compared to the experimental value. However, the use of these
interactions leads to the overbinding of the ground state of
${}^{13}_{\Lambda}{\rm C}$ by about $3$ MeV, while the original parameter set
of odd state part of the $\Lambda$N interaction, shown in the parenthesis in
Table.1, with $k_{F}=1.062$ fm, gives much better agreement with the
experimental data Hiyama1997ptep . This is because the ground state of
${}^{13}_{\Lambda}{\rm C}$ has a higher denisty with more compact shell-model-
like struture than ${}^{9}_{\Lambda}{\rm Be}$ and the $0_{2}^{+}$ state of
${}^{13}_{\Lambda}{\rm C}$. We then replace the ground state of
${}^{13}_{\Lambda}{\rm C}$ with the one obtained by the more appropriate
interaction (see Fig. 2).

##  III Results and Discussions

###  III.1 Energy spectra of the $0^{+}$ states of ${}^{13}_{\Lambda}{\rm C}$

Figure 2:  The calculated energy levels of 12C and ${}^{13}_{\Lambda}$C. The
left two columns show the $J^{\pi}=0^{+}$ spectra of ${}^{13}_{\Lambda}$C and
12C, which are calculated with the $3\alpha$ and $3\alpha+\Lambda$ THSR
ansatz, respectively Funaki2018plb . The right two columns show the
$J^{\pi}=0^{+}$ spectra of ${}^{13}_{\Lambda}$C and 12C, which are calculated
with the $3\alpha$ and $3\alpha+\Lambda$ OCM, respectively. The experimental
values of 12C are shown in the the middle column Cook1957 ; Itoh_2013 .

In Fig. 2, we show the calculated $0^{+}$ states of ${}^{13}_{\Lambda}{\rm
C}$, together with the $0^{+}$ states of ${}^{12}{\rm C}$, which are obtained
with the $3\alpha$ OCM with the same $\alpha$-$\alpha$ interaction as used in
${}^{13}_{\Lambda}{\rm C}$. The results of Ref. Funaki2018plb  obtained by
using the H-THSR wave function and those of ${}^{12}{\rm C}$ by the same THSR
ansatz in Ref. funaki2015hoyleband , together with the corresponding
experimental data, are also shown.

First, we discuss the $0^{+}$ states of ${}^{12}{\rm C}$, which are shown in
the middle of Fig. 2. While the ground state of ${}^{12}{\rm C}$ has a shell-
model-like structure, the $0^{+}_{2}$ state, i.e. the Hoyle state, has a
gaslike structure of the $3\alpha$ clusters HoriuchiOCM ; THSR2001 ;
Funaki2003 . The $0^{+}_{3}$ state is considered to have a higher nodal
structure for the ${}^{8}{\rm Be}+\alpha$ relative wave function. It is also
emphasized that this state has an analogous structure to the Hoyle state, as
excited from the Hoyle state with a large monopole transition strength. On the
other hand, the $0_{4}^{+}$ state is considered to have a linear-chain-like
structure of the $3\alpha$ clusters Refs.AMD2007 ; FMD2007 ; funaki2016 ,
which is the different cluster structure from those of the Hoyle and
$0^{+}_{3}$ states. We consider that all these four $0^{+}$ states are well
reproduced by the present $3\alpha$ OCM calculation. The $0_{2}^{+}$,
$0_{3}^{+}$, and $0_{4}^{+}$ states are obtained at $0.8$ MeV, $1.8$ MeV, and
$4.9$ MeV above the $3\alpha$ threshold, respectively, within the bound state
approximation. The calculated root mean square (rms) radii of these states are
shown in Table. 2. The rms radius of the Hoyle state, $4.2$ fm, is much larger
than that of the more compact ground state, $2.4$ fm, and is smaller than
those of the $0_{3}^{+}$ and $0_{4}^{+}$ states, $5.6$ fm and $5.4$ fm,
respectively. This tendency is consistent with the previous calculations with
the THSR ansatz funaki2016 .

Let us then discuss the $0^{+}$ states of ${}^{13}_{\Lambda}$C, which are also
shown in Fig. 2. As we mentioned in the previous section, the ground state is
calculated with the original parameter sets of $\Lambda N$ interaction
Yamamoto1994yng . The calculated binding energy, $E=-19.15$ MeV, measured from
the $3\alpha+\Lambda$ threshold, and also $B_{\Lambda}=11.86$ MeV Tamura2006 ,
are in good agreement with the corresponding experimental values, $E=-18.65\pm
0.05$ MeV and $B_{\Lambda}=11.38\pm 0.05$ MeV, respectively. We can also see
that the $0_{2}^{+}$ state is slightly bound below the lowest ${}^{12}{\rm
C}+\Lambda$ threshold and the other higher three $0^{+}$ states are obtained
as resonant states. The binding energies for the $0_{2}^{+}$, $0_{3}^{+}$,
$0_{4}^{+}$, and $0_{5}^{+}$ states are calculated as $E=-7.49$ MeV, $E=-5.0$
MeV, $E=-3.5$ MeV, and $E=-1.2$ MeV, respectively. We should note that, as
mentioned in Introduction, apart from ${}^{12}{\rm C}$, many thresholds such
as ${}^{9}_{\Lambda}{\rm Be}(0^{+}_{1})+\alpha$, ${}^{12}{\rm
C}(2^{+}_{1})+\Lambda$, ${}^{9}_{\Lambda}{\rm Be}(2^{+}_{1})+\alpha$,
${}^{5}_{\Lambda}{\rm He}+^{8}{\rm Be}(0^{+})$, and ${}^{5}_{\Lambda}{\rm
He}+2\alpha$, are newly open in ${}^{13}_{\Lambda}{\rm C}$, which may give a
variety of cluster structures.

For comparison, in the left most column of Fig. 2, we show the energy spectra
of $0^{+}$ states of ${}^{13}_{\Lambda}{\rm C}$ calculated with the H-THSR
wave function Funaki2018plb , where the four $0^{+}$ states are obtained below
the $3\alpha+\Lambda$ threshold, while we obtain the five $0^{+}$ states in
the present calculation. The authors in Ref. Funaki2018plb  show that the
$0^{+}_{\rm{I}}$, $0^{+}_{\rm{II}}$, $0^{+}_{\rm{III}}$, and $0^{+}_{\rm{IV}}$
states keep the structures of the $0_{1}^{+}$, $0_{2}^{+}$, $0_{4}^{+}$, and
$0_{3}^{+}$ states of ${}^{12}{\rm C}$, to which the $\Lambda$ particle
couples, respectively. In particular, it is argued that the $0^{+}_{\rm{IV}}$
state includes the ${}^{9}_{\Lambda}{\rm Be}(0_{2}^{+})+\alpha$ component
dominantly, where the $0_{2}^{+}$ state of ${}^{9}_{\Lambda}{\rm Be}$ is
identified as a ${}^{5}_{\Lambda}{\rm He}+\alpha$ resonant state. It is thus
concluded that the $0^{+}_{\rm{IV}}$ state has an analogous structure to the
Hoyle analog state, a gaslike ${}^{5}_{\Lambda}{\rm He}+2\alpha$ cluster
structure. However, as we will discuss later, it should be noted that in our
calculation with the OCM, we cannot find the $0_{2}^{+}$ state of
${}^{9}_{\Lambda}{\rm Be}$ around this energy region qian2020 .

Table 2: Rms radii of the $0^{+}$ states of ${}^{13}_{\Lambda}$C and 12C. The units are in fm. ${}^{13}_{\Lambda}$C |  ${\rm R}_{\rm rms}^{{\rm c}-\Lambda}$ |  ${\rm R}_{\rm rms}$ |  12C |  ${\rm R}_{\rm rms}$
---|---|---|---|---
${}^{13}_{\Lambda}$C$(0_{1}^{+})$ |  2.3 |  2.3 |  12C$(0_{1}^{+})$ |  2.4
${}^{13}_{\Lambda}$C$(0_{2}^{+})$ |  3.1 |  3.1 |  12C$(0_{2}^{+})$ |  4.2
${}^{13}_{\Lambda}$C$(0_{3}^{+})$ |  4.3 |  5.4 |  12C$(0_{3}^{+})$ |  5.6
${}^{13}_{\Lambda}$C$(0_{4}^{+})$ |  4.3 |  4.1 |  12C$(0_{4}^{+})$ |  5.4
${}^{13}_{\Lambda}$C$(0_{5}^{+})$ |  5.2 |  5.6 |  |

###  III.2 $0^{+}_{1}$, $0^{+}_{2}$ and $0^{+}_{4}$ in ${}^{13}_{\Lambda}{\rm
C}$

In order to investigate the structures of the obtained $0^{+}$ states and the
analogous nature between ${}^{13}_{\Lambda}$C and 12C, we calculate the
$S^{2}$-factors of the states, which are obtained by integrating the square of
the RWAs, as follows:

$S^{2}_{i}=\int\mathscr{Y}_{i}(r)^{2}r^{2}dr,$ (8)

where $\mathscr{Y}_{i}(r)$ are the RWAs defined below,

$\displaystyle{\cal Y}_{i}(r)=\left\\{\begin{aligned}
&\sqrt{\frac{3!}{3!1!}}\left\langle\left[\frac{\delta(\rho_{3}-r)}{\rho_{3}^{2}}Y_{0}(\hat{\rho}_{3}),\phi_{i}(^{12}{\rm{C}})\right]_{00}\bigg{|}\Psi(^{13}_{\Lambda}\rm{C})\right\rangle,\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(i=1,\cdots,5),\\\
&\sqrt{\frac{3!}{2!1!}}\left\langle\left[\frac{\delta(\rho_{1}-r)}{\rho_{1}^{2}}Y_{0}(\hat{\rho}_{1}),\phi_{i}(^{9}_{\Lambda}{\rm{Be}})\right]_{00}\bigg{|}\Psi(^{13}_{\Lambda}\rm{C})\right\rangle,\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(i=6,7,10),\\\
&\sqrt{\frac{3!}{2!1!}}\\!\left\langle\left[\frac{\delta(\rho_{4}-r)}{\rho_{4}^{2}}Y_{0}(\hat{\rho}_{4}),\\!\phi(^{5}_{\Lambda}{\rm{He}})\phi_{i}(^{8}{\rm{Be}})\right]_{00}\bigg{|}\Psi(^{13}_{\Lambda}\rm{C})\\!\right\rangle,\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(i=8,9).\end{aligned}\right.$

Here, $\rho_{3}$, $\rho_{1}$ and $\rho_{4}$ are the relative coordinates shown
in Fig. 1. $\phi_{i}(^{12}{\rm{C}})$, with the channel numbers $i=1,\cdots,5$,
denote the wave functions of the $0_{1}^{+}$, $0^{+}_{2}$, $0^{+}_{3}$,
$0^{+}_{4}$ and $2^{+}_{1}$ states of 12C, respectively, which are calculated
with the $3\alpha$ OCM. $\phi_{i}(^{9}_{\Lambda}{\rm{Be}})$ with $i=6,7,10$
denote the wave functions of the $0_{1}^{+}$, $2^{+}_{1}$, and $0_{2}^{+}$
states of ${}^{9}_{\Lambda}{\rm{Be}}$, respectively, which are calculated with
the same framework of the $2\alpha+\Lambda$ OCM as the present
$3\alpha+\Lambda$ OCM. We will explain the detail of the $0_{2}^{+}$ state of
${{}^{9}_{\Lambda}{\rm Be}}$ later in Sec. III.3. $\phi_{i}(^{8}{\rm{Be}})$
with $i=8,9$ denote the wave functions of the $0^{+}$ and $2^{+}$ states of
8Be, respectively, which are calculated with the $2\alpha$ OCM. The
$\Psi(^{13}_{\Lambda}\rm{C})$ is the wave function of the $0^{+}$ state of
${}^{13}_{\Lambda}{\rm C}$ in Eq. (2).

Figure 3:  The $S^{2}$-factors of the $0_{1}^{+}$ state in the present
calculation and of the $0^{+}_{\rm{I}}$ state in Fig. 2 calculated with the
H-THSR ansatz Funaki2018plb , are shown by red and blue bars, respectively.
The labels 1,2,3,4, and 5 represent the ${}^{12}{\rm C}(0^{+}_{1})+\Lambda$,
${}^{12}{\rm C}(0^{+}_{2})+\Lambda$, ${}^{12}{\rm C}(0^{+}_{3})+\Lambda$,
${}^{12}{\rm C}(0^{+}_{4})+\Lambda$, and ${}^{12}{\rm C}(2^{+}_{1})+\Lambda$
channels, respectively, and the labels 6, 7, 8, 9, 10 represent the
${}^{9}_{\Lambda}{\rm Be}(0^{+}_{1})+\alpha$, ${}^{9}_{\Lambda}{\rm
Be}(2^{+}_{1})+\alpha$, ${}^{5}_{\Lambda}{\rm He}+^{8}{\rm Be}(0^{+})$,
${}^{5}_{\Lambda}{\rm He}+{{}^{8}{\rm Be}(2^{+})}$, ${}^{9}_{\Lambda}{\rm
Be}(0^{+}_{2})+\alpha$ channels, respectively.  Figure 4:  The $S^{2}$-factors
of the $0_{2}^{+}$ state in the present calculation and of the
$0^{+}_{\rm{II}}$ state in Fig. 2 calculated with the H-THSR ansatz
Funaki2018plb , are shown by red and blue bars, respectively. The labels
1,2,3,4, and 5 represent the ${}^{12}{\rm C}(0^{+}_{1})+\Lambda$, ${}^{12}{\rm
C}(0^{+}_{2})+\Lambda$, ${}^{12}{\rm C}(0^{+}_{3})+\Lambda$, ${}^{12}{\rm
C}(0^{+}_{4})+\Lambda$, and ${}^{12}{\rm C}(2^{+}_{1})+\Lambda$ channels,
respectively, and the labels 6, 7, 8, 9, 10 represent the
${}^{9}_{\Lambda}{\rm Be}(0^{+}_{1})+\alpha$, ${}^{9}_{\Lambda}{\rm
Be}(2^{+}_{1})+\alpha$, ${}^{5}_{\Lambda}{\rm He}+^{8}{\rm Be}(0^{+})$,
${}^{5}_{\Lambda}{\rm He}+{{}^{8}{\rm Be}(2^{+})}$, ${}^{9}_{\Lambda}{\rm
Be}(0^{+}_{2})+\alpha$ channels, respectively. Figure 5:  The $S^{2}$-factors
of the $0_{4}^{+}$ state in the present calculation and of the corresponding
$0^{+}_{\rm{III}}$ state in Fig. 2 calculated with the H-THSR ansatz
Funaki2018plb , are shown by red and blue bars, respectively. The labels
1,2,3,4, and 5 represent the ${}^{12}{\rm C}(0^{+}_{1})+\Lambda$, ${}^{12}{\rm
C}(0^{+}_{2})+\Lambda$, ${}^{12}{\rm C}(0^{+}_{3})+\Lambda$, ${}^{12}{\rm
C}(0^{+}_{4})+\Lambda$, and ${}^{12}{\rm C}(2^{+}_{1})+\Lambda$ channels,
respectively, and the labels 6, 7, 8, 9, 10 represent the
${}^{9}_{\Lambda}{\rm Be}(0^{+}_{1})+\alpha$, ${}^{9}_{\Lambda}{\rm
Be}(2^{+}_{1})+\alpha$, ${}^{5}_{\Lambda}{\rm He}+{{}^{8}{\rm Be}(0^{+})}$,
${}^{5}_{\Lambda}{\rm He}+{{}^{8}{\rm Be}(2^{+})}$, ${}^{9}_{\Lambda}{\rm
Be}(0^{+}_{2})+\alpha$ channels, respectively. Figure 6:  The RWAs of the
$0_{2}^{+}$ state of ${}^{12}{\rm C}$ in the ${}^{8}{\rm Be}(0^{+})+\alpha$
channel is shown in the black solid line. The red solid and red dash-dotted
lines represent the RWAs of the $0_{2}^{+}$ state of ${}^{13}_{\Lambda}{\rm
C}(0^{+}_{2})$ in the ${}^{9}_{\Lambda}{\rm Be}(0^{+}_{1})+\alpha$ and
${}^{5}_{\Lambda}{\rm He}+^{8}{\rm Be}(0^{+})$ channels, respectively.

The calculated $S^{2}$-factors of the $0_{1}^{+}$, $0_{2}^{+}$, and
$0_{4}^{+}$ states are shown in Fig. 3, Fig. 4 and Fig. 5, respectively,
colored in red, for the 10 channels. For comparison, we also show the
$S^{2}$-factors calculated by Funaki et al. Funaki2018plb  side by side,
colored in blue.

We can see that in Fig. 3, the $0_{1}^{+}$ state has the largest fraction from
the 12C$(0^{+}_{1})+\Lambda$ channel (channel 1), which means that the
$0^{+}_{1}$ state dominantly has the structure that the $\Lambda$ particle
couples to the 12C$(0^{+}_{1})$ core in an $S$-wave, keeping the shell-model-
like structure. This state also has large components from the channels
${}^{9}_{\Lambda}{\rm Be}(0^{+}_{1})+\alpha$ (channel 6) and
${}^{9}_{\Lambda}{\rm Be}(2^{+}_{1})+\alpha$ (channel 7), which again show
that it keeps the SU(3)-like nature of the $0_{1}^{+}$ state of ${}^{12}{\rm
C}$, with the $\Lambda$ particle coupling to ${}^{8}{\rm Be}$ core in an
$S$-wave. All these distributions are consistent with those obtained in Ref.
Funaki2018plb  colored in blue. We also mention that the rms radius, which is
shown in Table 2 for this state, $2.3$ fm is almost the same as that for the
$0_{1}^{+}$ state of ${}^{12}{\rm C}$, $2.4$ fm. This indicates that there is
almost no shrinkage by adding the $\Lambda$ particle to the 12C$(0^{+}_{1})$
core, since the density of the ground state of ${}^{12}{\rm C}$ is close to
saturation.

The dominant channels for the $0^{+}_{2}$ state of ${}^{13}_{\Lambda}$C are
${}^{9}_{\Lambda}{\rm Be}(0^{+}_{1})+\alpha$ (channel 6) and
${}^{5}_{\Lambda}{\rm He}+^{8}{\rm Be}(0^{+})$ (channel 8), and the channel
${}^{12}{\rm C}(0^{+}_{2})+\Lambda$ (channel 2) gives smaller contribution,
although the component of this channel is the largest among the five
${}^{12}{\rm C}+\alpha$ channels (channels 1,2,3,4,5). On the other hand, we
can see that in Table 2, the rms radius of this state, $3.1$ fm, is much
smaller than that of the Hoyle state, $4.2$ fm. This is because the density of
the Hoyle state is much lower than saturation, and therefore the $\Lambda$
particle can shrink the Hoyle state so much to be the more compact $0_{2}^{+}$
state of ${}^{13}_{\Lambda}{\rm C}$. This core shrinkage gives smaller overlap
between the $0_{2}^{+}$ state of ${}^{13}_{\Lambda}{\rm C}$ and ${}^{12}{\rm
C}(0^{+}_{2})+\Lambda$ in the correponding RWA, leading to the smaller
$S^{2}$-factor in this channel. It is also interesting to consider the energy
gain between the $0_{2}^{+}$ state of ${}^{13}_{\Lambda}$C and the Hoyle
state, $B_{\Lambda}=8.3$ MeV, which is $2.7$ times larger than
$B_{\Lambda}=3.12$ MeV for ${}^{5}_{\Lambda}{\rm He}$ and is larger than
$B_{\Lambda}=6.71$ MeV for ${}^{9}_{\Lambda}{\rm Be}$. This indicates that the
$\Lambda$ particle strongly couples to the ${}^{8}{\rm Be}$ core as well as
the ${}^{4}{\rm He}$ core, i.e. all the $\alpha$ clusters, which is consistent
with the fact that this state has large $S^{2}$-factors from the channels 6
and 8. We should also mention that the distributions of the $S^{2}$-factors in
the present calculations and in Ref. Funaki2018plb  are in good agreement with
each other.

In Fig. 6, we show the RWAs of the $0_{2}^{+}$ state of ${}^{13}_{\Lambda}{\rm
C}$ in the ${}^{9}_{\Lambda}{\rm Be}(0^{+}_{1})+\alpha$ (${\cal Y}_{i=6}(r)$)
and ${}^{5}_{\Lambda}{\rm He}+{{}^{8}{\rm Be}(0^{+})}$ (${\cal Y}_{i=8}(r)$)
channels. For comparison, we also show the RWAs of the Hoyle state in the
${}^{8}{\rm Be}(0^{+})+\alpha$ channel. The RWAs for ${}^{13}_{\Lambda}{\rm
C}$ in both channels are pulled into inner region, in comparison with that for
the Hoyle state, due to the shrinkage effect of the $\Lambda$ particle.
However, the behaviours of oscillation in inner region of the RWA in the
${}^{5}_{\Lambda}{\rm He}+^{8}{\rm Be}(0^{+})$ channel and the one for the
Hoyle state in the ${}^{8}{\rm Be}(0^{+})+\alpha$ channel are similar to each
other. In both cases, the nodes in the inner region disappear and change to
the oscillation in a similar way, since the core ${}^{8}{\rm Be}$ dissolves
into $2\alpha$ clusters and the $\alpha$ or ${}^{5}_{\Lambda}{\rm He}$ cluster
does not feel the effect of the antisymmetrization from the core sufficiently.
On the other hand, the three nodes recover for the RWA in the
${}^{9}_{\Lambda}{\rm Be}(0^{+}_{1})+\alpha$ channel, since the core
${}^{9}_{\Lambda}{\rm Be}(0^{+}_{1})$ is a compact object, again due to the
shrinkage effect of the $\Lambda$ particle.

Let us then discuss the $0^{+}_{4}$ state. The $S^{2}$-factors for this state
are shown in Fig. 5, denoted by red bars. In comparison with them, we also
show those for the state calculated as the $0^{+}_{\rm{III}}$ state with the
H-THSR ansatz by blue bars. Although the component of ${}^{9}_{\Lambda}{\rm
Be}(0_{1}^{+})+\alpha$ channel (channel 6) in our calculation is about 2 times
larger than in the H-THSR calculation, both distributions are in good
agreement with each other, and hence we can say that our $0_{4}^{+}$ state
corresponds to the $0^{+}_{\rm{III}}$ state in the H-THSR calculation, which
is shown to have a structure that the $\Lambda$ particle moves around a
linear-chain-like configuration of the $3\alpha$ clusters in Ref.
Funaki2018plb . The largest component of the ${}^{12}{\rm C}+\Lambda$ channels
is of ${}^{12}{\rm C}(0^{+}_{4})+\alpha$ channel. Many previous cluster model
calculations show that the $0_{4}^{+}$ state of ${}^{12}{\rm C}$ dominantly
has the ${}^{8}{\rm Be}(2_{1}^{+})+\alpha$ component and this state is
considered to have a linear-chain-like structure, as mentioned in Introduction
Fujiwara1980-hoyleband ; AMD2007 ; FMD2007 . We can see that the present
$0_{4}^{+}$ state of ${}^{13}_{\Lambda}{\rm C}$ also has the largest component
from the ${}^{9}_{\Lambda}{\rm Be}(2_{1}^{+})+\alpha$ channel as well as large
component from the ${}^{5}_{\Lambda}{\rm He}+^{8}{\rm Be}(2^{+})$ channel,
which also supports the idea that in our $0_{4}^{+}$ state the $\Lambda$
particle moves in an $S$-wave around the $0_{4}^{+}$ state of ${}^{12}{\rm C}$
with the linear-chain-like configuration.

Concerning the shrinkage effect between the core 12C and the
${}^{13}_{\Lambda}$C, as shown in Table. 2, the density of the $0_{4}^{+}$
state is $(5.4{\rm fm}/4.1{\rm fm})^{3}=2.3$ times smaller than the
$0_{4}^{+}$ state of ${}^{12}{\rm C}$. This amount of shrinkage is similar to
that between the $0_{2}^{+}$ state and the Hoyle state in ${}^{12}{\rm C}$,
$(4.2{\rm fm}/3.1{\rm fm})^{3}=2.5$. The energy gain of this state from the
$0_{4}^{+}$ state of ${}^{12}{\rm C}$, $8.1$ MeV, is also close to that of the
$0_{2}^{+}$ state from the Hoyle state, which is $8.3$ MeV. Thus as in the
same way as the $0_{2}^{+}$ state, in the $0_{4}^{+}$ state, the $\Lambda$
particle couples to all of the $\alpha$ clusters, keeping a linear-chain-like
arrangement of the $3\alpha$ clusters.

###  III.3 $0^{+}_{3}$ and $0^{+}_{5}$ in ${}^{13}_{\Lambda}{\rm C}$

In Fig. 7, we show the $S^{2}$-factors for the $0_{5}^{+}$ state, which is
denoted by red bars, and for the $0^{+}_{\rm{IV}}$ state obtained in Ref.
Funaki2018plb  with the H-THSR ansatz, for comparison, by blue bars, side by
side. We can see that both the distributions are qualitatively in good
agreement with each other, and therefore we can conclude that our $0_{5}^{+}$
state corresponds to the $0_{\rm IV}^{+}$ state obtained in Ref. Funaki2018plb
. As mentioned in Introduction, the $0_{\rm IV}^{+}$ state in Ref.
Funaki2018plb  can be considered to be the Hoyle analog state, where the
$\Lambda$ particle strongly couples with an $\alpha$ cluster to form
${}^{5}_{\Lambda}{\rm He}$ and mutually $2\alpha$ and ${}^{5}_{\Lambda}{\rm
He}$ clusters loosely couple like a gas. The authors in Ref. Funaki2018plb
also discussed the existence of the $0_{2}^{+}$ state of ${}^{9}_{\Lambda}{\rm
Be}$, which is a two-body resonant state of a well developed
${}^{5}_{\Lambda}{\rm He}$ and $\alpha$ cluster structure, though
experimentally the state is still unknown. Thus, the fact that the $0_{\rm
IV}^{+}$ state of ${}^{13}_{\Lambda}{\rm C}$ in Ref. Funaki2018plb  dominantly
has the ${}^{9}_{\Lambda}{\rm Be}(0^{+}_{2})+\alpha$ channel (channel 10)
component, as shown in Fig. 7, as well as a large component from the
${{}^{5}_{\Lambda}{\rm He}}+{{}^{8}{\rm Be}(0^{+})}$ channel (channel 8),
gives the evidence that this state has a gaslike ${}^{5}_{\Lambda}{\rm
He}+2\alpha$ cluster structure as the Hoyle analog state.

Figure 7:  The $S^{2}$-factors of the $0_{5}^{+}$ state in the present
calculation and of the $0^{+}_{\rm{IV}}$ state in Fig. 2 calculated with the
H-THSR ansatz Funaki2018plb , are shown by red and blue bars, respectively.
The labels 1,2,3,4, and 5 represent the ${}^{12}{\rm C}(0^{+}_{1})+\Lambda$,
${}^{12}{\rm C}(0^{+}_{2})+\Lambda$, ${}^{12}{\rm C}(0^{+}_{3})+\Lambda$,
${}^{12}{\rm C}(0^{+}_{4})+\Lambda$, and ${}^{12}{\rm C}(2^{+}_{1})+\Lambda$
channels, respectively, and the labels 6, 7, 8, 9, 10 represent the
${}^{9}_{\Lambda}{\rm Be}(0^{+}_{1})+\alpha$, ${}^{9}_{\Lambda}{\rm
Be}(2^{+}_{1})+\alpha$, ${}^{5}_{\Lambda}{\rm He}+^{8}{\rm Be}(0^{+})$,
${}^{5}_{\Lambda}{\rm He}+{{}^{8}{\rm Be}(2^{+})}$, ${}^{9}_{\Lambda}{\rm
Be}(0^{+}_{2})+\alpha$ channels, respectively. Figure 8:  The RWAs of the
$0_{1}^{+}$ and $0_{2}^{+}$ state of ${}^{9}_{\Lambda}{\rm Be}$ in the
${}^{5}_{\Lambda}{\rm He}+\alpha$ channel are shown in black and blue dashed
lines, respectively. The solid line represents the $\alpha$-$\alpha$ relative
wave function of the $0^{+}$ state of ${}^{8}{\rm Be}$. See the text for the
details of the $0_{2}^{+}$ state of ${}^{9}_{\Lambda}{\rm Be}$.

However, in the present full-channel three-body $\alpha$-$\alpha$-$\Lambda$
calculation by the OCM, we could not obtain the $0_{2}^{+}$ state of
${}^{9}_{\Lambda}{\rm Be}$ around the same energy region as they obtained.
Nevertheless, we could obtain a $0_{2}^{+}$ state of ${}^{9}_{\Lambda}{\rm
Be}$ at $0.7$ MeV below the $2\alpha+\Lambda$ threshold, where the limited
model space including ${}^{5}_{\Lambda}{\rm He}+\alpha$ channel are only taken
into account qian2020 . In Fig. 8, we show the RWA of this artificial
$0_{2}^{+}$ state of ${}^{9}_{\Lambda}{\rm Be}$ in the ${}^{5}_{\Lambda}{\rm
He}+\alpha$ channel (blue dotted line). For comparison, we also show the RWA
of the $0_{1}^{+}$ state of ${}^{9}_{\Lambda}{\rm Be}$ in the same channel
(black dotted line) and the $\alpha$-$\alpha$ relative wave function of the
$0^{+}$ state of ${}^{8}{\rm Be}$ (black solid line). The largest amplitude
for the ${}^{9}_{\Lambda}{\rm Be}(0_{2}^{+})$ spreads outside up to around
$10$ fm, which indicates that this state has a very well developed
${}^{5}_{\Lambda}{\rm He}+\alpha$ cluster structure. We thus make use of this
state as the ${}^{9}_{\Lambda}{\rm Be}(0_{2}^{+})+\alpha$ channel (channel
10). Figure 7 shows that this channel component gives the largest contribution
to the $S^{2}$-factors for the $0_{5}^{+}$ state, in the same way as the
$0_{\rm IV}^{+}$ state of Ref. Funaki2018plb , placed side by side, by blue
bars. We can also see that the second largest component is from the
${{}^{5}_{\Lambda}{\rm He}}+{{}^{8}{\rm Be}(0^{+})}$ channel (channel 8).
Therefore, we can say again that our $0_{5}^{+}$ state corresponds to the
$0_{4}^{+}$ state in Ref. Funaki2018plb  and this state has the structure that
the ${}^{5}_{\Lambda}{\rm He}$ and $2\alpha$ clusters weakly interact with
each other in relative $S$ waves, like a gas.

Besides, among the five ${}^{12}{\rm C}+\Lambda$ channels, this state has the
largest component from the ${}^{12}{\rm C}(0_{3}^{+})+\Lambda$ channel. As far
as both the $0_{5}^{+}$ state of ${}^{13}_{\Lambda}{\rm C}$ and the
$0_{3}^{+}$ state of ${}^{12}{\rm C}$ can be associated with each other, the
energy gain of the former state from the latter is $3.0$ MeV, as shown in Fig.
2, which is close to the binding energy of ${}^{5}_{\Lambda}{\rm He}$, $3.12$
MeV. The calculated rms radius of this state, $5.6$ fm, is as large as the
$0_{3}^{+}$ state of ${}^{12}{\rm C}$, as shown in Table 2. The $\Lambda$
particle namely gives no shrinkage of the core ${{}^{12}{\rm C}}$ nucleus. All
these results again indicate that the $\Lambda$ particle couples with only one
$\alpha$ cluster to form ${}^{5}_{\Lambda}{\rm He}$, without any shrinkage of
the core, and the ${}^{5}_{\Lambda}{\rm He}$ and $2\alpha$ clusters move
freely like a gas, analogous to the dilute $3\alpha$ clustering of the Hoyle
state. It is also interesting to mention that the $0_{3}^{+}$ state of
${}^{12}{\rm C}$, which is likely to be a precursor of the Hoyle state, is
substituted for the Hoyle state, as the ${}^{5}_{\Lambda}{\rm He}+2\alpha$
gas, when the $\Lambda$ particle is added, while the Hoyle state is converted
to the $0_{2}^{+}$ state of ${}^{13}_{\Lambda}{\rm C}$ with much more compact
${}^{9}_{\Lambda}{\rm Be}+\alpha$ structure.

Figure 9:  The $S^{2}$-factors of the $0_{3}^{+}$ state in the present
calculation are shown. The labels 1,2,3,4, and 5 represent the ${}^{12}{\rm
C}(0^{+}_{1})+\Lambda$, ${}^{12}{\rm C}(0^{+}_{2})+\Lambda$, ${}^{12}{\rm
C}(0^{+}_{3})+\Lambda$, ${}^{12}{\rm C}(0^{+}_{4})+\Lambda$, and ${}^{12}{\rm
C}(2^{+}_{1})+\Lambda$ channels, respectively, and the labels 6, 7, 8, 9, 10
represent the ${}^{9}_{\Lambda}{\rm Be}(0^{+}_{1})+\alpha$,
${}^{9}_{\Lambda}{\rm Be}(2^{+}_{1})+\alpha$, ${}^{5}_{\Lambda}{\rm
He}+{{}^{8}{\rm Be}(0^{+})}$, ${}^{5}_{\Lambda}{\rm He}+{{}^{8}{\rm
Be}(2^{+})}$, ${}^{9}_{\Lambda}{\rm Be}(0^{+}_{2})+\alpha$ channels,
respectively.

Next let us discuss the $0^{+}_{3}$ state. We should mention that this is a
new state and has never been obtained in Ref. Funaki2018plb  as well as in any
other papers. In Fig.9, we show the $S^{2}$-factors for this state. We can see
that the component of the ${}^{9}_{\Lambda}{\rm Be}(0^{+}_{1})+\alpha$ channel
is prominently large. This clearly means that this state dominantly has the
${}^{9}_{\Lambda}{\rm Be}(0^{+}_{1})+\alpha$ cluster structure. Very large rms
radius of this state, $5.4$ fm, as shown in Table 2, suggests that the
${}^{9}_{\Lambda}{\rm Be}(0_{1}^{+})+\alpha$ cluster structure is very much
developed in this state. On the other hand, it should also be mentioned that
the ${}^{12}{\rm C}(0^{+}_{3})+\Lambda$ channel gives the second largest
contribution to the $S^{2}$-factors, indicating that this state can most
likely be associated with the $0_{3}^{+}$ state of ${}^{12}{\rm C}$, in a
sense that the $\Lambda$ particle can be added to the ${}^{12}{\rm C}$ core as
just an impurity. This assumption of the correspondence with the $0_{3}^{+}$
state of ${}^{12}{\rm C}$ is also consistent with the energy gain from the
$0^{+}_{3}$ state of ${}^{12}{\rm C}$, $6.7$ MeV (see Fig. 2), which coincides
with the $\Lambda$ binding energy for ${}^{9}_{\Lambda}{\rm Be}$,
$B_{\Lambda}=6.71$ MeV. Namely, the $0_{3}^{+}$ state of
${}^{13}_{\Lambda}{\rm C}$ is given by coupling the $\Lambda$ particle with
the ${}^{8}{\rm Be}$ part in the ${}^{12}{\rm C}(0_{3}^{+})$ core, to form the
well developed ${}^{9}_{\Lambda}{\rm Be}+\alpha$ cluster structure. Thus we
propose the idea that the existence of the $0_{3}^{+}$ state of ${}^{12}{\rm
C}$ gives birth to two types of cluster states of ${}^{13}_{\Lambda}{\rm C}$,
the $0_{5}^{+}$ and $0_{3}^{+}$ states, where the $\Lambda$ particle couples
with only the one $\alpha$ cluster to form ${}^{5}_{\Lambda}{\rm He}$ for the
former state and with the two $\alpha$ clusters to form ${}^{9}_{\Lambda}{\rm
Be}$ for the latter.

In order to investigate more deeply all these structures of the $0_{3}^{+}$
and $0_{5}^{+}$ states, we show in Fig. 10 the RWAs of these states in their
important channels, together with relevant RWAs of the $0_{2}^{+}$ state of
${}^{12}{\rm C}$ and the $0_{3}^{+}$ state of ${{}^{13}_{\Lambda}{\rm C}}$.
For the $0^{+}_{3}$ state of ${}^{13}_{\Lambda}$C, the RWA of
${}^{9}_{\Lambda}{\rm Be}+\alpha$ channel, ${\cal Y}_{i=6}(r)$ in Eq. (III.2),
is denoted by a blue dashed line. It resembles the RWA in the ${}^{8}{\rm
Be}+\alpha$ channel of the $0_{3}^{+}$ state of ${}^{12}{\rm C}$, denoted by a
black solid line, while the former is drawn into an inner region, compared
with the latter, because of a shrinkage effect of the additional $\Lambda$
particle. This clearly indicates that the $0_{3}^{+}$ state takes over a
higher nodal structure of the relative wave function between the ${}^{8}{\rm
Be}$ and $\alpha$ cluster for the $0_{3}^{+}$ state of ${}^{12}{\rm C}$. In
fact, this RWA has four nodes and hence one more oscillation than the one of
the $0_{2}^{+}$ state in the ${}^{9}_{\Lambda}$Be+$\alpha$ channel (red dash-
dotted line). That is why the $0_{3}^{+}$ state has the larger rms radius than
that of the $0_{2}^{+}$ state. These results again support our idea that in
the $0_{3}^{+}$ state the $\Lambda$ particle is strongly coupled with the
${}^{8}{\rm Be}$ part in the $0_{3}^{+}$ state of ${}^{12}{\rm C}$ to form the
higher nodal ${}^{9}_{\Lambda}{\rm Be}+\alpha$ cluster structure, due to the
orthogonalization to the $0_{2}^{+}$ state, with the large component of
${}^{9}_{\Lambda}{\rm Be}+\alpha$ cluster structure.

On the other hand, in the $0_{5}^{+}$ state the RWA in the
${{}^{5}_{\Lambda}{\rm He}}+{{}^{8}{\rm Be}(0^{+})}$ channel, ${\cal
Y}_{i=8}(\rho)$ in Eq. (III.2), which is denoted by blue solid line, has a
different nodal behavior, in which the nodes disappear in an outer region more
than about $2$ fm and only a few small oscillations remain. Since the outer
most nodal position corresponds to a range of the core giving the effect of
antisymmetrization to the remaining cluster, this result means that the
${}^{5}_{\Lambda}{\rm He}$ cluster does not feel the effect of the
antisymmetrization from the ${}^{8}{\rm Be}$ core. In other words, the core
${}^{8}{\rm Be}$ is dissolved into the $2\alpha$ clusters, and a dilute
${}^{5}_{\Lambda}{\rm He}+2\alpha$ cluster structure is realized.

Figure 10:  The RWAs of the $0_{3}^{+}$ state of ${}^{12}{\rm C}$ in the
${}^{8}{\rm Be}(0^{+})+\alpha$ channel are shown in black solid line. The red
dash-dotted and blue dashed lines represent the RWAs of the $0_{2}^{+}$ and
$0_{3}^{+}$ states of ${}^{13}_{\Lambda}{\rm C}$ in the ${}^{9}_{\Lambda}{\rm
Be}+\alpha$ channel, respectively. The blue solid line is the RWAs of the
$0_{5}^{+}$ state of ${}^{13}_{\Lambda}{\rm C}$ in the ${}^{8}{\rm
Be}(0^{+})+^{5}_{\Lambda}{\rm He}$ channel.

##  IV Summary

We investigate the structures of the $0^{+}$ states in ${}^{13}_{\Lambda}$C by
using the $3\alpha+\Lambda$ OCM, which is the semi-microscopic four-body
cluster model. In order to solve the Schrödinger equation of the
$3\alpha+\Lambda$ four-body system, we apply the Gaussian expansion method. We
use the $\alpha$-$\alpha$ interaction, which reproduces the observed
$\alpha$-$\alpha$ scattering data, and an effective three-body repulsive
force, in order to fit the binding energy of the ground state of 12C to the
empirical value. The Pauli forbidden states $(0s,1s,0d)$ between the $2\alpha$
clusters are ruled out by introducing the Pauli exclusion operator in the OCM.
We employed the $\alpha\Lambda$ potential by folding procedure of YNG-NF
$\Lambda N$ interaction with the $\alpha$ cluster wave function with
$(0s)^{4}$ configuration. We adjust even- and odd-states of the $\Lambda N$
interaction, so as to reproduce the observed binding energies of ground states
of ${}^{5}_{\Lambda}$He and ${}^{9}_{\Lambda}$Be. We then obtain the five
$0^{+}$ states.

In order to investigate the structures of the five $0^{+}$ states and discuss
the binding mechanism between the $\Lambda$ particle and core nucleus, we
calculate the energy spectrum, rms radii, the RWAs, and $S^{2}$-factors of
these states. The results are compared with the ones calculated by Funaki et
al. in Ref. Funaki2018plb  by using the so-called H-THSR ansatz, where the
four $0^{+}$ states are obtained.

The $0_{1}^{+}$ state is analogous to the ground state of ${}^{12}{\rm C}$,
where the structure of the ${}^{12}{\rm C}$ core in ${}^{13}_{\Lambda}{\rm C}$
is almost the same as the one in the ground state of 12C. There is no
shrinkage effect between the $0_{1}^{+}$ state of ${{}^{13}_{\Lambda}{\rm C}}$
and the ground state of 12C. The $0_{2}^{+}$ state is found to correspond to
the Hoyle state, where the $\Lambda$ particle couples strongly with all the
$3\alpha$ clusters and causes an outstanding reduction of the spatial size.

The $0_{4}^{+}$ state in ${}^{13}_{\Lambda}$C corresponds to the $0_{4}^{+}$
state of ${}^{12}{\rm C}$, which has a linear-chain-like structure. We find
that the $D$-wave component between ${}^{9}_{\Lambda}{\rm Be}$ and $\alpha$
cluster is largely included in this state, which indicates that this state may
still keep a linear-chain-like structure as the $0_{4}^{+}$ state of
${}^{12}{\rm C}$.

We find that the $0_{5}^{+}$ state may correspond to the $0_{3}^{+}$ state of
${}^{12}{\rm C}$ and has a large $S^{2}$-factor in the ${}^{9}_{\Lambda}{\rm
Be}(0^{+}_{2})+\alpha$ channel. Considering the fact that the $0_{2}^{+}$
state of ${}^{9}_{\Lambda}{\rm Be}$ has the well developed
${}^{5}_{\Lambda}{\rm He}+\alpha$ cluster structure, we can say that this
state has a structure that the $\Lambda$ particle strongly couples with only
one $\alpha$ cluster to form ${}^{5}_{\Lambda}{\rm He}$, while it loosely
couples with the other $2\alpha$ clusters. The energy gain of this state from
the $0_{3}^{+}$ state of ${}^{12}{\rm C}$, $3.0$ MeV, is very close to the
binding energy of ${}^{5}_{\Lambda}{\rm He}$, $3.12$ MeV, and also supports
this conclusion.

By comparing the $S^{2}$-factors between our calculated states in
${}^{13}_{\Lambda}$C and the ones obtained in Ref. Funaki2018plb , we find
that our $0_{1}^{+}$, $0_{2}^{+}$, $0_{4}^{+}$ and $0_{5}^{+}$ correspond to
the $0_{\rm I}^{+}$, $0_{\rm II}^{+}$, $0_{\rm III}^{+}$ and $0_{\rm IV}^{+}$
states in Ref. Funaki2018plb .

Finally we discuss the $0_{3}^{+}$ state, which has never been discussed
before. We find that it has a dominant component of $S^{2}$-factor in the
${}^{9}_{\Lambda}{\rm Be}(0^{+}_{1})+\alpha$ channel, as well as a large
component in the ${}^{12}{\rm C}(0_{3}^{+})+\Lambda$ channel. The RWA in the
${}^{9}_{\Lambda}{\rm Be}(0^{+}_{1})+\alpha$ channel shows a higher nodal
behaviour, as built on the $0_{2}^{+}$ state with the ${}^{9}_{\Lambda}{\rm
Be}(0^{+}_{1})+\alpha$ structure. The result that the energy gain of this
state from the $0_{3}^{+}$ state of ${}^{12}{\rm C}$, $6.7$ MeV, coincides
with the binding energy of ${}^{9}_{\Lambda}{\rm Be}$, $6.71$ MeV, also
supports the idea that the $\Lambda$ particle strongly couples with the
$2\alpha$ clusters to form ${}^{9}_{\Lambda}{\rm Be}$, and higher nodal
structure is realized, due to the orthogonalization to the $0_{2}^{+}$ state
of dominant ${}^{9}_{\Lambda}{\rm Be}+\alpha$ cluster structure.

###### Acknowledgements.

The authors would like to thank Prof. Emiko Hiyama for useful discussions and
support for the code. This work is supported by the Strategic Priority
Research Program of Chinese Academy of Sciences under the Grant NO.
XDB34030301, the National Natural Science Foundation of China under the Grant
NOs. 12005266, 12075288, 11735003, 11961141012 and Guangdong Major Project of
Basic and Applied Basic Research No. 2020B0301030008. It is also supported by
the Youth Innovation Promotion Association CAS. YF acknowledges the financial
support by JSPS KAKENHI Grant No. 25400288.

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# What’s happening in your neighborhood? A Weakly Supervised Approach to
Detect Local News

Deven Santosh Shah
Microsoft
<EMAIL_ADDRESS>
Shiying He
Microsoft
<EMAIL_ADDRESS>
Gosuddin Kamaruddin Siddiqi
Microsoft
<EMAIL_ADDRESS>
Radhika Bansal
Microsoft
<EMAIL_ADDRESS>

###### Abstract

Local news articles are a subset of news that impact users in a geographical
area, such as a city, county, or state. Detecting local news (Step 1) and
subsequently deciding its geographical location as well as radius of impact
(Step 2) are two important steps towards accurate local news recommendation.
Naive rule-based methods, such as detecting city names from the news title,
tend to give erroneous results due to lack of understanding of the news
content. Empowered by the latest development in natural language processing,
we develop an integrated pipeline that enables automatic local news detection
and content-based local news recommendations. In this paper, we focus on Step
1 of the pipeline, which highlights: (1) a weakly supervised framework
incorporated with domain knowledge and auto data processing, and (2)
scalability to multi-lingual settings. Compared with Stanford CoreNLP NER
model, our pipeline has higher precision and recall evaluated on a real-world
and human-labeled dataset. This pipeline has potential to more precise local
news to users, helps local businesses get more exposure, and gives people more
information about their neighborhood safety.

##  1 Introduction

Local news has always been a constant source of interest because of more
relevancy to the individuals comparing with national or international news.
People love to remain informed about the news and events happening in and
around their neighborhood and find ways to connect with the community.
Detecting this local news and showcasing it to the right audience will help
them to achieve this. Not only does it benefit the local users and news
publishers, but showcasing geolocation-specific news articles can drive user
engagement (Robindro et al., 2017) for the products of digital news
recommendation. These local news articles could be of different types like
crime, events, food and drink, healthcare, politics, college-level sports,
real estate, etc. Some examples of these types of local news articles are:

  * •

Crime: “San Jose Police arrest 74-year-old Fresno man in connection to
homicide”111https://www.cbsnews.com/sanfrancisco/news/san-jose-police-
arrest-74-year-old-fresno-man-in-connection-to-homicide/

  * •

Food and restaurants: “Carmel’s Much-Anticipated New Fine Dining Restaurant
Chez Noir Opens Friday”222https://sf.eater.com/2022/10/5/23389267/chez-noir-
open-new-carmel-restaurant-jonny-black

  * •

Real estate: “See where home prices have been rising the fastest in
Washington”333https://www.msn.com/en-us/money/realestate/see-where-home-
prices-have-been-rising-the-fastest-in-washington/ss-AA153W3H

  * •

Law: “Legislature must remake water laws for a drier
California”444https://calmatters.org/commentary/2022/10/legislature-must-step-
up-for-water-rights-of-all-californians/

  * •

Sports: “Samson Ebukam’s strong second year continues in win over
Rams”555https://sports.yahoo.com/samson-ebukam-strong-second-
continues-120006770.html

Hence, keeping the users informed is a two-step process, (1) detecting whether
an article is a local news article, (2) determining the geolocation and the
impact radius of the local news article, so that we could serve right news
articles to the right audience. In this paper, we will primarily focus on the
former task. We define local news articles that impact a specific set of users
at the city/county/state level.

Many papers have researched local news. These research papers rely on the
geolocation mentioned in the article to serve the local news to their
respective users (Tahmasebzadeh et al., 2021; Bell et al., 2015; Robindro et
al., 2017; Sankaranarayanan et al., 2009). However, having a geolocation
mentioned in the article doesn’t necessarily mean the article is local and is
impacting the local population. We came across multiple issues on relying on
geolocation extraction to treat the article as a local news article. These
are:

  1. 1.

Articles of National Importance: We came across multiple news articles in
which the geolocation is mentioned. However, it impacted more than just the
local population of the geolocation. These news articles were indifferent to
the location name present in them. For instance: “Ryvid Anthem Launch Edition
Electric Bike Preorders Are Now Open”; this article has Irvine, CA mentioned
in its body, but the article impacts more than just the local population of
Irvine, CA666https://www.rideapart.com/news/604729/ryvid-anthem-launch-
preorders-open/.

  2. 2.

Articles reported from a location: The geolocation were the locations from
which the news article is being reported from but the news is not about that
location. For instance: “Laboratory to study dark matter opens 1km under
Australian town”777https://www.pressreader.com/usa/the-guardian-
usa/20220820/282248079355210 is a science and space related article with
Melbourne and Australia in its body and title. “Prince Harry makes surprise
visit to Mozambique ahead of trip back to UK” is apparently talking about a
celebrity888https://metro.co.uk/2022/08/18/prince-harry-makes-surprise-visit-
to-mozambique-ahead-of-trip-back-to-uk-17208363/. These articles will
certainly raise interests from broader readers more than the residents from
the localized areas mentioned in the context. Therefore it is limited to just
showcasing the news to the local audience.

  3. 3.

Difficult to detect geolocation: We also found multiple news articles in which
the geolocation wasn’t present. Still, acronyms of those locations are
present, for instance: “WWU students receive racist emails encouraging
violence against Black students”999https://www.kiro7.com/news/local/video-wwu-
students-receive-racist-emails-encouraging-violence-against-black-
students/ab9d52e5-75e1-4b13-b8b3-6e0c352ee4d4/, “SPD updates employee policies
on tattoos, jewelry, hair styles, gender
language”101010https://komonews.com/news/local/spd-seattle-police-department-
employee-policy-tattoo-jewelry-hair-style-beard-gender-language-recruiting-
application-officer-king-county, WWU for Western Washington University, or SPD
for Seattle Police Department. Detecting these acronyms and mapping them to
the correct location is a difficult task. Techniques (Tahmasebzadeh et al.,
2021; Robindro et al., 2017; Bell et al., 2015; Sankaranarayanan et al., 2009)
relying on geolocation in the article would miss out on these types of local
news articles to be showcased to the right audience.

Detecting whether an article is a local news article is not only a mere
extraction of location names or a certain piece of keywords. It needs a
comprehensive systematization of the contextual information, summarization of
the article, and then predicting the possibility of the news documents that
will attract interest from users in particular location affinity. It is
essential to develop an advanced algorithm to understand human language.

Thus, we propose a solution to train a local news classifier. Our major
contributions include: (1) serving local news as a two-step process, (2) a
weakly supervised framework to gather weakly supervised data along with click
statistics of users to train a deep learning model, and (3) an approach to
scale the model to non-English languages.

##  2 Related Work

While this is a first attempt at defining the local news and determining it by
developing a deep learning model, other techniques exist that do not precisely
differentiate between local and non-local news articles but primarily focuses
on showcasing news article based on the geolocation of the user. Tahmasebzadeh
et al. (2021); Bell et al. (2015); Robindro et al. (2017), rely on extracting
the local news as news articles having any geolocation information present in
them. As we mentioned in section 1, not all geolocation-mentioned news
articles are local and have a local impact. These studies are theoretical and
not in production to serve the geolocation-specific local news. Tahmasebzadeh
et al. (2021) proposed using geolocation and structural type extracted from an
image to showcase local news of that area. Their geolocation extraction as a
classification task makes it difficult to scale it worldwide. Bell et al.
(2015), focused on Automatic Speech Recognition (ASR) to convert the audio
news from news broadcasting channels to textual content to showcase to the
users. Robindro et al. (2017) proposed a study where showcasing news articles
belonging to the same geolocation as the user would drive user engagement.

Sankaranarayanan et al. (2009) proposed using tweets on the Twitter platform
to gather breaking news in the area. Unlike Google News, Bing News, and Yahoo!
News, they gather breaking news from User Generated Content (UGC). They also
proposed an importance score defining a particular news article’s importance
to a neighborhood. Selecting the users manually wouldn’t help scale the system
worldwide. This technique was theoretical and not in production.

In the field of Journalism, Gonçalves et al. (2021); Väätäjä et al. (2012)
focused on the idea of participatory journalism to help crowdsource the local
news to the community’s people. They would rely on the geolocation extracted
from the mobile application as the location of the post/news (Alt et al.,
2010). Gonçalves et al. (2021) focused on coping with the challenges local
journalism is facing in Portugal. The neighborhood’s people will be encouraged
to share photos, videos, and posts about the news, and local journalists will
pick it up if it seems a critical issue like crime. These techniques will
require the manual intervention of determining the needle in a haystack of
posts, figuring out the credibility of the users posting, and it would be
difficult to scale it worldwide.

Kliman-Silver et al. (2015) presents a study showcasing the importance of user
personalization using Google search results based on user’s geolocation. They
showcase that the queries that are indifferent to the location, for instance,
“Joe Biden” or “abortion” only show a minor change in the search results for
different user locations. However, maximum changes are observed for queries
like “Starbucks” and “KFC”. This can further be extended to showcase local
news articles based on the precise location of the user (Alt et al., 2010).

Detecting local news would be the first step in moving forward in the user
geolocation-based personalization in the news domain, followed by the
geolocation detection and recognition from the article. In this paper, we
primarily focus on the first step, i.e., detecting local news articles. If we
don’t detect local news and don’t show it to the right audience, it will
degrade the user experience.

##  3 Methodology

###  3.1 Problem formulation

As we mentioned earlier, we define serving local news to the users as a two-
step process, (1) detecting whether an article is a local news article, (2)
determining the geolocation of the article to be served to the right audience.
These two tasks form the basis for informing users about their
city/county/state. We define local news as news articles that impact a
specific segment of users at the city/county/state level. Our approach to
determining local news is to develop a scalable multi-lingual local news
classifier trained on a weakly supervised dataset. We will discuss various
techniques used to curate the weakly supervised training dataset in Section 4.

###  3.2 Model Overview

We trained a binary classification model by using XLM-RoBERTa (XLM-R)(Conneau
et al., 2019) model as our base model. We chose XLM-R as it is a transformer-
based multi-lingual model with pre-learned multi-lingual associations in 100
different languages, allowing us to quickly transfer learn the model learned
in the English language to other languages. These multi-lingual associations
enable us to scale the classifier to different languages instead of having a
separate local classifier model for different languages without losing much on
the precision and recall of the English language data. We fine-tuned XLM-R by
attaching convolution filters to capture 2, 3, and 4 grams which are
eventually connected to a dense layer to calculate the probability of the
article being a local news. More details on the model’s scalability to other
languages can be found in section 4.

###  3.3 Prediction Algorithm / Features Used

Four different features were used in the binary classification model for
training and inference. These are:

  * •

Topics: To capture topics of the article, we ran various off-the-shelf and in-
house trained topic models, like important keywords extractor, Term Frequency-
Inverse Document Frequency(TF-IDF), and Latent Dirichlet Analysis(LDA).

  * •

Tag-line: We use an in-house trained extractive summarization model to extract
the tagline from the body of the article.

  * •

Title: The title is extracted from the article.

  * •

URL Features: Features extracted from the URL proved fruitful in gaining
higher precision and recall of the classifier model. The URL is split by ‘/’.
With initial model training and analysis, we found the model is biased towards
the publisher names instead of basing the decision on the article’s content.
Hence, we filtered out the publisher name (recognized as the domain name in
the URL) and all the numbers present in the URL. Some URLs also contain the
title of the article. In case 80% of the words overlap between the last
segment of the URL, which normally contains the title, and the extracted title
of the article, we filter out the title segment of the URL.

These features are then concatenated and fed into the Binary Local News
classification model for training and inference.

Figure 1: Publisher to Location Affinity: Depicting steps to figure out the
publisher to location affinity.

##  4 Dataset Preparation

To gather weakly supervised labeled training data, we used multiple weak
supervision techniques and various knowledge constraints (Shah et al., 2021).
The weak supervision techniques are illustrated as follows:

  * •

Publisher marked local articles: We used the publisher marked local articles
as positives and non-local articles as negatives. However, these labels had a
lot of noise. Below are the techniques used to refine the dataset further.

  * •

Publisher-to-location affinity: We mined the user click logs and calculated
the affinity of the publisher to a location. The steps to calculate the
publisher-to-location affinity are (as shown in figure 1):

    * –

Gather all the articles published by the publisher in a time frame.

    * –

Calculate an aggregated click counts across all articles published by a
publisher grouped by the cities where they were showcased.

    * –

Pick only those cities which have number of clicks > 50 and calculate it’s
click distribution.

    * –

Calculate the gap ratio between the values of the city with max distribution
and every other city (discussed in Section 1).

    * –

Choose cities with a gap ratio < 0.25 as the cities the publisher has an
affinity for.

    * –

Classify the publisher as a strong local publisher if the number of chosen
cities in same state is < 10\. For instance, KOMO-TV Seattle is a local
publisher with an affinity to the cities in King county, Washington, US.

    * –

Classify the publisher as a strong non-local publisher if the chosen cities
are across more than two states. For instance, FOX News is a national news
agency having an affinity to various cities across different states like New
York, California, Michigan, etc.

    * –

Classify the rest publishers as ambiguous, such as Associated Press, because
they broadcast both local and non-local news articles.

    * –

Label the articles published by strong local and non-local publishers as local
and non-local, respectively.

After applying these techniques, the dataset will still have noise, for
example, local publishers may also cover national news articles and vice
versa. However, by combining all the knowledge constraints, the noise can be
further reduced.

The publisher-to-location affinity was used to train the binary classification
model only on the English data for bootstrapping the data of non-English
languages (discussed in Section • ‣ 4). We came up with article-to-location
affinity similar to publisher-to-location affinity to correct the labels and
generate an all-language weakly supervised dataset to eventually train a
multi-lingual local classifier model.

  * •

Distant Supervision: From our raw data, we matched the canonical URL and title
of the licensed articles with the URL and the title of the non-licensed
articles, respectively. Matching aims to get similar content from non-licensed
news and increase the dataset size and diversity, as the features extracted
from the non-licensed content would differ.

  * •

Bootstrapping:  Bootstrapping helped us further reduce the noise from the non-
English language training dataset. We trained a binary local news classifier
model on the English language data. The Precision-Recall numbers of this model
are presented in the Table 3. We used the model trained on English language to
detect whether articles from any other languages are local by translating them
into English using GPT-3 (Brown et al., 2020). Suppose the article from a
different language is already marked as local, however, the new trained model
predicts with a high probability that it is non-local (local probability <
0.2), then we corrected the label to non-local and vice versa.

  * •

Neural Machine Translations (NMT): Due to the scarcity of news articles in
other languages, we had to increase the dataset size to make the model
understand the local characteristics of a particular language to avoid label
bias in the training dataset (Shah et al., 2019). To increase the dataset size
further for languages like German, Italian, Spanish, Japanese, French,
Portuguese, Russian, Chinese, and Korean, we used GPT-3 (Brown et al., 2020)
for translations. Both front and back translations are used to generate the
dataset (as shown in Figure 2):

    * –

Front translations: Translate local contents from the English to the target
languages. Front translations helped maintain the precision of the model
trained on English language data.

    * –

Back translations: Translate the data from different languages to English and
then back to their original language. This keeps the same semantics of the
news articles while the changed words allows the model to expand its
vocabulary and improve the precision and recall on non-English data.

Figure 2: Neural Machine Translation; Introduce diversity in the style of
writing news articles in the dataset.

This resulted noise-reduced weakly supervised dataset was used for the
proposed multi-lingual binary classification model.

###  4.1 Test Dataset

We manually labeled and created UHRS111111https://prod.uhrs.playmsn.com/UHRS/
hitapp to crowdsource the labeling of news articles from both English and non-
English languages. The local-news distribution of the test set across
different markets is shown in the table 1.

Language | Local news distribution
---|---
English | 37.21%
non-English | 46.56%
Market | Local news distribution
EN-AU | 23.31%
EN-CA | 42.07%
EN-GB | 38.49%
EN-IN | 39.47%
EN-US | 42.22%
DE-DE | 74.46%
ES-MX | 32.74%
ES-US | 51.60%
FR-CA | 79.61%
FR-FR | 47.40%
IT-IT | 84.33%
JA-JP | 38.89%
Table 1: Local news test set distribution aggregated and market breakdown.

Gap Ratio: Gap Ratio is the metric that we utilize to filter out outliers in
the data based on their distributions. The steps to calculate the gap ratio
are as follows:

  1. 1.

Calculate the distribution of values for a key/column/index.

  2. 2.

Calculate the gap between the distributions of a key with the key having the
max distribution. We formulated it as:

$x_{gap}=\frac{x-x_{maxDistribution}}{x_{maxDistribution}}$

where $x$ is the distribution of a key, $x_{maxDistribution}$ is the
distribution value of the key having the maximum distribution share, $x_{gap}$
is the gap between the distribution of a particular key, and the key having
the maximum distribution value.

  3. 3.

Choose those keys whose gap ratio is < 0.25; Increasing this threshold will
result in capturing more outliers, decreasing the threshold will make the
outlier selection stricter.

##  5 Experiment Details

We trained the model on an Azure compute server with a 2.4GHz CPU and 1 Tesla
V100 GPU. We extracted 5.2 million articles across 10 markets and 6 different
languages and then split the dataset into 90%-10% as training and validation
sets. Table 2 shows the training data distribution across different markets.

Language | w/o NMT | w/ NMT
---|---|---
Data size | % of local articles | Data size | % of local articles
English | 4466533 | 23.50% | 4466533 | 23.50%
non-English | 726718 | 49.41% | 952287 | 60.85%
Market | Data size | % of local articles | Data size | % of local articles
EN-AU | 562242 | 9.26% | 562242 | 9.26%
EN-CA | 706091 | 17.80% | 706091 | 17.80%
EN-GB | 725554 | 13.22% | 725554 | 13.22%
EN-IN | 633173 | 1.40% | 633173 | 1.40%
EN-US | 1839473 | 41.70% | 1839473 | 41.70%
DE-DE | 104080 | 27.46% | 216721 | 64.69%
ES-MX | 148324 | 57.52% | 181527 | 64.78%
FR-FR | 161953 | 54.22% | 171666 | 56.29%
IT-IT | 211942 | 59.13% | 213525 | 58.88%
JA-JP | 100419 | 31.92% | 168848 | 58.86%
Table 2: Local news training set distribution aggregated and market-breakdown.

We didn’t perform hyperparameter tuning and used the default parameters to
fine-tune the Roberta-XLM model (Conneau et al., 2019). We used cross-entropy
loss as our loss function with Adam optimizer.

##  6 Results

The model performance on the test set is measured by precision and recall as
the metric. Since we want to improve user engagement in the local news
segment, our primary focus is improving the precision at the recall cost. We
chose Stanford’s CoreNLP NER model Manning et al. (2014) where geolocation
presents in the articles as local news as baseline. Table 3 and 4 depict
precision-recall of the baseline and trained local news classifier model. The
performances reported are at the prediction score cut-off of 0.5. From the
table, the multi-lingual local classifier model outperforms the NER model on
precision. Interestingly, on the non-English segment, the NER model has a
slightly better recall. This is explainable as local news are supposed to have
a geolocation name presented in the article, however the inverse isn’t true.
Geolocation presents in the article is only a sufficient condition for an
article to be classified as a local content as we introduced in Section 1.
Therefore, this local classifier model achieves a strong performance on
precision while keeping a similar recall as the NER model.

The detailed performance on per market are shown in table 4. The local
classifier model trained on the multi-lingual data set with NMT maintains good
performance on local news identification in most of the market segments.

Language |  English Local Classifier |  Local Classifier w/o NMT |  Local Classifier |  NER Model
---|---|---|---|---
P | R | P | R | P | R | P | R
English | 0.902 |  0.518 | 0.914 |  0.472 | 0.883 |  0.763 | 0.52 |  0.747
Non-English | - |  - | 0.873 |  0.298 | 0.895 |  0.386 | 0.687 |  0.421
Table 3: Aggregated precision/recall numbers on English and non-English language data sets Market |  English Local Classifier |  Local Classifier w/o NMT |  Local Classifier |  NER Model
---|---|---|---|---
P | R | P | R | P | R | P | R
en-us |  0.937 |  0.894 |  0.909 |  0.909 |  0.952 |  0.902 |  68.27 |  0.744
en-ca |  0.918 |  0.647 |  0.904 |  0.541 |  0.864 |  0.836 |  0.540 |  0.779
en-gb |  0.843 |  0.654 |  0.881 |  0.485 |  0.881 |  0.766 |  0.511 |  0.636
en-au |  1.000 |  0.260 |  0.901 |  0.666 |  0.875 |  0.710 |  0.377 |  0.826
en-in |  0.600 |  0.044 |  0.729 |  0.518 |  0.801 |  0.720 |  0.500 |  0.770
de-de |  - |  - |  0.905 |  0.627 |  0.923 |  0.438 |  0.820 |  0.532
es-mx |  - |  - |  0.870 |  0.643 |  0.900 |  0.739 |  0.468 |  0.507
es-us |  - |  - |  0.824 |  0.628 |  0.820 |  0.659 |  0.588 |  0.722
fr-ca |  - |  - |  0.862 |  0.788 |  0.895 |  0.695 |  0.826 |  0.329
fr-fr |  - |  - |  0.878 |  0.318 |  0.869 |  0.220 |  0.536 |  0.494
it-it |  - |  - |  0.969 |  0.524 |  0.951 |  0.530 |  0.860 |  0.437
ja-jp |  - |  - |  0.769 |  0.109 |  0.846 |  0.121 |  0.000 |  0.000
Table 4: Precision/recall numbers per market

##  7 Conclusion

In this paper, we propose an integrated pipeline for local news recommendation
in two steps (1) detecting local news and (2) determining the geolocation and
the impact radius of the article. The first step is the primary focus of this
paper. This is achieved by a multi-lingual model incorporated with multiple
weakly supervised methods. We conducted comprehensive experiments on a real-
world dataset from 6 different languages. The results show that our model is
able to detect the local news precisely with the scalability to multiple
languages.

## Limitations

There is good potential to improve the local news classifier for non-English
languages. Lack of clean training data couldn’t help us achieve higher recall
in non-English languages. We hope this paper encourages further research in
improving the local news classifier in non-English languages to keep people
informed.

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##  Appendix A Appendices

###  A.1 Case Studies

Several research papers rely on the geolocation information present in the
article to detect it as a local news article. Tahmasebzadeh et al. (2021)
proposed using geolocation and structural type extracted from an image to
showcase local news of that area. However, their focus is on news articles
belonging to 14 different structural entities like Skyscraper, Square,
Historic Site, Waterfall, Tourist Attraction, Museum, Building, Religious
Building, Tower, Castle, Bridge, and Monument. The predicted geolocation and
the structure type are used to determine the entity from the Wikidata. Hence,
their recall decreases to capture only those entities in the Wikidata. They
have used ResNets to determine the geolocation in the image, which they have
developed as a classification problem making it difficult to scale it
worldwide. The entities determined from the Wikidata are used to retrieve news
articles from Event Registry. The news articles would get restricted to the
structural entities in an area. They won’t be able to capture the local news
articles about crime, real estate, science, politics, and weather, to name a
few.

Bell et al. (2015) also presented a technique in which extraction of local
news relies only on the geolocation mentioned in the article. Their primary
focus was on using Automatic Speech Recognition(ASR) to convert the audio news
published by news broadcasting agencies to text and use extractive
summarization techniques to extract ten crucial sentences from the text. They
also extract the geolocation information using a Named Entity Recognition
(NER) model and match it with Open Street Map. However, they do not mention
how they would determine if the news article is local. They didn’t mention the
process of training the NER model. If they used an off-the-shelf NER (Manning
et al., 2014), they would miss out on local articles having geolocation
information in the form of acronyms like SFPD, WWU, etc., as we mentioned in
the Section 1.

Sankaranarayanan et al. (2009) proposed using tweets(User Generated Content)
on the Twitter platform to gather breaking news in the area. They also
proposed an importance score defining a particular news article’s importance
to a neighborhood. They manually select the users who post the news. They used
Naïve Bayes to differentiate it as news or junk. They cluster the tweets and
use the geolocation mentioned in the tweet and the user’s location to
determine the geolocation foci of the cluster. This foci is considered the
geolocation of all the tweets falling in that cluster. The tweet may not have
geolocation present; it might not belong to any cluster but will be forced to
join a particular cluster and get it geotagged. Or the tweet may belong to a
different geolocation than the one it is assigned to in a cluster. It is
difficult to scale the framework worldwide as it involves manually selecting
the users who post the news. It is difficult to determine the credibility of
the users who post on Twitter. This technique is theoretical and not in
Production.

###  A.2 Use Case

Another critical use case of this model across news recommendation products is
to identify local news from national publishers and non-local news from local
publishers. This provides insights on analyzing the documents and our partners
which brings more business impacts for the users and partners. The aggregated
precision/recall numbers on these two segments are shown in table 5. As the
publisher segments can be treated as one of the priors of the news articles to
be classified as local or non-local, for the segment of national publishers,
we use the cut-off of 0.7 to report the performance and 0.5 for local
publishers as prediction cut-off. The model shows the high precision on these
two segments.

Language |  Publisher Segment | Baseline Model | Local Classifier
---|---|---|---
P |  R |  P |  R
English |  Local Publishers |  0.936 |  0.762 |  0.925 |  0.896
National Publishers |  0.763 |  0.121 |  0.813 |  0.512
non-English |  Local Publishers |  - |  - |  0.902 |  0.420
National Publishers |  - |  - |  0.875 |  0.094
Table 5: Precision/recall numbers on publisher segments