# Low energy reaction $K^{-}p\rightarrow\Lambda\eta$ and the negative parity
$\Lambda$ resonances

Li-Ye Xiao and Xian-Hui Zhong 111E-mail<EMAIL_ADDRESS>Department of
Physics, Hunan Normal University, and Key Laboratory of Low-Dimensional
Quantum Structures and Quantum Control of Ministry of Education, Changsha
410081, China

###### Abstract

The reaction $K^{-}p\rightarrow\Lambda\eta$ at low energies is studied with a
chiral quark model approach. Good descriptions of the existing experimental
data are obtained. It is found that $\Lambda(1670)$ dominates the reaction
around threshold. Furthermore, $u$\- and $t$-channel backgrounds play crucial
roles in this reaction as well. The contributions from the $D$-wave state
$\Lambda(1690)$ are negligibly small for its tiny coupling to $\eta\Lambda$.
To understand the strong coupling properties of the low-lying negative parity
$\Lambda$ resonances extracted from the $\bar{K}N$ scattering, we further
study their strong decays. It is found that these resonances are most likely
mixed states between different configurations. Considering these low-lying
negative parity $\Lambda$ resonances as mixed three-quark states, we can
reasonably understand both their strong decay properties from Particle Data
Group and their strong coupling properties extracted from the $\bar{K}N$
scattering. As a byproduct, we also predict the strong decay properties of the
missing $D$-wave state $|\Lambda\frac{3}{2}^{-}\rangle_{3}$ with a mass of
$\sim 1.8$ GeV. We suggest our experimental colleagues search it in the
$\Sigma(1385)\pi$ and $\Sigma\pi$ channels.

###### pacs:

12.39.Jh, 13.75.Jz, 13.30.-a, 14.20.Jn

##  I INTRODUCTION

Our knowledge about $\Lambda$ resonances is much poorer than that of nucleon
resonances Beringer:2012 . Even for the well-established low-lying negative
parity states, such as $\Lambda(1405)S_{01}$, $\Lambda(1520)D_{03}$ and
$\Lambda(1670)S_{01}$, their properties are still controversial Klempt:2009pi
. Up to now we can not clarify whether these states are excited three quark
states, dynamically generated resonances, three quark states containing multi-
quark components or the other explanations, although there are extensive
discussions about their natures Chen:2009de ; Isgur78 ; Isgur:1977ky ;
Capstick:1986bm ; Melde:2008 ; Bijker:2000 ; Glozman:1997ag ; Loring:2001ky ;
Schat:2001xr ; Goity:2002pu ; Menadue:2011pd ; Engel:2012qp ; Koniuk:1979vy ;
Melde:2006yw ; Melde:2007 ; An:2010wb ; Hyodo:2011ur ; Oller:2005ig ;
Roca:2006sz ; Oller:2006hx ; Borasoy:2005ie ; Hyodo:2003qa ; Oset:2001cn ;
GarciaRecio:2002td ; Jido:2003cb ; Oller:2000fj ; Oset:1997it ; Oller:2006jw ;
Borasoy:2006sr ; Roca:2008kr ; Bouzasa;2008 ; Martin:1969ud ; Manley:2002 ;
Lutz:2001yb ; Buttgen:1985yz ; MuellerGroeling:1990cw ; Hamaie:1995wy ;
Zhong:2009 ; Martin:1980qe ; Gensini:1997fp ; Guo:2012vv ; Zhang:2013cua ;
Zhang:2013sva ; Geng:2007vm ; Geng:2007hz ; Geng:2008er ; Liu:2012ge ;
Liu:2011sw ; Xie:2013wfa ; GarciaRecio:2003ks ; Zou:2008be .

Recently, we systematically studied the reactions
$K^{-}p\rightarrow\Sigma^{0}\pi^{0},\Lambda\pi^{0},\bar{K}^{0}n$ in a chiral
quark model approach Zhong:2013oqa . Obvious roles of the low-lying negative
parity states, $\Lambda(1405)$, $\Lambda(1520)$ and $\Lambda(1670)$, are found
in the $K^{-}p\rightarrow\Sigma^{0}\pi^{0},\bar{K}^{0}n$ reactions, where we
have extracted their strong coupling properties. For example, we found that
$\Lambda(1670)$ should have a very weak coupling to $\bar{K}N$, while
$\Lambda(1520)$ needs a strong coupling to $\bar{K}N$, which can not be well
explained with the symmetry constituent quark model in the $SU(6)\otimes O(3)$
limit Zhong:2013oqa .

To obtain more strong coupling properties and better understandings of these
low-lying $\Lambda$ resonances, in this work, we will continue to study
another important $\bar{K}N$ reaction $K^{-}p\rightarrow\eta\Lambda$. This
reaction provides us a very clear place to study the low-lying $\Lambda$
resonances, because only the $\Lambda$ resonances contribute here due to the
isospin selection rule. Especially, the poorly known strong coupling of
$\Lambda(1670)$ to $\eta\Lambda$ might be reliably obtained from the
$K^{-}p\rightarrow\eta\Lambda$, for this reaction at threshold is dominated by
formation of the $\Lambda(1670)$ Starostin:2001zz . Recently, the new data of
the $K^{-}p\rightarrow\eta\Lambda$ reaction from Crystal Ball Collaboration
Starostin:2001zz  were analyzed with an effective Lagrangian model by Liu and
Xie Liu:2012ge ; Liu:2011sw . They might find some evidence of a exotic
$D$-wave resonance with mass $M\simeq 1669$ MeV and width $\Gamma\simeq 1.5$
MeV in the reaction, which will be discussed in present work as well.

Furthermore, to understand the natures of these strong coupling properties
extracted from the $\bar{K}N$ scattering, we will further carry out a
systematical study of the strong decays of the low-lying negative parity
$\Lambda$ resonances in the chiral quark model approach as well. Combing the
strong coupling properties extracted from the $\bar{K}N$ scattering with the
strong decay properties from the Particle Data Group (PDG) Beringer:2012 , we
expect to obtain more reliable understandings of the natures for these low-
lying negative parity $\Lambda$ resonances.

This work is organized as follows. In Sec. II, the model is reviewed. Then,
the numerical results are presented and discussed in Sec. III. Finally, a
summary is given in Sec. IV.

##  II FRAMEWORK

In this work, we study the $K^{-}p\rightarrow\eta\Lambda$ reaction in a chiral
quark model. This model has been well developed and widely applied to meson
photoproduction reactions qkk ; Li:1997gda ; zhao-kstar ; qk3 ; qk4 ; qk5 ;
He:2008ty ; Saghai:2001yd ; Zhong:2011ht . Its recent extension to describe
the $\pi N$ Zhong:2007fx  and $\bar{K}N$ Zhong:2009 ; Zhong:2013oqa  reactions
also turns out to be successful and inspiring.

In the calculations, we consider three basic Feynman diagrams, i.e., $s$-,
$u$\- and $t$-channels at the tree level. The reaction amplitude is expressed
as

$\mathcal{M}=\mathcal{M}_{s}+\mathcal{M}_{u}+\mathcal{M}_{t},$ (1)

where the $s$\- and $u$-channel reaction amplitudes $\mathcal{M}_{s}$ and
$\mathcal{M}_{u}$ are given by

$\displaystyle\mathcal{M}_{s}=\sum_{j}\langle
N_{f}|H^{f}_{m}|N_{j}\rangle\langle
N_{j}|\frac{1}{E_{i}+\omega_{i}-E_{j}}H^{i}_{m}|N_{i}\rangle,$ (2)
$\displaystyle\mathcal{M}_{u}=\sum_{j}\langle
N_{f}|H^{i}_{m}\frac{1}{E_{i}-\omega_{f}-E_{j}}|N_{j}\rangle\langle
N_{j}H^{f}_{m}|N_{i}\rangle.$ (3)

In the above equations, $H_{m}$ stands for the quark-meson coupling, which
might be described by the effective chiral Lagrangian qkk ; Li:1997gda

$H_{m}=\sum_{j}\frac{1}{f_{m}}\overline{\psi}_{j}\gamma^{j}_{u}\gamma^{j}_{5}\psi_{j}\vec{\tau}\cdot\partial^{u}\vec{\phi}_{m},$
(4)

where $\psi_{j}$ represents the $j$th quark field in a baryon, and $f_{m}$ is
the meson’s decay constant. The pseudoscalar meson octet $\phi_{m}$ is written
as

$\phi_{m}=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&\pi^{+}&K^{+}\cr\pi^{-}&-\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&K^{0}\cr
K^{-}&\bar{K}^{0}&-\sqrt{\frac{2}{3}}\eta\end{array}\right).$ (5)

In Eqs. (2) and (3), $\omega_{i}$ and $\omega_{f}$ are the energies of the
incoming and outgoing mesons, respectively. $|N_{i}\rangle$, $|N_{j}\rangle$
and $|N_{f}\rangle$ stand for the initial, intermediate and final states,
respectively, and their corresponding energies are $E_{i}$, $E_{j}$ and
$E_{f}$, which are the eigenvalues of the nonrelativistic Hamiltonian of
constituent quark model $\hat{H}$ Isgur78 ; Isgur:1977ky ; Capstick:1986bm .

The resonance transition amplitudes of the $s$-channel can be generally
expressed as Zhong:2007fx

$\displaystyle\mathcal{M}^{s}_{R}=\frac{2M_{R}}{s-M^{2}_{R}+iM_{R}\Gamma_{R}}\mathcal{O}_{R}e^{-(\textbf{k}^{2}+\textbf{q}^{2})/(6\alpha^{2})},$
(6)

where $M_{R}$ and $\Gamma_{R}$ stand for the mass and width of the resonance,
respectively. The Mandelstam variable $s$ is defined as
$s\equiv(P_{i}+k)^{2}$. The single-resonance-excitation amplitude,
$\mathcal{O}_{R}$, can be obtain by the relation Zhong:2013oqa

$\displaystyle\mathcal{O}(n,l,J)=\sum_{R}\mathcal{O}_{R}(n,l,J)=\sum_{R}g_{R}\mathcal{O}(n,l,J),$
(7)

where $g_{R}$ stands for the relative strength of a single-resonance in the
partial amplitude $\mathcal{O}(n,l,J)$. The $g_{R}$ factors are determined by
the structure of each resonance and their couplings to the meson and baryon.
The partial amplitudes, $\mathcal{O}(n,l,J)$, up to $n=2$ shell have been
derived in our previous work Zhong:2013oqa , where the details can be found.
For example, the important partial amplitude for the $S$ waves is given by
Zhong:2013oqa

$\displaystyle\mathcal{O}_{1}(S)$ $\displaystyle=$
$\displaystyle\left(g_{s1}-\frac{1}{2}g_{s2}\right)\Big{(}|\mathbf{A}_{out}|\cdot|\mathbf{A}_{in}|\frac{|\mathbf{k}||\mathbf{q}|}{9\alpha^{2}}+\frac{\omega_{i}}{6\mu_{q}}\mathbf{A}_{out}\cdot\mathbf{q}$
(8)
$\displaystyle+\frac{\omega_{f}}{6\mu_{q}}\mathbf{A}_{in}\cdot\mathbf{k}+\frac{\omega_{i}\omega_{f}}{4\mu_{q}\mu_{q}}\alpha^{2}\Big{)},$

where $\mathbf{k}$ and $\mathbf{q}$ stand for the three-momenta of the
incoming and outgoing mesons, respectively, and $\alpha$ is the harmonic
oscillator parameter. The reduced mass $\mu_{q}$ at the quark level is defined
by $1/\mu_{q}=1/m_{q}^{i}+1/m_{q}^{f}$, where $m_{q}^{i}$ and $m_{q}^{f}$
correspond to the initial and final quark masses, respectively.
$\mathbf{A}_{in}$ and $\mathbf{A}_{out}$ are the same variables defined in
Zhong:2013oqa . The $g$-factors in the partial amplitudes, such as $g_{s1}$
and $g_{s2}$, have been defined in Zhong:2013oqa  as well. These $g$-factors
can be derived in the SU(6)$\otimes$O(3) symmetry limit. In Tab. 1, we have
listed the $g$-factors for the reaction $K^{-}p\rightarrow\eta\Lambda$.

And the transition amplitudes of the $u$-channel are given by Zhong:2007fx ;
Zhong:2009

$\displaystyle\mathcal{M}^{u}_{n}=-\frac{2M_{n}}{u-M^{2}_{n}}\mathcal{O}_{n}e^{-(\textbf{k}^{2}+\textbf{q}^{2})/(6\alpha^{2})}.$
(9)

In Eq.(9), the amplitude $\mathcal{O}_{n}$ is determined by the structure of
each resonance and their couplings to the meson and baryon, which has been
derived in our previous work Zhong:2013oqa . The Mandelstam variable $u$ are
defined as $u\equiv(P_{i}-q)^{2}$.

In the calculations, we consider the vector- and scalar-exchanges for the
$t$-channel backgrounds. The vector meson-quark and scalar meson-quark
couplings are given by

$\displaystyle H_{V}$ $\displaystyle=$
$\displaystyle\bar{\psi}_{j}\left(a\gamma^{\nu}+\frac{b\sigma^{\nu\lambda}\partial_{\lambda}}{2m_{q}}\right)V_{\nu}\psi_{j},$
(10) $\displaystyle H_{S}$ $\displaystyle=$ $\displaystyle
g_{Sqq}\bar{\psi}_{j}\psi_{j}S,$ (11)

where $V$ and $S$ stand for the vector and scalar fields, respectively. The
constants $a$, $b$ and $g_{Sqq}$ are the vector, tensor and scalar coupling
constants, respectively. They are treated as free parameters in this work.

On the other hand, the $VPP$ and $SPP$ couplings ($P$ stands for a
pseudoscalar-meson) are adopted as

$\displaystyle H_{VPP}$ $\displaystyle=$ $\displaystyle-
iG_{V}Tr([\phi_{m},\partial_{\mu}\phi_{m}]V^{\mu}),$ (12) $\displaystyle
H_{SPP}$ $\displaystyle=$
$\displaystyle\frac{g_{SPP}}{2m_{\pi}}\partial_{\mu}\phi_{m}\partial^{\mu}\phi_{m}S,$
(13)

where $G_{V}$ and $g_{SPP}$ are the $VPP$ and $SPP$ coupling constants to be
determined by experimental data.

For the vector meson exchange, the $t$-channel amplitude in the quark model is
written as Zhong:2013oqa

$\mathcal{M}^{V}_{t}=\mathcal{O}^{t}_{V}\frac{1}{t-M^{2}_{V}}e^{-(\mathbf{q}-\mathbf{k})^{2}/(6\alpha^{2})},$
(14)

where $e^{-(\mathbf{q}-\mathbf{k})^{2}/(6\alpha^{2})}$ is a form factor
deduced from the quark model, and $M_{V}$ is the vector-meson mass. The
amplitude $\mathcal{O}^{t}_{V}$ is given by Zhong:2013oqa

$\displaystyle\mathcal{O}^{t}_{V}$ $\displaystyle=$ $\displaystyle
G_{v}a[g^{s}_{t}(\mathcal{H}_{0}+\mathcal{H}_{1}\mathbf{q}\cdot\mathbf{k})+g^{v}_{t}\mathcal{H}_{2}i\mathbf{\sigma}\cdot(\mathbf{q}\times\mathbf{k})]$
(15) $\displaystyle+\text{tensor term},$

where the factors $g^{s}_{t}$ and $g^{v}_{t}$ are defined by
$g^{s}_{t}\equiv\langle N_{f}|\sum^{3}_{j=1}I^{ex}_{j}|N_{i}\rangle$ and
$g^{v}_{t}\equiv\langle
N_{f}|\sum^{3}_{j=1}\sigma_{j}I^{ex}_{j}|N_{i}\rangle$, where, $I^{ex}_{j}$ is
the isospin operator of exchanged meson. These factors can be deduced from the
quark model.

For the scalar meson exchange, the $t$-channel amplitude in the quark model is
given by Zhong:2013oqa

$\mathcal{M}^{S}_{t}=\mathcal{O}^{t}_{S}\frac{1}{t-M^{2}_{S}}e^{-(\mathbf{q}-\mathbf{k})^{2}/(6\alpha^{2})},$
(16)

where $M_{S}$ is the scalar-meson mass, and the $\mathcal{O}^{t}_{S}$ is
written as Zhong:2013oqa

$\displaystyle\mathcal{O}^{t}_{S}$ $\displaystyle\simeq$
$\displaystyle\frac{g_{SPP}g_{Sqq}}{2m_{\pi}}(\omega_{i}\omega_{f}-\mathbf{q}\cdot\mathbf{k})[g^{s}_{t}(\mathcal{A}_{0}+\mathcal{A}_{1}\mathbf{q}\cdot\mathbf{k})$
(17) $\displaystyle
g^{v}_{t}\mathcal{A}_{2}i\mathbf{\sigma}\cdot(\mathbf{q}\times\mathbf{k})].$

In Eqs.(15) and (17), the variables $\mathcal{H}_{i}$ and $\mathcal{A}_{i}$
($i=0,1,2$) are the same definitions as in Zhong:2013oqa .

In this work, we consider the $K^{*}$\- and $\kappa$-exchanges in the
$K^{-}p\rightarrow\Lambda\eta$ process. The factors $g^{s}_{t}$ and
$g^{v}_{t}$ derived from the quark model have been listed in Tab. 1 as well.

Table 1: The $g$-factors appearing in the $s$-, $u$\- and $t$-channel amplitudes of the $K^{-}p\rightarrow\Lambda\eta$ process obtained in in the SU(6)$\otimes$O(3) symmetry limit. $\phi_{p}$ is the $\eta$-$\eta^{\prime}$ mixing angle defined in DiDonato:2011kr ; Ke:2009mn . $g_{s1}=-\frac{\sqrt{6}}{6}\sin{\phi_{p}}$ | $g_{v1}=-\frac{\sqrt{6}}{4}\sin{\phi_{p}}$
---|---
$g^{u}_{s1}=\frac{\sqrt{3}}{2}\cos{\phi_{p}}$ | $g^{u}_{v1}=\frac{\sqrt{3}}{2}\cos{\phi_{p}}$
$g^{s}_{t}=\frac{\sqrt{6}}{2}$ | $g^{v}_{t}=\frac{\sqrt{6}}{2}$

##  III RESULT AND ANALYSIS

###  III.1 Parameters

With the transition amplitudes derived within the quark model, the
differential cross section can be calculated by Zhong:2013oqa

$\displaystyle\frac{d\sigma}{d\Omega}$ $\displaystyle=$
$\displaystyle\frac{(E_{i}+M_{i})(E_{f}+M_{f})}{64\pi^{2}s(2M_{i})(2M_{f})}\frac{|\mathbf{q}|}{|\mathbf{k}|}\frac{M_{N}^{2}}{2}$
(18)
$\displaystyle\times\sum_{\lambda_{i},\lambda_{f}}\left|[\frac{\delta_{m_{i}}}{f_{m_{i}}}\frac{\delta_{m_{f}}}{f_{m_{f}}}(\mathcal{M}_{s}+\mathcal{M}_{u})+\mathcal{M}_{t}]_{\lambda_{f},\lambda_{i}}\right|^{2},$

where $\lambda_{i}=\pm 1/2$ and $\lambda_{f}=\pm 1/2$ are the helicities of
the initial and final state $\Lambda$ baryons, respectively. $f_{m_{i}}$ and
$f_{m_{f}}$ are the initial and final meson decay constants, respectively.
$\delta_{m_{i}}\delta_{m_{f}}$ is a global parameter accounting for the flavor
symmetry breaking effects arising from the quark-meson couplings, which will
be determined by experimental data.

In the calculation, the universal value of harmonic oscillator parameter
$\alpha=0.4$ GeV is adopted. The masses of the $u$, $d$, and $s$ constituent
quarks are set as $m_{u}=m_{d}=330$ MeV, and $m_{s}=450$ MeV, respectively.
The decay constants for $\eta$ and $K$ are adopted as $f_{\eta}=f_{K}=160$
MeV.

In present work, the resonance transition amplitude, $\mathcal{O}_{R}$, is
derived in the $SU(6)\otimes O(3)$ symmetric quark model limit. In reality,
due to e.g. spin-dependent forces in the quark-quark interaction, the symmetry
of $SU(6)\otimes O(3)$ is generally broken. As a result, configuration mixing
would occur Isgur78 ; Isgur:1977ky ; Capstick:1986bm ; Schat:2001xr . To take
into account the breaking of that symmetry, a set of coupling strength
parameters, $C_{R}$, should be introduced for each resonance amplitude,

$\mathcal{O}_{R}\rightarrow C_{R}\mathcal{O}_{R},$ (19)

where $C_{R}$ should be determined by fitting the experimental observation.
One finds that $C_{R}=1$ in the $SU(6)\otimes O(3)$ symmetry limit, while
deviation of $C_{R}$ from unity implies the $SU(6)\otimes O(3)$ symmetry
breaking. The determined values of $C_{R}$ for the
$K^{-}p\rightarrow\Lambda\eta$ process have been listed in Table 2. These
strength parameters $C_{R}$ for the main resonances will be further discussed
in Sec. III.4.

Table 2: The determined values for the parameters $C_{R}$, $\delta_{m_{i}}\delta_{m_{f}}$ and $\phi_{P}$ in the $K^{-}p\rightarrow\Lambda\eta$ scatting process. Parameter | $C_{S_{01}(1405)}$ | $C_{D_{03}(1520)}$ | $C_{S_{01}(1670)}$ | $C_{D_{03}(1690)}$ | $\delta_{m_{i}}\delta_{m_{f}}$ | $\phi_{P}$
---|---|---|---|---|---|---
Value | 1.17 | 1.18 | 34.70 | 38.58 | 1.24 | $35^{\circ}$

In the $t$ channel, there are two parameters, $G_{V}a$ and $g_{SPP}g_{Sqq}$,
which come from $K^{*}$\- and $\kappa$-exchanges, respectively. By fitting the
data, we obtain $G_{V}a\simeq 4.8$ and $g_{SPP}g_{Sqq}\simeq 105$, which are
consistent with our previous determinations in Zhong:2013oqa .

In the $u$ channel, the intermediate states are nucleon and nucleon
resonances. One finds that the contributions from $n\geq 1$ shell are
negligibly small, and are insensitive to the degenerate masses for these
shells. In present work, we take $M_{1}=1650$ MeV and $M_{2}=1750$ MeV for the
degenerate masses of $n=1$ and $n=2$ shell nucleon resonances, respectively.

In the $s$-channel of the $K^{-}p\rightarrow\Lambda\eta$ process, there are
only the contributions from $\Lambda$ resonances. The low-lying $\Lambda$
resonances classified in the quark model up to $n=2$ shell are listed in Tab.
3. From the table, we can see that in $n=0$ shell, only the $\Lambda$ pole
contribute to the process, while in $n=1$ shell, two $S$-waves (i.e.,
$[70,^{2}1]\Lambda(1405)S_{01}$, $[70,^{2}8]\Lambda(1670)S_{01}$) and two
$D$-waves (i.e., $[70,^{2}1]\Lambda(1520)D_{03}$,
$[70,^{2}8]\Lambda(1690)D_{03}$) contribute to the reaction. The excitations
of $[70,^{4}8]$ are forbidden for the $\Lambda$-selection rule Zhao:2006an .
In the calculations, the $n=2$ shell $\Lambda$ resonances in the $s$ channel
are treated as degeneration, and their degenerate mass and width are taken as
$M$=1800 MeV and $\Gamma$=100 MeV, since in the low-energy region the
contributions from the $n=2$ shell are not significant.

By fitting the experimental data, we obtain their Breit-Wigner masses and
widths, which are listed in Tab. 4. From the table, it is seen that the
extracted resonances’ parameters are compatible with the data from PDG
Beringer:2012 .

Table 3: The classification of $\Lambda$ resonances in the quark model up to n=2 shell. The ”?” denotes the resonances being unestablished. $l_{I,2J}$ is the PDG notation of baryons. $N_{6}$ and $N_{3}$ denote the SU(6) and SU(3) representation. $L$ and $S$ stand for the total orbital momentum and spin of a baryon, respectively. $|N_{6},^{2S+1}N_{3},n,L\rangle$ | $l_{I,2J}$ | $|N_{6},^{2S+1}N_{3},n,L\rangle$ | $l_{I,2J}$
---|---|---|---
$|56,^{2}8,0,0\rangle$ | $P_{01}(1116)$ | … | …
$|70,^{2}1,1,1\rangle$ | $S_{01}(1405)$ | $|56,^{2}8,2,2\rangle$ | $P_{03}(?)$
| $D_{03}(1520)$ |  | $F_{05}(?)$
$|70,^{2}8,1,1\rangle$ | $S_{01}(1670)$ | $|70,^{2}1,2,2\rangle$ | $P_{03}(?)$
| $D_{03}(1690)$ |  | $F_{05}(?)$
$|70,^{4}8,1,1\rangle$ | $S_{01}(1800)$ | $|70,^{2}8,2,2\rangle$ | $P_{03}(?)$
| $D_{03}(?)$ |  | $F_{05}(?)$
| $D_{05}(1830)$ |  |
$|56,^{2}8,2,0\rangle$ | $P_{01}(1600)$ | $|70,^{4}8,2,2\rangle$ | $P_{01}(?)$
$|70,^{2}1,2,0\rangle$ | $P_{01}(1810)$ |  | $P_{03}(?)$
$|70,^{2}8,2,0\rangle$ | $P_{01}(?)$ |  | $F_{05}(?)$
$|70,^{4}8,2,0\rangle$ | $P_{03}(?)$ |  | $F_{07}(?)$
Table 4: Breit-Wigner masses $M_{R}$ (MeV) and widths $\Gamma_{R}$ (MeV) for the resonances in the $s$-channel compared with the experimental data from PDG. Resonance | $M_{R}$ | $\Gamma_{R}$ |  $M_{R}$ (PDG) |  $\Gamma_{R}$ (PDG)
---|---|---|---|---
$\Lambda(1405)S_{01}$ | $1405.0$ | $53.37$ | $1405.0^{+1.3}_{-1.0}$ | $50\pm 2$
$\Lambda(1520)D_{03}$ | $1519.5$ | $15.6$ | $1519.5\pm 1.0$ | $15.6\pm 1.0$
$\Lambda(1670)S_{01}$ | $1676.0$ | $35.0$ | $1670\pm 10$ | $25\sim 50$
$\Lambda(1690)D_{03}$ | $1682.4$ | $70.0$ | $1690\pm 5$ | $50\sim 70$

###  III.2 Total cross section

The total cross section as a function of the beam momentum is shown in Fig. 1,
where we find that the observations can be well described within the chiral
quark model.

Figure 1: $K^{-}p\rightarrow\Lambda\eta$ total cross sections compared with
the data Starostin:2001zz . The bold solid curves are for the full model
calculations. In the upper panel, exclusive cross sections for
$\Lambda(1405)S_{01}$, $\Lambda(1670)S_{01}$, $t$-channel and $u$-channel are
indicated explicitly by the legends in the figures. In the lower panel, the
results by switching off the contributions of $\Lambda(1405)S_{01}$,
$\Lambda(1670)S_{01}$, $t$-channel and $u$-channel are indicated explicitly by
the legends in the figures.

It is found that $\Lambda(1670)S_{01}$ should play a dominant role in the
reaction. $\Lambda(1670)S_{01}$ is responsible for the bump structure in the
cross section around its threshold. To well describe the data, a large
amplitude of $\Lambda(1670)S_{01}$ in the reaction is needed, which is about
34 times (i.e., $C_{S_{01}(1670)}=34$) larger than that derived in the
$SU(6)\otimes O(3)$ limit. In our previous work, we found
$\Lambda(1670)S_{01}$ should have a weaker coupling to $\bar{K}N$ than that
derived in the $SU(6)\otimes O(3)$ limit Zhong:2013oqa , thus,
$\Lambda(1670)S_{01}$ should have a much stronger coupling to $\eta\Lambda$
than that derived from the symmetry quark model. These phenomenologies might
be explained by the configuration mixing between the $S$-wave states
$\Lambda(1405)S_{01}$, $\Lambda(1670)S_{01}$ and $\Lambda(1800)S_{01}$, which
will be further studied in Sec. III.4.

Furthermore, a sizeable contribution from $\Lambda(1405)$ also can be seen in
the cross section. The total cross section around the peak is slightly
underestimated without its contribution.

No obvious contributions from the $D$-wave states, $\Lambda(1520)D_{03}$ and
$\Lambda(1690)D_{03}$, are found in the reaction.

It should be emphasized that both $u$\- and $t$-channel backgrounds play
crucial roles in the reactions. Switching off their contributions, the cross
section changes significantly. The important roles of $u$\- and/or $t$-channel
backgrounds are also found in the other $\bar{K}N$ reactions
$K^{-}p\rightarrow\Sigma^{0}\pi^{0},\Lambda\pi^{0},\bar{K}^{0}n$ Zhong:2013oqa
.

Figure 2: Differential cross sections of the $K^{-}p\rightarrow\eta\Lambda$
compared with the data from Starostin:2001zz . The bold solid curves are for
the full model calculations. The results by switching off the contributions
from $\Lambda(1405)S_{01}$, $\Lambda(1670)S_{01}$ and $u$\- and $t$-channel
backgrounds are indicated explicitly by the legend in the figures.

###  III.3 Differential cross section

The differential cross sections (DCS) compared with the data are shown in Fig.
2. From the figure, it is seen that the data in the low energy region from
threshold to $P_{K}=770$ MeV/c can be reasonably described within our chiral
quark model. However, it should be remarked that our theoretical results seem
to slightly underestimate the DCS at both forward and backward angles in the
beam momenta region of $P_{K}=730\sim 742$ MeV/c. Improved measurements in
this beam momenta region are needed to clarify the discrepancies.

To explore the contribution of individual resonances and the $u$\- and
$t$-channel backgrounds to the DCS, we have shown the predictions by switching
off one of their contributions in Fig. 2 as well. From the figure, the
dominant roles of $\Lambda(1670)S_{01}$ and $u$-, $t$-channel backgrounds are
significantly seen in the DCS. Switching off the contribution of
$\Lambda(1670)S_{01}$, we find that the cross sections will be underestimated
draftily. Without the $u$-channel contribution, the DCS will be significantly
underestimated around $\eta$ production threshold. While, switching off
$t$-channel contribution, we can see that the DCS are strongly overestimated
at both forward and backward angles. Furthermore, slight contributions of
$\Lambda(1405)S_{01}$ can be seen in the DCS around $\eta$ production
threshold, where $\Lambda(1405)S_{01}$ has a constructive interference with
$\Lambda(1670)S_{01}$. However, $\Lambda(1405)S_{01}$ is not a crucial
contributor in the reaction, which can explain the reason why the contribution
of $\Lambda(1405)S_{01}$ can be neglected in some studies at the hadron level
Manley:2002 ; Liu:2012ge ; Liu:2011sw .

From Fig. 2, it is seen that a bowl structure seems to appear in the data
around $\eta$ production threshold. As we know, the bowl structures are the
typical effects of the interferences between the $S$\- and $D$-wave states. In
this energy region, the bowl structures might be caused by the interferences
between $\Lambda(1670)S_{01}$ and $\Lambda(1690)D_{03}$. Considering
$\Lambda(1690)D_{03}$ as the conventional three-quark state classified in the
constituent quark model, we can not obtain a bowl structure in the DCS for the
too small contributions of $\Lambda(1690)D_{03}$ in the reaction. In Refs.
Liu:2012ge ; Liu:2011sw , Liu and Xie had carefully studied these bowl
structures appearing in the DCS, they need an exotic $D$-wave state
$\Lambda(1669)$ with a very narrow width of $\Gamma\simeq 1.5$ MeV to
reproduce the bowl structures. Finally, it should be pointed out that for the
rather large uncertainties of the present data, we can not confirm whether
there are obvious bowl structures in the DCS or not. Thus, more accurate
measurements are needed.

As a whole, $\Lambda(1670)S_{01}$ plays a dominant role in the reaction.
$\Lambda(1670)S_{01}$ should have a much stronger coupling to $\eta\Lambda$,
while have a weaker coupling to $\bar{K}N$ than that derived in the
$SU(6)\otimes O(3)$ limit. The $u$\- and $t$-channel backgrounds also play
crucial roles in the reaction. Furthermore, slight contributions of
$\Lambda(1405)S_{01}$ can be seen in the DCS around $\eta$ production
threshold. No obvious evidence from the $D$-wave states, $\Lambda(1520)D_{03}$
and $\Lambda(1690)D_{03}$, is found in the reaction.

###  III.4 Configuration mixing and strong couplings

To further understand the strength parameters $C_{R}$ in the
$K^{-}p\rightarrow\Lambda\eta$ reaction, and explain the strong coupling
properties of the $\Lambda$ resonances extracted from the $\bar{K}N$
scattering, e.g., the weak coupling of $\Lambda(1670)S_{01}$ to $\bar{K}N$ and
strong coupling of $\Lambda(1670)S_{01}$ to $\eta\Lambda$, in this subsection
we study the configuration mixing effects in the low-lying negative $\Lambda$
resonances.

####  III.4.1 Configuration mixing and strong decays

If there is configuration mixing in several resonances with the same $J^{P}$
values, their strong coupling properties might be very different from the pure
states classified in the constituent quark model. Here, we study the strong
decays of low-lying negative $\Lambda$ resonances and test whether the
configuration mixing can explain the strong couplings of these resonances or
not.

In this work, the strong decays of the $\Lambda$ resonances also studied with
the chiral quark model. This approach has been successfully used to study the
strong decays of charmed baryons, $\Xi$ baryons, and heavy-light mesons
Zhong:2007gp ; Zhong:2010vq ; Liu:2012sj ; Xiao:2013xi . The details of how to
describe the strong decays of the baryon resonances in the chiral quark model
can be found in Xiao:2013xi .

As we know, $\Sigma(1385)$ is a well-estimated strangeness-1 hyperon state.
According to the classification of the quark model, it is assigned to the pure
$|56,^{4}10,0,0,\frac{3}{2}^{+}\rangle$ representation. In this work, the
measured width of $\Sigma^{0}(1385)$ as an input (i.e., $\Gamma=36$ MeV
Beringer:2012 ) is used to determine the overall parameter $\delta$($=0.654$)
in the decay amplitudes. With this determined parameter, we can calculate the
strong decays of the other strangeness-1 hyperon states.

#####  $S$-wave states

Firstly, we study the strong decay properties of the $S$-wave states
$\Lambda(1405)S_{01}$, $\Lambda(1670)S_{01}$ and $\Lambda(1800)S_{01}$. If
they are pure states, according to the classification of the constituent quark
model, they should be assigned to the $|70,^{2}1,1,1,\frac{1}{2}^{-}\rangle$,
$|70,^{2}8,1,1,\frac{1}{2}^{-}\rangle$ and
$|70,^{4}8,1,1,\frac{1}{2}^{-}\rangle$, respectively Xiao:2013xi .

Table 5: The predicted total and partial decay widths (MeV) and partial decay width ratios of $\Lambda(1670)S_{01}$ as a pure state of $|70,^{2}8,1,1,\frac{1}{2}^{-}\rangle$. $\Gamma^{th}$ denotes our prediction, while $\Gamma^{exp}$ denotes the data from PDG.  Channel | $\Gamma^{th}_{i}$ | $\Gamma^{th}_{total}$ | $\Gamma^{exp}_{total}$ | $\frac{\Gamma_{i}}{\Gamma_{total}}|_{th}$ | $\frac{\Gamma_{i}}{\Gamma_{total}}|_{exp}$
---|---|---|---|---|---
$\Sigma\pi$ | $15.4$ | $123.4$ |  $25$ to $50(\approx 35)$ | $0.12$ | $0.25\sim 0.55$
$NK$ | $103.1$ |  |  | $0.84$ | $0.20\sim 0.30$
$\Lambda\eta$ | $0.28$ |  |  | $0.002$ | $0.10\sim 0.25$
$\Sigma(1385)\pi$ | $4.7$ |  |  | $0.04$ | $\cdot\cdot\cdot$

Considering $\Lambda(1670)S_{01}$ as the pure state
$|70,^{2}8,1,1,\frac{1}{2}^{-}\rangle$, we calculate its strong decay
properties, which are listed in Tab. 5. From the table, we see that the total
decay width in theory ($\Gamma^{th}_{total}=123.4$ MeV) are much larger than
the data ($\Gamma^{exp}_{total}\simeq 35$ MeV). Meanwhile, according to our
calculation, the branching ratio of the $\Lambda\eta$ channel is too small,
while the branching ratio of the $N\bar{K}$ channel is too large to compare
with the data. Thus, as a pure state, the $\Lambda(1670)S_{01}$ strong decays
can not be described at all.

It should be remarked that several different representations with the same
$J^{P}$ numbers might be coupled together via some kind of interactions
Isgur78 ; Isgur:1977ky ; Capstick:1986bm ; Schat:2001xr . Thus,
$\Lambda(1670)S_{01}$ may be a mixed state between three different
representations $|70,^{2}1,1,1\rangle$, $|70,^{2}8,1,1\rangle$ and
$|70,^{4}8,1,1\rangle$ with $J^{P}=1/2^{-}$. Based on the standard Cabibbo-
Kobayashi-Maskawa (CKM) matrix method, the physical states might be expressed
as

$\left(\begin{array}[]{c}|\Lambda(1800)\frac{1}{2}^{-}\rangle\cr|\Lambda(1670)\frac{1}{2}^{-}\rangle\cr|\Lambda(1405)\frac{1}{2}^{-}\rangle\end{array}\right)=U\left(\begin{array}[]{c}|70,^{2}1\rangle\cr|70,^{2}8\rangle\cr|70,^{4}8\rangle\end{array}\right),$
(20)

with

$U=\left(\begin{array}[]{ccc}c_{12}c_{13}&s_{12}c_{13}&s_{13}\cr-
s_{12}c_{23}-c_{12}s_{23}s_{13}&c_{12}c_{23}-s_{12}s_{23}s_{13}&s_{23}c_{13}\cr
s_{12}s_{23}-c_{12}c_{23}s_{13}&-c_{12}s_{23}-s_{12}c_{23}s_{13}&c_{23}c_{13}\end{array}\right),$
(21)

where $c_{ij}\equiv\cos{\theta}_{ij}$ and $s_{ij}\equiv\sin{\theta}_{ij}$.
$\theta_{ij}$ stands for the mixing angles, which could be determined by
fitting the strong decay data of $\Lambda(1670)S_{01}$.

By fitting the experiment data from PDG Beringer:2012 , we have obtained that
$\theta_{12}\simeq 75^{0}$, $\theta_{13}\simeq 50^{0}$ and $\theta_{23}\simeq
125^{0}$. With these mixing angles, the strong decay properties of
$\Lambda(1670)S_{01}$ can be reasonably described. The theoretical results
compared with the data are listed in Tab. 6. From the table, it is seen that
with the configuration mixing the $\Lambda\eta$ branching ratio is enhanced
obviously, while the $N\bar{K}$ branching ratio is suppressed, which can
naturally explain the weak coupling of $\Lambda(1670)S_{01}$ to $\bar{K}N$ and
strong coupling of $\Lambda(1670)S_{01}$ to $\eta\Lambda$ needed in the
$\bar{K}N$ reactions.

Table 6: The predicted total and partial decay widths (MeV) and partial decay width ratios of $\Lambda(1670)S_{01}$ as a mixed state compared with the experiment data from PDG. Channel | $\Gamma^{th}_{i}$ | $\Gamma^{th}_{total}$ | $\Gamma^{exp}_{total}$ | $\frac{\Gamma_{i}}{\Gamma_{total}}|_{th}$ | $\frac{\Gamma_{i}}{\Gamma_{total}}|_{exp}$
---|---|---|---|---|---
$\Sigma\pi$ | $11.8$ | $44.7$ |  $25$ to $50(\approx 35)$ | $0.26$ | $0.25\sim 0.55$
$NK$ | $13.6$ |  |  | $0.30$ | $0.20\sim 0.30$
$\Lambda\eta$ | $18.2$ |  |  | $0.41$ | $0.10\sim 0.25$
$\Sigma(1385)\pi$ | $1.1$ |  |  | $0.02$ | $\cdot\cdot\cdot$

According to the determined mixing angles, Eq.(20) can be explicitly expressed
as

$\left(\begin{array}[]{c}|\Lambda(1800)\frac{1}{2}^{-}\rangle\cr|\Lambda(1670)\frac{1}{2}^{-}\rangle\cr|\Lambda(1405)\frac{1}{2}^{-}\rangle\cr\end{array}\right)=\left(\begin{array}[]{ccc}0.17&0.62&0.77\cr
0.39&-0.76&0.53\cr
0.90&0.21&-0.37\end{array}\right)\left(\begin{array}[]{c}|70,^{2}1\rangle\cr|70,^{2}8\rangle\cr|70,^{4}8\rangle\cr\end{array}\right),$
(22)

where, we find that the main component of $\Lambda(1670)S_{01}$ is
$|70,^{2}8\rangle(\sim 58\%$). Meanwhile, the $|70,^{2}1\rangle$ and
$|70,^{4}8\rangle$ components also have a sizable proportion, which are $\sim
15\%$ and $\sim 28\%$, respectively. $\Lambda(1405)S_{01}$ is dominated by the
$|70,^{2}1\rangle$($\sim 81\%$), while it contains significant octet
components of $|70,^{2}8\rangle(\sim 4\%$) and $|70,^{4}8\rangle$($\sim
14\%$). $\Lambda(1800)S_{01}$ is dominated by both the
$|70,^{4}8\rangle$($\sim 59\%$) and $|70,^{2}8\rangle(\sim 38\%$) components.

With these mixing schemes, we have calculated the strong decay properties of
$\Lambda(1405)S_{01}$ and $\Lambda(1800)S_{01}$. The calculated decay width of
$\Lambda(1405)S_{01}$ is $\Gamma\simeq 53$ MeV, which in good agreement with
the data ($\Gamma=50\pm 2$ MeV).

Considering the uncertainties in the mass of $\Lambda(1800)S_{01}$, we vary
its mass from 1700 MeV to 1870 MeV. The predicted strong decay properties of
$\Lambda(1800)S_{01}$ have been shown in Fig. 3. From the figure, we can see
that the strong decays of $\Lambda(1800)S_{01}$ are dominated by the
$\bar{K}N$, $\eta\Lambda$ and $\Sigma\pi$ decay modes. While the decay channel
$\Sigma(1385)\pi$ also has a significant contribution to the strong decays of
$\Lambda(1800)S_{01}$. It is found that our predicted strong decay properties
of $\Lambda(1800)S_{01}$ are compatible with the data of ALSTON (see Tab. 7)
AlstonGarnjost:1977rs . About $\Lambda(1800)S_{01}$, more measurements are
needed in experiments.

As a whole the configuration mixing is needed to understand the strong decay
properties of the $S$-wave resonances $\Lambda(1405)S_{01}$,
$\Lambda(1670)S_{01}$ and $\Lambda(1800)S_{01}$.

Figure 3: The strong decay properties of $\Lambda(1800)S_{01}$, which is taken as a mixed state in Eq.(20). Table 7: The predicted total and partial decay widths (MeV) of $\Lambda(1800)S_{01}$ compared with the experiment data from ALSTON AlstonGarnjost:1977rs . We set the mass of $\Lambda(1800)S_{01}$ as M=1725 MeV, which is the observed value from ALSTON. Channel | $N\bar{K}$ | $\Sigma\pi$ | $\Lambda\eta$ | $\Sigma(1385)\pi$
---|---|---|---|---
$\Gamma^{th}_{i}$ | $51.1$ | $39.5$ | $56.1$ | $13.2$
$\Gamma^{exp}_{i}$ | $52\pm 9$ | … | … | …

#####  $D$-wave states

Then, we will further study whether the configuration mixing is necessary to
explain the strong decays of the well-established $D$-wave resonances
$\Lambda(1520)D_{03}$ and $\Lambda(1690)D_{03}$ or not. If
$\Lambda(1520)D_{03}$ and $\Lambda(1690)D_{03}$ are pure states, they should
be classified as the $|70,^{2}1,1,1,\frac{3}{2}^{-}\rangle$ and
$|70,^{4}8,1,1,\frac{3}{2}^{-}\rangle$ configurations, respectively, in the
constituent quark model.

Table 8: The predicted total and partial decay widths (MeV) and partial decay width ratios of $\Lambda(1520)D_{03}$ as a pure state $|70,^{2}1,1,1,\frac{3}{2}^{-}\rangle$ compared with the experiment data from PRD. Channel | $\Gamma^{th}_{i}$ | $\Gamma^{th}_{total}$ | $\Gamma^{exp}_{total}$ | $\frac{\Gamma_{i}}{\Gamma_{total}}|_{th}$ | $\frac{\Gamma_{i}}{\Gamma_{total}}|_{exp}$
---|---|---|---|---|---
$\Sigma\pi$ | $10.7$ | $14.5$ | $15.6\pm 1.0$ | $0.74$ | $0.42\pm 0.01$
$NK$ | $3.81$ |  |  | $0.26$ | $0.45\pm 0.01$
Table 9: The predicted total and partial decay widths (MeV) and partial decay width ratios of $\Lambda(1690)D_{03}$ as a pure state of $|70,^{2}8,1,1,\frac{3}{2}^{-}\rangle$ compared with the experiment data from PRD. Channel | $\Gamma^{th}_{i}$ | $\Gamma^{th}_{total}$ | $\Gamma^{exp}_{total}$ | $\frac{\Gamma_{i}}{\Gamma_{total}}|_{th}$ | $\frac{\Gamma_{i}}{\Gamma_{total}}|_{exp}$
---|---|---|---|---|---
$\Sigma\pi$ | $9.74$ | $117.2$ | $50\sim 70(\approx 60)$ | $0.08$ | $0.20\sim 0.40$
$NK$ | $58.31$ |  |  | $0.50$ | $0.20\sim 0.30$
$\Lambda\eta$ | $0.001$ |  |  | $0.00$ | $\cdot\cdot\cdot$
$\Sigma(1385)\pi$ | $49.1$ |  |  | $0.42$ | $\cdot\cdot\cdot$

Firstly, we study the decay properties of $\Lambda(1520)D_{03}$ and
$\Lambda(1690)D_{03}$ as pure states. The predictions compared with the data
are listed in Tabs. 8 and 9, respectively.

From Tab. 8, we find that as a pure state the strong decay coupling of
$\Lambda(1520)D_{03}$ to $\Sigma\pi$ is overestimated. However, the strong
coupling of $\Lambda(1520)D_{03}$ to $N\bar{K}$ is underestimated, which is
also found in the $\bar{K}N$ scattering Zhong:2013oqa .

While considering $\Lambda(1690)D_{03}$ as a pure state
$|70,^{4}8,1,1,\frac{3}{2}^{-}\rangle$, from Tab. 9 we find that the
theoretical predictions are inconsistent with the experimental observations.
The predicted total decay width is much larger than the data. In addition, the
partial decay width ratio for $\Sigma\pi$ is too small, while, that for
$N\bar{K}$ is too large to compare with the data. Thus, as pure states, the
strong decay properties of $\Lambda(1520)D_{03}$ and $\Lambda(1690)D_{03}$
can’t be understood reasonably.

For these reasons, it is naturally for us to take $\Lambda(1520)$ and
$\Lambda(1690)$ as two mixing states between
$|70,^{2}1,1,1,\frac{3}{2}^{-}\rangle$, $|70,^{2}8,1,1,\frac{3}{2}^{-}\rangle$
and $|70,^{4}8,1,1,\frac{3}{2}^{-}\rangle$. By the using of the CKM matrix
method again, and fitting the strong decay data of $\Lambda(1690)$, we obtain

$\left(\begin{array}[]{c}|\Lambda(1520)\frac{3}{2}^{-}\rangle\cr|\Lambda(1690)\frac{3}{2}^{-}\rangle\cr|\Lambda\frac{3}{2}^{-}\rangle_{3}\cr\end{array}\right)=\left(\begin{array}[]{ccc}0.94&0.34&0.09\cr
0.31&-0.92&0.26\cr
0.17&-0.21&-0.96\end{array}\right)\left(\begin{array}[]{c}|70,^{2}1\rangle\cr|70,^{2}8\rangle\cr|70,^{4}8\rangle\cr\end{array}\right).$
(23)

From Eq.(23), it is seen that $\Lambda(1690)$ has sizable components of
$|70,^{2}1\rangle$ ($\sim 9\%$) and $|70,^{4}8\rangle$ ($\sim 7\%$), except
for the main component $|70,^{2}8\rangle$ ($\sim 85\%$). The predicted strong
decay properties of $\Lambda(1690)D_{03}$ compared with the data are listed in
Tab. 11, where we find that with the configuration mixing effects, the strong
decays of $\Lambda(1690)D_{03}$ can be well described. It should be emphasized
that $\Lambda(1690)$ has a very weak coupling to $\Lambda\eta$, although it
has been draftily enhanced by considering the configuration mixing effects,
which gives an explanation why the contribution of $\Lambda(1690)D_{03}$ to
the reaction $K^{-}P\rightarrow\Lambda\eta$ is tiny even though
$\Lambda(1690)D_{03}$ has a large $C_{R}$-factor.

Table 10: The predicted total and partial decay widths (MeV) and partial decay width ratios of $\Lambda(1520)$ as a mixed state compared with the experiment data from PDG. Channel | $\Gamma^{th}_{i}$ | $\Gamma^{th}_{total}$ | $\Gamma^{exp}_{total}$ | $\frac{\Gamma_{i}}{\Gamma_{total}}|_{th}$ | $\frac{\Gamma_{i}}{\Gamma_{total}}|_{exp}$
---|---|---|---|---|---
$\Sigma\pi$ | $7.0$ | $13.5$ | $15.6\sim 1.0$ | $0.52$ | $0.42\pm 0.01$
$NK$ | $6.2$ |  |  | $0.46$ | $0.45\pm 0.01$
$\Sigma(1385)\pi$ | $0.3$ |  |  | $0.02$ | $\cdot\cdot\cdot$
Figure 4: The strong decay properties of $|\Lambda\frac{3}{2}^{-}\rangle_{3}$
as a counterpart of $\Lambda(1690)$.

Furthermore, with the mixing scheme determined in Eq.(23), we study the strong
decay of $\Lambda(1520)D_{03}$. The predicted results compared with the data
are listed in Tab. 10, where we find both the total decay width and the
partial decay width ratios are in good agreement with the data. The $N\bar{K}$
branching ratio is about a factor 2 larger than that derived in the
$SU(6)\otimes O(3)$ limit, which is consistent with our previous analysis of
the $\bar{K}N$ scattering in Zhong:2013oqa . From Eq.(23), we can see that the
main component of $\Lambda(1520)D_{03}$ is still the $|70,^{2}1\rangle$
configuration ($\sim 88\%$), while it contains significant octet component of
$|70,^{2}8\rangle$ ($\sim 12\%$).

Table 11: The predicted total and partial decay widths(MeV) and partial decay width ratios of $\Lambda(1690$ as a mixed state compared with the experiment data from PDG.  Channel | $\Gamma^{th}_{i}$ | $\Gamma^{th}_{total}$ | $\Gamma^{exp}_{total}$ | $\frac{\Gamma_{i}}{\Gamma_{total}}|_{th}$ | $\frac{\Gamma_{i}}{\Gamma_{total}}|_{exp}$
---|---|---|---|---|---
$\Sigma\pi$ | $27.5$ | $70.6$ | $50\sim 70(\approx 60)$ | $0.39$ | $0.20\sim 0.40$
$NK$ | $21.4$ |  |  | $0.30$ | $0.20\sim 0.30$
$\Lambda\eta$ | $0.01$ |  |  | $0.00$ | $\cdot\cdot\cdot$
$\Sigma(1385)\pi$ | $21.6$ |  |  | $0.30$ | $\cdot\cdot\cdot$

Finally, we give our predictions of the third $D$-wave resonance
$|\Lambda\frac{3}{2}^{-}\rangle_{3}$, which is still not established in
experiment. According to the quark model prediction, its mass is around $1800$
MeV Isgur78 ; Isgur:1977ky ; Capstick:1986bm ; Schat:2001xr . Varying its mass
from 1700 MeV to 1900 MeV, we calculate the strong decays of
$|\Lambda\frac{3}{2}^{-}\rangle_{3}$. Our predictions have been shown in Fig.
4. It is found that the strong decays of $|\Lambda\frac{3}{2}^{-}\rangle_{3}$
are dominated by $\Sigma(1385)\pi$ and $\Sigma\pi$, while the $N\bar{K}$ and
$\Lambda\eta$ branching ratios are negligibly small. Thus, we suggest the our
experimental colleagues find this missing $D$-wave state in the
$\Sigma(1385)\pi$ and $\Sigma\pi$ channels.

In a word, the configuration mixing is also needed to understand the strong
decay properties of the $D$-wave resonances $\Lambda(1520)D_{03}$,
$\Lambda(1690)D_{03}$.

####  III.4.2 Interpretation of $C_{R}$ with configuration mixing

If the configuration mixing effects are included, the single-resonance-
excitation amplitude given in Eq. (7) should be rewritten as

$\displaystyle\mathcal{O}(n,l,J)=\sum_{R}g_{R}^{\prime}\mathcal{O}(n,l,J)\equiv\sum_{R}C_{R}g_{R}\mathcal{O}(n,l,J),$
(24)

where $g_{R}^{\prime}$ ($g_{R}$) stands for the relative strength of a single-
resonance with (without) configuration mixing effects in the partial amplitude
$\mathcal{O}(n,l,J)$. The $C_{R}$ parameters can be derived by

$C_{R}=\frac{g^{\prime}_{R}}{g_{R}}.$ (25)

In the following work, we study the $C_{R}$ parameters of the important
resonances $\Lambda(1405)S_{01}$, $\Lambda(1670)S_{01}$, $\Lambda(1520)D_{03}$
and $\Lambda(1690)D_{03}$ for the $K^{-}p\rightarrow\eta\Lambda$ process.

Taking $\Lambda(1405)S_{01}$, $\Lambda(1670)S_{01}$, $\Lambda(1520)D_{03}$ and
$\Lambda(1690)D_{03}$ as pure states in the constituent quark model, we can
derive the couplings of the transition amplitudes for these resonances, which
are given by

$\displaystyle R_{\Lambda(1405)}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{3}}{108}(\sqrt{2}\sin\phi_{P}+\cos\phi_{P}),$ (26)
$\displaystyle R_{\Lambda(1670)}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{3}}{108}(\sqrt{2}\sin\phi_{P}-\cos\phi_{P}),$ (27)
$\displaystyle R_{\Lambda(1520)}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{3}}{54}(\sqrt{2}\sin\phi_{P}+\cos\phi_{P}),$ (28)
$\displaystyle R_{\Lambda(1690)}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{3}}{54}(\sqrt{2}\sin\phi_{P}-\cos\phi_{P}),$ (29)

where the $\phi_{P}$ is the $\eta$-$\eta^{\prime}$ mixing angle. Then the
$g_{R}$ parameters for these states can be obtained by

$\displaystyle g_{\Lambda(1405)~{}\mathrm{or}~{}\Lambda(1670)}$
$\displaystyle=$
$\displaystyle\frac{R_{\Lambda(1405)}~{}\mathrm{or}~{}R_{\Lambda(1670)}}{R_{\Lambda(1405)}+R_{\Lambda(1670)}},$
(30) $\displaystyle g_{\Lambda(1520)~{}\mathrm{or}~{}\Lambda(1690)}$
$\displaystyle=$
$\displaystyle\frac{R_{\Lambda(1520)}~{}\mathrm{or}~{}R_{\Lambda(1690)}}{R_{\Lambda(1520)}+R_{\Lambda(1690)}}.$
(31)

Considering the configuration mixing effects, the wave functions of the $S$\-
and $D$-wave states $\Lambda(1405)S_{01}$, $\Lambda(1670)S_{01}$,
$\Lambda(1520)D_{03}$ and $\Lambda(1690)D_{03}$ can be generally written as

$\displaystyle|\Lambda(1405)\rangle=a_{1}|70,^{2}1\rangle_{S}+b_{1}|70,^{2}8\rangle_{S}+c_{1}|70,^{4}8\rangle_{S},$
(32)
$\displaystyle|\Lambda(1670)\rangle=a_{2}|70,^{2}1\rangle_{S}+b_{2}|70,^{2}8\rangle_{S}+c_{2}|70,^{4}8\rangle_{S},$
(33)
$\displaystyle|\Lambda(1520)\rangle=a_{3}|70,^{2}1\rangle_{D}+b_{3}|70,^{2}8\rangle_{D}+c_{3}|70,^{4}8\rangle_{D},$
(34)
$\displaystyle|\Lambda(1690)\rangle=a_{4}|70,^{2}1\rangle_{D}+b_{4}|70,^{2}8\rangle_{D}+c_{4}|70,^{4}8\rangle_{D},$
(35)

where $a_{i}$, $b_{i}$ and $c_{i}$ ($i=1,2,3,4$) have been determined in Eqs.
(22) and (23). Then we can derive the couplings of the transition amplitudes
for these mixed states, they are

$\displaystyle R^{\prime}_{\Lambda(1405)}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{3}}{108}(\sqrt{2}\sin\phi_{P}+\cos\phi_{P})(a_{1}^{2}+a_{1}b_{1})$
(36)
$\displaystyle-\frac{\sqrt{3}}{108}(\sqrt{2}\sin\phi_{P}-\cos\phi_{P})(b_{1}^{2}+a_{1}b_{1}),$
$\displaystyle R^{\prime}_{\Lambda(1670)}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{3}}{108}(\sqrt{2}\sin\phi_{P}+\cos\phi_{P})(a_{2}^{2}+a_{2}b_{2})$
(37)
$\displaystyle-\frac{\sqrt{3}}{108}(\sqrt{2}\sin\phi_{P}-\cos\phi_{P})(b_{2}^{2}+a_{2}b_{2}),$
$\displaystyle R^{\prime}_{\Lambda(1520)}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{3}}{54}(\sqrt{2}\sin\phi_{P}+\cos\phi_{P})(a_{3}^{2}+a_{3}b_{3})$
(38)
$\displaystyle-\frac{\sqrt{3}}{54}(\sqrt{2}\sin\phi_{P}-\cos\phi_{P})(b_{3}^{2}+a_{3}b_{3}),$
$\displaystyle R^{\prime}_{\Lambda(1690)}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{3}}{54}(\sqrt{2}\sin\phi_{P}+\cos\phi_{P})(a_{4}^{2}+a_{4}b_{4})$
(39)
$\displaystyle-\frac{\sqrt{3}}{54}(\sqrt{2}\sin\phi_{P}-\cos\phi_{P})(b_{4}^{2}+a_{4}b_{4}).$

Finally, we obtain the relative strength parameters $g_{R}^{\prime}$ for these
mixed states:

$\displaystyle g_{\Lambda(1405)~{}\mathrm{or}~{}\Lambda(1670)}^{\prime}$
$\displaystyle=$
$\displaystyle\frac{R_{\Lambda(1405)}^{\prime}~{}\mathrm{or}~{}R_{\Lambda(1670)}^{\prime}}{R_{\Lambda(1405)}^{\prime}+R_{\Lambda(1670)}^{\prime}},$
(40) $\displaystyle g_{\Lambda(1520)~{}\mathrm{or}~{}\Lambda(1690)}^{\prime}$
$\displaystyle=$
$\displaystyle\frac{R_{\Lambda(1520)}^{\prime}~{}\mathrm{or}~{}R_{\Lambda(1690)}^{\prime}}{R_{\Lambda(1520)}^{\prime}+R_{\Lambda(1690)}^{\prime}}.$
(41)

With these extracted $g_{R}$ and $g_{R}^{\prime}$ parameters, the $C_{R}$
parameters can be easily worked out according to Eq. (25). It is found that
$C_{R}$ is a function of the $\eta$-$\eta^{\prime}$ mixing angle $\phi_{P}$,
which might be in the range $\phi_{P}\simeq(30^{\circ},47^{\circ})$ Ke:2009mn
; DiDonato:2011kr . Considering the uncertainties of $\phi_{P}$, we plot
$C_{R}$ as a function of $\phi_{P}$ in Fig. 5. From the figure, one can find
that the $C_{R}$ parameters for $\Lambda(1670)S_{01}$ and
$\Lambda(1690)D_{03}$ are sensitive to the $\eta$-$\eta^{\prime}$ mixing angle
$\phi_{P}$. When takeing a small $\eta$-$\eta^{\prime}$ mixing angle
$\phi_{P}=35^{\circ}$, we obtain a large value $C_{\Lambda(1670)}\simeq 34$
for $\Lambda(1670)S_{01}$, which can naturally explain the large contributions
of $\Lambda(1670)S_{01}$ found in the $K^{-}P\rightarrow\Lambda\eta$ process.

Using the determined $\eta$-$\eta^{\prime}$ mixing angle
$\phi_{P}=35^{\circ}$, we also obtain a large value of
$C_{\Lambda(1690)}\simeq 39$ for $\Lambda(1690)D_{03}$. It is should be
mentioned that although the configuration mixing effects have largely enhanced
the contribution of $\Lambda(1690)$ in the $K^{-}P\rightarrow\Lambda\eta$
process (about a factor of $39$), the contribution of $\Lambda(1690)D_{03}$ in
the reaction is still negligibly small for the very weak coupling to
$\eta\Lambda$.

As a whole, the configuration mixing effects are crucial to understand the
strong decay properties of the low-lying negative $\Lambda$ resonances. These
resonances are most likely mixed states between different configurations,
which is consistent with the predictions in large $N_{c}$ QCD Schat:2001xr .
Considering configuration mixing effects, we can reasonably explain the weak
coupling of $\Lambda(1670)S_{01}$ to $\bar{K}N$ and strong coupling of
$\Lambda(1670)S_{01}$ to $\eta\Lambda$, and the large strength parameter
$C_{\Lambda(1670)}\simeq 34$. The contribution of $\Lambda(1690)D_{03}$ to the
$K^{-}P\rightarrow\Lambda\eta$ process is too small to give a bowl structure
in the DCS, even we consider the configuration mixing effects in these
$D$-wave states.

Figure 5: The coupling strength parameter, $C_{R}$, as a function of the
$\eta$-$\eta^{\prime}$ mixing angle $\phi_{P}$.

##  IV Summary

In this work, we have studied the low energy reaction
$K^{-}p\rightarrow\Lambda\eta$ with a chiral quark model approach. A
reasonable description of the measurements has been achieved. It is found that
$\Lambda(1670)S_{01}$ dominates the reaction around at the low energy regions,
and the $t$\- and $u$-channel backgrounds also play crucial roles. Slight
contributions of $\Lambda(1405)S_{01}$ are found, however,
$\Lambda(1405)S_{01}$ does not obviously affect the shapes of the differential
cross sections. No obvious roles of the $D$-wave states $\Lambda(1520)D_{03}$
and $\Lambda(1690)D_{03}$ are found in the reaction.

Furthermore, by the study of the $K^{-}p\rightarrow\Lambda\eta$ process, we
have extracted the strong interaction properties of $\Lambda(1670)S_{01}$. We
find that a much large amplitude of $\Lambda(1670)S_{01}$ in the reaction is
needed, which is about 34 times (i.e., $C_{S_{01}(1670)}\simeq 34$) larger
than that derived from the symmetry quark model. Combing this with our
previous study in Zhong:2013oqa , we conclude that $\Lambda(1670)S_{01}$
should have a much weaker coupling to $\bar{K}N$, while a much stronger
coupling to $\eta\Lambda$ than that predicted in the symmetry quark model.

To understand these strong interaction properties of $\Lambda(1670)S_{01}$, we
further study the strong decay properties of the low-lying negative parity
$\Lambda$-resonances. It is found that the configuration mixing effects are
crucial to understand the strong decay properties of the low-lying negative
$\Lambda$ resonances. These resonances are most likely mixed states between
different configurations. Considering configuration mixing effects, we can
reasonably explain the strong interaction properties of $\Lambda(1670)S_{01}$
extracted from the $K^{-}P\rightarrow\Lambda\eta$.

The data of the $K^{-}p\rightarrow\Lambda\eta$ process show that there seems
to exist a bowl structure in the DCS in a narrow energy region near the
$\eta\Lambda$ threshold, which indicates a strong $D$-wave contribution there.
However, the contribution of $\Lambda(1690)D_{03}$ to
$K^{-}P\rightarrow\Lambda\eta$ process too small to give a bowl structure in
the DCS. Although with the configuration mixing effects in these $D$-wave
states, the amplitude of $\Lambda(1690)D_{03}$ in the reaction could be
enhanced a factor of $\sim 38$, the contribution of $\Lambda(1690)D_{03}$ is
still tiny for the very weak coupling of $\Lambda(1690)D_{03}$ to
$\eta\Lambda$. Based on the bowl structures in the DCS, Liu and Xie believed
there might exist a exotic $D$-wave state $\Lambda(1669)D_{03}$ with a very
narrow width of $\Gamma=1.5$ MeV. To clarify whether there are contributions
of a narrow $D$-wave state or not, more accurate measurements are needed.

As a byproduct, we also have predicted the strong decay properties of the
unestablished $D$-wave state $|\Lambda\frac{3}{2}^{-}\rangle_{3}$. This
resonance mainly decay into $\Sigma(1385)\pi$ and $\Sigma\pi$ channels. We
hope the experimentalists can search this missing $D$-wave state in the
$\Sigma(1385)\pi$ and $\Sigma\pi$ channels.

##  Acknowledgements

This work is supported, in part, by the National Natural Science Foundation of
China (Grants No. 11075051 and No. 11375061), Program for Changjiang Scholars
and Innovative Research Team in University (PCSIRT, Grant No. IRT0964), the
Program Excellent Talent Hunan Normal University, the Hunan Provincial Natural
Science Foundation (Grants No. 11JJ7001 and No. 13JJ1018), and the Hunan
Provincial Innovation Foundation For Postgraduate.

## References

  * (1) J. Beringer et al. [Particle Data Group], Phys. Rev. D 86, 010001 (2012).
  * (2) E. Klempt and J. -M. Richard, Rev. Mod. Phys. 82, 1095 (2010).
  * (3) N. Isgur and G. Karl, Phys. Rev. D 18, 4187 (1978).
  * (4) N. Isgur and G. Karl, Phys. Lett. B72, 109 (1977); Phys. Rev. D 19, 2653 (1979) [Erratum-ibid. 23, 817 (1981)]; 20, 1191 (1979).
  * (5) S. Capstick and N. Isgur, Phys. Rev. D 34, 2809 (1986).
  * (6) Y. Chen and B. -Q. Ma, Nucl. Phys. A 831, 1 (2009).
  * (7) T. Melde, W. Plessas and B. Sengl, Phys. Rev. D 77, 114002 (2008).
  * (8) R. Bijker, F. Iachello and A. Leviatan, Annals Phys. 284, 89 (2000).
  * (9) L. Y. Glozman, W. Plessas, K. Varga and R. F. Wagenbrunn, Phys. Rev. D 58, 094030 (1998).
  * (10) U. Loring, B. C. Metsch and H. R. Petry, Eur. Phys. J. A 10, 447 (2001).
  * (11) C. L. Schat, J. L. Goity and N. N. Scoccola, Phys. Rev. Lett. 88, 102002 (2002) [hep-ph/0111082].
  * (12) J. L. Goity, C. L. Schat and N. N. Scoccola, Phys. Rev. D 66, 114014 (2002) [hep-ph/0209174].
  * (13) B. J. Menadue, W. Kamleh, D. B. Leinweber and M. S. Mahbub, Phys. Rev. Lett. 108, 112001 (2012) [arXiv:1109.6716 [hep-lat]].
  * (14) G. P. Engel, C. B. Lang and A. Schafer, Phys. Rev. D 87, no. 3, 034502 (2013) [arXiv:1212.2032 [hep-lat]].
  * (15) R. Koniuk and N. Isgur, Phys. Rev. D 21, 1868 (1980) [Erratum-ibid. D 23, 818 (1981)].
  * (16) T. Melde, W. Plessas and B. Sengl, Phys. Rev. C 76, 025204 (2007) [nucl-th/0612053].
  * (17) T. Melde and W. Plessas, Eur. Phys. J. A 35, 329 (2008).
  * (18) C. S. An, B. Saghai, S. G. Yuan and J. He, Phys. Rev. C 81, 045203 (2010) [arXiv:1002.4085 [nucl-th]].
  * (19) T. Hyodo and D. Jido, Prog. Part. Nucl. Phys. 67, 55 (2012).
  * (20) J. A. Oller, J. Prades and M. Verbeni, Phys. Rev. Lett. 95, 172502 (2005).
  * (21) L. Roca, S. Sarkar, V. K. Magas and E. Oset, Phys. Rev. C 73, 045208 (2006).
  * (22) B. Borasoy, R. Nissler and W. Weise, Eur. Phys. J. A 25, 79 (2005).
  * (23) T. Hyodo, S. i. Nam, D. Jido and A. Hosaka, Prog. Theor. Phys. 112, 73 (2004).
  * (24) D. Jido, J. A. Oller, E. Oset, A. Ramos and U. G. Meissner, Nucl. Phys. A 725, 181 (2003).
  * (25) E. Oset, A. Ramos and C. Bennhold, Phys. Lett. B 527, 99 (2002) [Erratum-ibid. B 530, 260 (2002)].
  * (26) C. Garcia-Recio, J. Nieves, E. R. Arriola and M. J. Vicente Vacas, Phys. Rev. D 67, 076009 (2003).
  * (27) J. A. Oller and U. G. Meissner, Phys. Lett. B 500, 263 (2001).
  * (28) E. Oset and A. Ramos, Nucl. Phys. A 635, 99 (1998).
  * (29) J. A. Oller, Eur. Phys. J. A 28, 63 (2006).
  * (30) J. A. Oller, J. Prades and M. Verbeni, Eur. Phys. J. A 31, 527 (2007).
  * (31) B. Borasoy, U. G. Meissner and R. Nissler, Phys. Rev. C 74, 055201 (2006).
  * (32) L. Roca, T. Hyodo and D. Jido, Nucl. Phys. A 809, 65 (2008).
  * (33) Antonio O. Bouzasa, Eur. Phys. J. A 37, 201 (2008).
  * (34) A. D. Martin, N. M. Queen and G. Violini, Nucl. Phys. B 10, 481 (1969).
  * (35) D. M. Manley et al., Phys. Rev. Lett. 88, 012002 (2002).
  * (36) M. F. M. Lutz and E. E. Kolomeitsev, Nucl. Phys. A 700, 193 (2002).
  * (37) R. Buttgen, K. Holinde and J. Speth, Phys. Lett. B 163, 305 (1985).
  * (38) A. Mueller- Groeling, K. Holinde and J. Speth, Nucl. Phys. A 513, 557 (1990).
  * (39) T. Hamaie, M. Arima and K. Masutani, Nucl. Phys. A 591, 675 (1995).
  * (40) X. H. Zhong, Q. Zhao, Phys. Rev. C 79, 045202 (2009).
  * (41) A. D. Martin, Nucl. Phys. B 179, 33 (1981).
  * (42) P. M. Gensini, R. Hurtado and G. Violini, PiN Newslett. 13, 291 (1997).
  * (43) Z. -H. Guo and J. A. Oller, Phys. Rev. C 87, 035202 (2013) [arXiv:1210.3485 [hep-ph]].
  * (44) H. Zhang, J. Tulpan, M. Shrestha and D. M. Manley, arXiv:1305.3598 [hep-ph].
  * (45) H. Zhang, J. Tulpan, M. Shrestha and D. M. Manley, arXiv:1305.4575 [hep-ph].
  * (46) L. S. Geng and E. Oset, Eur. Phys. J. A 34, 405 (2007) [arXiv:0707.3343 [hep-ph]].
  * (47) L. S. Geng, E. Oset and M. Doring, Eur. Phys. J. A 32, 201 (2007) [hep-ph/0702093].
  * (48) L. S. Geng, E. Oset and B. S. Zou, Eur. Phys. J. A 38, 239 (2008) [arXiv:0807.4798 [hep-ph]].
  * (49) B. -C. Liu and J. -J. Xie, Phys. Rev. C 86, 055202 (2012) [arXiv:1211.4375 [hep-ph]].
  * (50) B. -C. Liu and J. -J. Xie, Phys. Rev. C 85, 038201 (2012).
  * (51) J. -J. Xie, B. -C. Liu and C. -S. An, Phys. Rev. C 88, 015203 (2013).
  * (52) C. Garcia-Recio, M. F. M. Lutz and J. Nieves, Phys. Lett. B 582, 49 (2004).
  * (53) B. S. Zou, eConf C 070910, 112 (2007) [arXiv:0802.0087 [nucl-ex]]; Chin. Phys. C 33, 1113 (2009).
  * (54) X. -H. Zhong and Q. Zhao, Phys. Rev. C 88, 015208 (2013).
  * (55) A. Starostin et al. [Crystal Ball Collaboration], Phys. Rev. C 64, 055205 (2001).
  * (56) Z. P. Li, Phys. Rev. D 48, 3070 (1993); 50, 5639 (1994); 52, 4961 (1995); 52, 1648 (1995).
  * (57) Z. P. Li, H. X. Ye and M. H. Lu, Phys. Rev. C 56, 1099 (1997).
  * (58) Q. Zhao, J. S. Al-Khalili and C. Bennhold, Phys. Rev. C 64, 052201(R) (2001).
  * (59) Q. Zhao, B. Saghai and Z. P. Li, J. Phys. G 28, 1293 (2002).
  * (60) Q. Zhao, Z. P. Li and C. Bennhold, Phys. Rev. C 58, 2393 (1998); Phys. Lett. B 436, 42 (1998).
  * (61) J. He, B. Saghai and Z. Li, Phys. Rev. C 78, 035204 (2008).
  * (62) B. Saghai and Z. P. Li, Eur. Phys. J. A 11, 217 (2001) [nucl-th/0104084].
  * (63) Q. Zhao, J. S. Al-Khalili, Z. P. Li and R. L. Workman, Phys. Rev. C 65, 065204 (2002).
  * (64) X. -H. Zhong and Q. Zhao, Phys. Rev. C 84, 065204 (2011);Phys. Rev. C 84, 045207 (2011).
  * (65) X. H. Zhong, Q. Zhao, J. He and B. Saghai, Phys. Rev. C 76, 065205 (2007).
  * (66) C. Di Donato, G. Ricciardi and I. Bigi, Phys. Rev. D 85, 013016 (2012) [arXiv:1105.3557 [hep-ph]].
  * (67) H. -W. Ke, X. -Q. Li and Z. -T. Wei, Eur. Phys. J. C 69, 133 (2010) [arXiv:0912.4094 [hep-ph]].
  * (68) Q. Zhao and F. E. Close, Phys. Rev. D 74, 094014 (2006) [hep-ph/0605336].
  * (69) X. H. Zhong and Q. Zhao, Phys. Rev. D 77, 074008 (2008); Phys. Rev. D 78, 014029 (2008); Phys. Rev. D 81, 014031 (2010).
  * (70) X. -H. Zhong, Phys. Rev. D 82, 114014 (2010).
  * (71) L. -H. Liu, L. -Y. Xiao and X. -H. Zhong, Phys. Rev. D 86, 034024 (2012).
  * (72) L. -Y. Xiao and X. -H. Zhong, Phys. Rev. D 87, 094002 (2013) [arXiv:1302.0079 [hep-ph]].
  * (73) M. Alston-Garnjost, R. W. Kenney, D. L. Pollard, R. R. Ross, R. D. Tripp, H. Nicholson and M. Ferro-Luzzi, Phys. Rev. D 18, 182 (1978).


Steven Weinberg was already my hero when I was an undergraduate student in my
home country Guatemala where I was fascinated by his First Three Minutes book
[1]. It was hard for me to believe that physics was so powerful as to be able
to describe, with some certainty, what happens in the very early universe.
Reading that this subject was treated by such a famous physicist (he had
written the classic textbook on gravitation [2] in the early 1970’s and
received the Nobel prize the same year I finished my undergraduate) made me
more confident that the content of the book was based on serious physics
rather than speculations.

I was extremely lucky to have started my PhD in 1981, just before Steve joined
the University of Texas at Austin (UT). Coming from Guatemala, UT was the only
place I could have hoped to be accepted in the US for my PhD, thanks to an
initiative started by UT Professor Robert Little to support Central American
physicists. I knew the high energy and relativity groups were already very
strong at UT, and I was very pleased to have been accepted with the help of
Robert Little and also Professor Mel Oakes whom I have met in a school in
Guatemala a few years before.

The first two years we have to take basic courses, called ‘the big seven’ but
we could sit in more advanced courses and Steve’s Quantum Field Theory course
was clearly the most popular. Attended by graduate students of all levels,
postdocs and some faculty. It was a pleasure to see Bryce DeWitt sitting
always on the first row and asking many difficult questions. I confess that
most of the times I could not understand the details of their discussions. One
instance that I remember vividly is when Steve was introducing the Fadeev-
Popov formalism and Bryce did not seem to follow the argument. At some point
Steve stopped and said: ‘but Bryce you invented this!’. The lectures were
outstanding and Steve took them very seriously. Since this was before his
field theory books were written [3, 4, 5], he distributed handwritten notes at
the beginning of each lecture.

One lecture clearly stood out. In early 1983 he came to the lecture room and
before starting he smiled and said something like ‘I have just been informed
from my contacts at CERN that the W-particle has been discovered and that
there are hints also for the Z-particle’. With my fellow students we remember
this moment as the highlight of our time in UT. We do not agree on his exact
words nor our immediate reaction, but we all agree the generalised feeling of
excitement after his words. It is hard to describe how we all felt and we
could only imagine how Steve was feeling that day. We were living science
history in the making. For me, the only moment that matched this was to be
present at the CERN auditorium when the Higgs particle was announced, almost
30 years later, on the 4th of July 2012. These moments are rare and touch
deeply all of us that have dedicated our career to high energy physics.

After finishing the ‘big seven’ courses I managed to get the courage to ask
Steve to be my PhD supervisor, he asked me if I knew Feynman diagrams and
after my positive answer he asked me to read one of his more recent papers [6]
and discuss it with him after a week. An impossible task since I did not know
supersymmetry at that time but I did my best to get an idea about that paper.
Luckily for me, the week after Steve was very busy and suggested me to talk
with his postdoc, Joe Lykken (former deputy director of Fermilab) who was
already working with him [7]. Joe assigned me a project which was an extension
of his work with Steve and he helped me with the calculations, so I managed to
pass my qualifying exam even though I was extremely nervous since the previous
student of Steve had failed and left the field. We published our results [8]
but before sending the preprint we needed Steve’s approval. Since he was
traveling we sent him a copy and one day Steve’s assistant (the legendary
Adele Traverso) came to my office telling me ‘Professor Weinberg said that you
messed it up and he wants to talk to you’. I checked the meaning of ‘messed-
up’ in the dictionary and then went to his office, extremely nervous, to
receive the phone call. He had found a typo in the paper (a low case $f$
instead of $F$ in one of the equations) and did not understand what this
variable was. I explained it was only a typo and he was happy. I was born
again.

After the qualifying exam I became a member of ‘Weinberg’s group’ and we met
every week for the legendary ‘brown bag lunches’ at his office. Each time one
or two members of the group reported on some original work in progress or some
review of recent papers. The atmosphere was tense for a PhD student but we
learned a lot from Steve’s continuous interruptions to get to understand each
subject in his own way and giving not only hard-time but also suggestions to
the speaker for further research. This experience together with the
interactions with all of Steve’s visitors and the new hires such as Willy
Fischler and Joe Polchinski (and the other theorists who were already at UT,
such as Cecil and Bryce DeWitt, Philip Candelas, Claudio Teitelboim (now
Bunster), George Sudarshan, Duane Dicus, John Wheeler and the great group of
postdocs and fellow students) made the PhD experience exceptionally rich.

During those years String Theory came to prominence and Steve and essentially
the whole group moved to work on this direction. They were very exciting times
and we felt UT was one of the top places in the world for the subject. This
allowed me to finish a good PhD thesis under Steve’s supervision although much
of my work I did with visitors like Paul Townsend and fellow students such as
Cliff Burgess and Anamarıa Font [9]. Joe Polchinski was one of my examiners
and, being the first time he had to examine a thesis, he came with many
questions to ask me after my standard presentation. Luckily for me Steve
stopped Joe and told him that the thesis was good and we should finish fast
the procedure. He could reserve his questions and ask them later in private. I
was very much relieved.

Steve helped me then to move on my career. I am sure it was because of his
special recommendation that I was given a nice postdoc position at CERN after
UT and for the further jobs I got after, the fact that I was a student of
Steve played a crucial role, until finishing with my professorship at
Cambridge and even my ten years as director of ICTP. For this I will always be
in debt to him.

During all the years since I left UT I was in touch with him and we met in
several occasions. Just to mention a few anecdotes:

  * •

In 1986 after finishing my PhD, I asked Steve if he would be willing to donate
his collection of scientific journals to the Guatemalan national university
(Universidad de San Carlos de Guatemala) which he kindly agreed. We needed
only to get someone to pay for the transportation and Abdus Salam immediately
agreed and arranged ICTP to cover the expenses. I used to call this as the
last Weinberg-Salam collaboration.

  * •

For my 60th birthday conference Steve recorded a 5 minutes video to talk about
me and to wish me a happy birthday. This video [10] is now one of my greatest
treasures.

  * •

On the exact day of the 50th anniversary of his Nobel prize paper (17 October
2017) I organised a special colloquium at ICTP that Steve kindly accepted to
give with the title ‘Reminiscences of the Standard Model’ [11]

Since Steve had constantly rejected all attempts to have a conference in his
honour111Whenever any of his former students or postdocs or the faculty
members of his group proposed it, his reply was that it was too early to
celebrate his achievements, giving the impression that the best was yet to
come., I took this opportunity to give a more than 15 minutes introduction to
the main scientific contributions of Steve. This of course does not make him
justice but at least illustrates that what for normal top scientists an
introduction of 1-5 minutes is enough to summarise their achievements, for
Steve this was impossible and I needed more than 15 minutes for a ‘short’
summary of his achievements. He was clearly the greatest physicist alive and
took my introduction with good humour.

  * •

A few years ago, after some email exchanges he told me he was talking with a
used-car dealer and realised that his last name was Quevedo. Steve was curious
if this last name was common in Spanish speaking countries and asked me about
it. I don’t remember how I answer but always wondered what was Steve doing
talking with a used-car dealer!

  * •

Even though I was Steve’s student we did not collaborate on scientific papers.
He usually preferred to work by himself. However due to the sad departure of
Joe Polchinski, we joined Raphael Bousso to write a scientific biography of
Joe for the National Academy. I appreciated a very human aspect of Steve while
writing this article [12].

  * •

The last time I met Steve in person was in February 2018 when I went to UT and
gave a colloquium. He introduced me very kindly (he needed only 1-2 minutes of
course). Here is a photograph that Professor Harry Swinney kindly took on that
special occasion:

Figure 1: Austin 2018.

His many contributions to science, including his outstanding textbooks and
popular science books, are well documented as well as his famous quotes on
many aspects of science and related subjects. Here I would like to share a
couple of Steve’s quotes that may be less known:

During the question period after his colloquium at ICTP someone asked him how
has he managed to get so much done in his career to what he replied: ‘very
simple, I don’t go to church and I don’t ski so I have plenty of free time’.
Finally, we were once having a conversation on religion with Don Page and he
emphasised that you do not need to be religious to have high moral standards
and said: ‘I know atheists that have the highest moral standards, for instance
myself!’.

After Steve passed away I was invited to give talks in different places
summarising Steve’s contributions to physics. I prepared a one hour talk
concentrating essentially on a small subclass of his renown papers (his almost
50 papers with more than 500 citations). I organised my presentation by 8
decades, since late 1950’s until he passed away (I received the copy of his
last book exactly the day he died). I did my best but it is actually
impossible to summarise in a talk all his contributions, especially if his 8
textbooks and also 8 popular science books are included (see for instance
[13]) 222Mike Duff did an excellent job collecting the selected papers of
Steve in [14]..

Steve keeps influencing me as well as to most of the high energy physics
community. As an example, during the past 4 years I lectured ‘The Standard
Model’ course at the Part III of the Cambridge Mathematical tripos[15]. After
much thinking and preparation trying to give an original presentation to the
subject I realised that at the end I was essentially following Steve’s logical
steps in his books (but not his notation). Starting with a chapter on the
history of the Standard Model, something that he always insisted in his
lectures. Followed by introducing particles as unitary representations of the
Poincaré group and only after that introduce fields in order to describe local
interactions among many particles that require assuming cluster decomposition
on top of special relativity and quantum mechanics. Quoting him[3] ‘If it
turned out that some physical system could not be described by a quantum field
theory, it would be a sensation; if it turned out that the system did not obey
the rules of quantum mechanics and relativity, it would be a cataclysm’.

In my lectures I also use his soft theorems [16] to restrict to massless
particles of helicity less than $2$ (and prove charge conservation and the
equivalence principle using his theorems for massless particles of helicity
$1$ and $2$ respectively). I describe the power of effective field theories
that he introduced over the years allowing us to consider non-renormalisable
field theories, as gravity, in the same class as renormalisable theories
except for their ultra violet completion. Using both the Fermi theory of weak
interactions and chiral perturbation theory as examples. All this on top of
introducing weak and strong interactions as gauge theories and gauge
symmetries as redundancies in field theories to describe helicity $1$
particles consistent with Lorentz invariance and of course the Glashow-
Weinberg-Salam model on which his less than 3 pages paper [17] may compete for
the title of the most confirmed predictive information per written word.
Predicting neutral currents, the $W^{\pm}$ bosons, the $Z$ boson, the Higgs
particle.

For addressing ideas beyond the Standard Model I mention his gauge coupling
unification work with Georgi and Quinn [18], his introduction of the axion
field [19], the famous Weinberg operator $H^{2}L^{2}/M$ to get neutrino masses
and the whole SMEFT that relies on his EFT formalism, etc.333For recent
discussions on his contributions to EFTs in his own words see [21, 20].
Essentially the whole course has Steve’s legacy written on it.

It has been my greatest honour to have somehow my career and my life
associated to such a giant scientific figure as Steve. It is impossible for me
to exaggerate how much admiration I have for him.

## References

  * [1] S. Weinberg, “The First Three Minutes. A Modern View of the Origin of the Universe,”
Basic Books (1977). Second edition (1993).

  * [2] S. Weinberg, “Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity,” John Wiley and Sons, 1972, ISBN 978-0-471-92567-5, 978-0-471-92567-5
  * [3] S. Weinberg, “The Quantum theory of fields. Vol. 1: Foundations,” Cambridge University Press, 2005, ISBN 978-0-521-67053-1, 978-0-511-25204-4 doi:10.1017/CBO9781139644167
  * [4] S. Weinberg, “The quantum theory of fields. Vol. 2: Modern applications,” Cambridge University Press, 2013, ISBN 978-1-139-63247-8, 978-0-521-67054-8, 978-0-521-55002-4 doi:10.1017/CBO9781139644174
  * [5] S. Weinberg, “The quantum theory of fields. Vol. 3: Supersymmetry,” Cambridge University Press, 2013, ISBN 978-0-521-67055-5, 978-1-139-63263-8, 978-0-521-67055-5
  * [6] S. Weinberg, “Does Gravitation Resolve the Ambiguity Among Supersymmetry Vacua?,” Phys. Rev. Lett. 48 (1982), 1776-1779 doi:10.1103/PhysRevLett.48.1776
  * [7] L. J. Hall, J. D. Lykken and S. Weinberg, “Supergravity as the Messenger of Supersymmetry Breaking,” Phys. Rev. D 27 (1983), 2359-2378 doi:10.1103/PhysRevD.27.2359
  * [8] J. D. Lykken and F. Quevedo, “Stable Hierarchies in O’Raifertaigh Models Coupled to $N=1$ Supergravity,” Phys. Rev. D 29 (1984), 293 doi:10.1103/PhysRevD.29.293
  * [9] C. P. Burgess, A. Font and F. Quevedo, “Low-Energy Effective Action for the Superstring,” Nucl. Phys. B 272 (1986), 661-676 doi:10.1016/0550-3213(86)90239-7
  * [10] https://mediacore.ictp.it/categories/00d285011e2b4ab25dc5f63a81ad9d9ec4a0291e-146303280/latest
  * [11] https://indico.ictp.it/event/8413/
  * [12] R. Bousso, F. Quevedo and S. Weinberg, “Joseph Polchinski: A Biographical Memoir,” [arXiv:2002.02371 [physics.hist-ph]]. https://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/polchinski-josephb.pdf
  * [13] https://meetings.aps.org/Meeting/TSF21/Session/L01.1
  * [14] M.J. Duff, “Steven Weinberg: Collected Papers” World Scientific (2024).
  * [15] F. Quevedo, A. Scahchner “Cambridge Lectures on The Standard Model” (to appear).
  * [16] S. Weinberg, “Infrared photons and gravitons,” Phys. Rev. 140 (1965), B516-B524 doi:10.1103/PhysRev.140.B516
  * [17] S. Weinberg, “A Model of Leptons,” Phys. Rev. Lett. 19 (1967), 1264-1266 doi:10.1103/PhysRevLett.19.1264
  * [18] H. Georgi, H. R. Quinn and S. Weinberg, “Hierarchy of Interactions in Unified Gauge Theories,” Phys. Rev. Lett. 33 (1974), 451-454 doi:10.1103/PhysRevLett.33.451
  * [19] S. Weinberg, “A New Light Boson?,” Phys. Rev. Lett. 40 (1978), 223-226 doi:10.1103/PhysRevLett.40.223
  * [20] S. Weinberg, “On the Development of Effective Field Theory,” Eur. Phys. J. H 46 (2021) no.1, 6 doi:10.1140/epjh/s13129-021-00004-x [arXiv:2101.04241 [hep-th]].
  * [21] S. Weinberg, “Effective field theory, past and future,” Int. J. Mod. Phys. A 31 (2016) no.06, 1630007 doi:10.1142/S0217751X16300076


# On-Demand Earth System Data Cubes

David Montero1,2,* César Aybar3 Chaonan Ji1 Guido Kraemer1
Maximilian Söchting1 Khalil Teber1 Miguel D. Mahecha1,2,4
1RSC4Earth, IEF, Leipzig University 2iDiv 3IPL, Universitat de València 4UFZ
*Corresponding author<EMAIL_ADDRESS>
###### Abstract

Advancements in Earth system science have seen a surge in diverse datasets.
Earth System Data Cubes (ESDCs) have been introduced to efficiently handle
this influx of high-dimensional data. ESDCs offer a structured, intuitive
framework for data analysis, organising information within spatio-temporal
grids. The structured nature of ESDCs unlocks significant opportunities for
Artificial Intelligence (AI) applications. By providing well-organised data,
ESDCs are ideally suited for a wide range of sophisticated AI-driven tasks. An
automated framework for creating AI-focused ESDCs with minimal user input
could significantly accelerate the generation of task-specific training data.
Here we introduce cubo, an open-source Python tool designed for easy
generation of AI-focused ESDCs. Utilising collections in SpatioTemporal Asset
Catalogs (STAC) that are stored as Cloud Optimised GeoTIFFs (COGs), cubo
efficiently creates ESDCs, requiring only central coordinates, spatial
resolution, edge size, and time range.

##  1 Introduction

Earth System Data Cubes (ESDCs) are multidimensional arrays encapsulating
analysis-ready Earth system data, defined by their dimensions, grids, data,
and attributes (Mahecha et al., 2020). Recent advances in cloud technologies,
such as the SpatioTemporal Asset Catalogs (STAC) specification, which
simplifies geospatial data description and indexing; and Cloud Optimised
GeoTIFF (COG), which allows for HTTP range requests; have enabled efficient
generation of ESDCs from cloud-stored data (Montero et al., 2023). Generated
ESDCs typically feature two spatial dimensions (such as $x$ and $y$), one
temporal dimension, and the variable dimension. As ESDCs are usually cuboids,
the length of the spatial grids can vary (e.g. global ESDCs have shorter
latitude grids than longitude grids). In the case of Artificial Intelligence
(AI) for local-scale applications, spatial grids of equal length are preferred
for vision AI tasks. Examples include BigEarthNet’s $120\times 120$, $60\times
60$, and $20\times 20$ image patches (Sumbul et al., 2019), and CloudSEN12’s
$509\times 509$ image patches (Aybar et al., 2022). We refer to ESDCs with
spatial grids of equal length as “AI-focused ESDCs”. Despite the availability
of tools leveraging cloud technologies for ESDC creation, a systematic
approach for producing AI-focused ESDCs on demand is lacking.

This paper introduces cubo111https://github.com/ESDS-Leipzig/cubo, an open-
source Python-based tool streamlined for creating AI-focused ESDCs. cubo
enables automatic and minimal-input ESDC generation from cloud-stored data,
greatly expanding the potential for generating comprehensive Earth system
datasets. The paper is structured as follows: Sec. 2 details the cubo
framework and its simplification of AI-focused ESDC generation; Sec. 3
presents examples of AI-focused ESDCs generated using cubo; and Sec. 4
provides our conclusions.

##  2 Framework

To streamline cubo’s functionality, we introduced a new ESDC characterisation
(Sec. 2.1). This significantly simplifies the input process, requiring only a
few input parameters from the user. Subsequently, cubo utilises these user-
defined parameters to construct an ESDC in a systematic manner (Sec. 2.2).

###  2.1 ESDC characterisation

cubo characterises AI-focused ESDCs using the parameters described in Box 1
and illustrated in Fig. 1.

Figure 1: cubo’s ESDC characterisation. Note that in this representation,
$t_{s}$ and $t_{e}$ represent values from the temporal dimension, while $\phi$
and $\lambda$ represent coordinates associated with the spatial centre of the
cube.

Box 1: Parameters characterising ESDCs • Central coordinates of the cube,
defined by latitude ($\phi$) and longitude ($\lambda$). • Edge size of the
cube in pixels ($s$). • Spatial resolution in meters ($r$). • Time range of
the cube, defined by start ($t_{s}$) and end ($t_{e}$) timestamps.

###  2.2 ESDC construction

Figure 2: Overview of the workflow for building an ESDC inside cubo. This
diagram presents a high-level summary of the ESDC construction process.

cubo builds ESDCs through a series of steps using the above-mentioned
parameters as user inputs (Fig. 2):

####  2.2.1 Bounding box calculation

cubo reprojects $\phi$ and $\lambda$ into their respective Universal
Transverse Mercator (UTM) zone coordinates, denoted as $y$ and $x$.
Additionally, cubo saves the Coordinate Reference System (CRS) as an
attribute. These coordinates are then aligned to the nearest pair divisible by
the spatial resolution $r$, calculated as:

$i_{r}=r\left\lfloor\frac{i}{r}+0.5\right\rfloor$ (1)

Here, $i$ represents either $x$ or $y$, and $i_{r}$ is the adjusted
coordinate. Next, the half-edge size in meters ($s_{h}$) is calculated as:

$s_{h}=\left\lfloor\frac{sr}{2}+0.5\right\rfloor$ (2)

This rounding process ensures that the edge size is an even number. Finally,
the bounding box coordinates are determined using:

$\begin{bmatrix}x_{\textrm{min}}&x_{\textrm{max}}\\\
y_{\textrm{min}}&y_{\textrm{max}}\end{bmatrix}=\begin{bmatrix}x_{r}&1\\\
y_{r}&1\end{bmatrix}\begin{bmatrix}1&1\\\ -s_{h}&s_{h}\end{bmatrix}$ (3)

Here, the tuples $(x_{\textrm{min}},y_{\textrm{max}})$ and
$(x_{\textrm{max}},y_{\textrm{min}})$ denote the upper left and lower right
coordinates of the bounding box, respectively.

####  2.2.2 ESDC creation

In its initial step, cubo accesses an STAC catalogue through an endpoint
specified by the user, defaulting to the Planetary Computer STAC catalogue’s
endpoint. Next, cubo feeds the bounding box parameters alongside $t_{s}$ and
$t_{e}$ to pystac-client to retrieve all STAC items from a user-defined
collection that intersect with these spatio-temporal constraints via a
“search” operation. “kwargs” arguments are an option, enabling users to
specify additional query parameters such as cloud cover percentage.
Subsequently, these items are transferred to stackstac, combined with $r$, the
bounding box values, and the CRS. Additionally, the user can define the bands
to retrieve. This process culminates in the generation of the ESDC as a “lazy”
xarray object (Hoyer et al., 2017), chunked via dask (Rocklin, 2015), with the
CRS specifically tailored to match the UTM zone associated with $\phi$ and
$\lambda$.

Table 1: Global attributes in the ESDC generated by cubo. Attribute |  Description
---|---
collection |  Identifier of the collection within the STAC catalogue.
stac |  Endpoint of the STAC catalogue used.
epsg |  EPSG code of the ESDC’s CRS, corresponding to a specific UTM zone.
resolution |  Spatial resolution, denoted by the value of $r$.
edge_size |  Edge size of the cube, given by the value of $s$.
central_lat |  Central latitude, indicated by $\phi$.
central_lon |  Central longitude, represented by $\lambda$.
central_y |  UTM coordinate y, corresponding to $y$.
central_x |  UTM coordinate x, corresponding to $x$.
time_coverage_start |  Start timestamp of the ESDC, indicated by $t_{s}$.
time_coverage_end |  End timestamp of the ESDC, given by $t_{e}$.

####  2.2.3 Attributes writing

After the creation of the ESDC, cubo inscribes a set of global attributes on
it, as outlined in Table 1. Additionally, cubo calculates the Euclidean
distance between the coordinates of each pixel in the ESDC and the projected
coordinate pair $(y,x)$. This distance array is stored within the ESDC,
adhering to the same spatial grids and dimensions, and can be accessed under
the coordinate cubo:distance_from_center.

##  3 Showcase

We illustrate cubo’s efficacy through two distinct examples: 1) creating
varied ESDCs with different parameters across multiple global locations, and
2) generating a standardised ESDC using different collections with identical
parameters in the same location, all sourced from the Planetary Computer STAC
catalogue.

Figure 3: Examples of ESDCs generated using cubo. Each row in the figure
depicts ESDCs’ example timesteps. The first column shows the parameters used
and the location of the central coordinates of the ESDC. The first row
displays an ESDC generated from Landsat-8 Collection 2 Level 2 data, capturing
a fire event in Indonesia. The second row shows an ESDC constructed from
Sentinel-1 Radiometrically Terrain Corrected (RTC) data, highlighting a flood
event in Pakistan. The third row presents an ESDC based on MODIS 17A2HGF gap-
filled 8-day GPP data, depicting a forest area in Germany. The final row
features an ESDC created using data from the ESA CCI LC product, focused on
Brasil.

In the first scenario, we harnessed various collections specialised for Earth
system research (Fig. 3). The examples were generated across various locations
globally, showcasing cubo’s versatility in handling data from any region. They
also emphasise the importance of spatio-temporal context in a range of
applications. For instance, the first three rows in Fig. 3 highlight potential
uses in studying climate extremes and their effects on both the natural
environment and human society. The first row features a Landsat-8 ESDC with a
30 m resolution, useful for detecting active fires and estimating burned
areas. The second row shows a Sentinel-1 ESDC at 10 m resolution, ideal for
flood detection and damage assessment. The third row introduces a MODIS-
derived Gross Primary Production (GPP) dataset (17A2HGF), with a 500 m
resolution, for analysing climate impacts on forest carbon sequestration.
Lastly, the fourth row illustrates an annual ESDC from the ESA Climate Change
Initiative (CCI) Land Cover (LC) product at 300 m resolution, beneficial as an
additional input in various Earth system projects.

Figure 4: Example of ESDCs aligned with a common spatio-temporal grid. The
displayed MODIS datasets are a) FPAR (15A2H), b) Red SR (09A1), c) Enhanced
Vegetation Index (EVI, 13Q1), d) NIR SR (09A1), e) Normalised Difference
Vegetation Index (NDVI, 13Q1), f) Nighttime LST (11A2), g) Daytime LST (11A2),
h) LAI (15A2H), and i) GPP (17A2HGF). The ESDCs were rendered using lexcube
(Söchting et al., 2023).

The ultimate goal of ESDCs is to integrate multiple datasets into a singular
comprehensive analysis, enhancing our understanding and insights into the
Earth system. In the second example, we focused on the same location as the
third row of Fig. 3 (DE-Hai Eddy Covariance ICOS site at Hainich National
Park, Germany) to create an extensive ESDC from various datasets, all
retrieved with a 500 m spatial resolution and a 32-pixel edge size, covering
data from 2022-08-01 to 2023-08-01. We gathered MODIS data on Surface
Reflectance (SR), Land Surface Temperature (LST), GPP, Leaf Area Index (LAI),
Fraction of Photosynthetically Active Radiation (FPAR), and Vegetation Indices
(VIs). Fig. 4 displays these ESDCs, aligned with the same spatio-temporal
grid. It’s noteworthy that datasets not matching the requested resolution
(e.g. LST, VIs, and SR products) were automatically resampled by cubo,
defaulting to the nearest neighbours method.

##  4 Conclusions

In this paper, we presented cubo, an open-source Python-based tool designed
for the straightforward generation of ESDCs on demand. cubo simplifies the
characterisation of AI-focused ESDCs, requiring minimal user input to create
these ESDCs. cubo is versatile, and compatible with any COG collection within
STAC. We anticipate cubo will be instrumental in various analytical processes
requiring spatio-temporal context in Earth system research, with a particular
emphasis on developing datasets for advanced AI tasks.

## References

  * Aybar et al. (2022) C. Aybar et al.  Cloudsen12, a global dataset for semantic understanding of cloud and cloud shadow in sentinel-2.  _Scientific Data_ , 9(1), Dec. 2022.  ISSN 2052-4463.  doi: 10.1038/s41597-022-01878-2.  URL http://dx.doi.org/10.1038/s41597-022-01878-2.
  * Hoyer et al. (2017) S. Hoyer et al.  xarray: N-d labeled arrays and datasets in python.  _Journal of Open Research Software_ , 5(1):10, Apr. 2017.  ISSN 2049-9647.  doi: 10.5334/jors.148.  URL http://dx.doi.org/10.5334/jors.148.
  * Mahecha et al. (2020) M. D. Mahecha et al.  Earth system data cubes unravel global multivariate dynamics.  _Earth System Dynamics_ , 11(1):201–234, Feb. 2020.  ISSN 2190-4987.  doi: 10.5194/esd-11-201-2020.  URL http://dx.doi.org/10.5194/esd-11-201-2020.
  * Montero et al. (2023) D. Montero et al.  Data cubes for earth system research: Challenges ahead.  _EarthArXiv_ , July 2023.  doi: 10.31223/x58m2v.  URL http://dx.doi.org/10.31223/X58M2V.
  * Rocklin (2015) M. Rocklin.  Dask: Parallel computation with blocked algorithms and task scheduling.  In K. Huff and J. Bergstra, editors, _Proceedings of the 14th Python in Science Conference_ , pages 130–136, 2015.
  * Söchting et al. (2023) M. Söchting et al.  Lexcube: Interactive visualization of large earth system data cubes.  _IEEE Computer Graphics and Applications_ , page 1–13, 2023.  ISSN 1558-1756.  doi: 10.1109/mcg.2023.3321989.  URL http://dx.doi.org/10.1109/MCG.2023.3321989.
  * Sumbul et al. (2019) G. Sumbul et al.  Bigearthnet: A large-scale benchmark archive for remote sensing image understanding.  In _IGARSS 2019 - 2019 IEEE International Geoscience and Remote Sensing Symposium_. IEEE, July 2019.  doi: 10.1109/igarss.2019.8900532.  URL http://dx.doi.org/10.1109/IGARSS.2019.8900532.