# Investigations on the flavor-dependent axial charges of the octet baryons

Rui Qi  School of Physical Science and Technology, Southwest University,
Chongqing 400715, China  Jin-Bao Wang  School of Physical Science and
Technology, Southwest University, Chongqing 400715, China  Gang Li
<EMAIL_ADDRESS>College of Physics and Engineering, Qufu Normal University,
Qufu 273165, China  Chun-Sheng An<EMAIL_ADDRESS>School of Physical Science
and Technology, Southwest University, Chongqing 400715, China  Cheng-Rong Deng
School of Physical Science and Technology, Southwest University, Chongqing
400715, China  Ju-Jun Xie  Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  School of Nuclear Science and Technology,
University of Chinese Academy of Sciences, Beijing 101408, China  School of
Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001,
China  Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical
Physics of Gansu Province, Lanzhou University, Lanzhou 730000, China

###### Abstract

We have investigated the axial charges of the ground octet baryons within the
extended chiral constituent quark model, where all the possible compact five-
quark Fock components $qqq(q\bar{q})(q=u,d,s)$ in the baryons are considered.
The transition couplings between the three- and five-quark components in the
baryons are assumed to be via the ${}^{3}P_{0}$ mechanism, which could
reproduce the sea asymmetry in proton very well. The numerical results for the
flavor-dependent axial charges of the octet baryons are comparable to those
predicted by other theoretical approaches. It is shown that the singlet axial
charges of the octet baryons, which should indicate total baryons spin arising
from the spin of the quarks, fall in the range $0.45-0.75$ in present model.
This is in consistent with the predictions by lattice QCD and chiral
perturbation theory. It’s also very interesting that the light quarks spin
$\Delta u$ and $\Delta d$ in the $\Lambda$ baryon are of small but negative
values, which exactly vanish in the traditional three-quark model.

##  I Introduction

The flavor-dependent axial charges of baryons, i.e. the singlet axial charge
$g^{(0)}_{A}$, the isovector axial charge $g^{(3)}_{A}$ and the $SU(3)$ octet
axial charge $g^{(8)}_{A}$, are fundamental observables in hadronic physics,
since they may provide us information about the spin structure and properties
of baryons. As we know, $g^{(0)}_{A}$ should indicate the total baryons spin
arising from the spin of quarks, and $g^{(3)}_{A}$ and $g^{(8)}_{A}$ should
govern the neutron and hyperons $\beta$ decays, and provide a quantitative
measure of spontaneous chiral symmetry breaking in low energy hadronic
physics. In addition, Goldberger-Treiman relation shows that the isovector
axial charge of nucleon could be directly related to the pion decay constant
Goldberger:1958tr . Therefore, one may establish a connection between the weak
and strong interactions by explicit investigations on the flavor-dependent
axial charges.

During the past years from 1990s, intensively experimental measurements on the
axial charges of nucleon have been performed, triggered by the renowned EMC
experiments, which raised up the proton spin puzzle Ashman:1987hv ;
Ashman:1989ig . Especially, the COMPASS collaboration has made great efforts
on corresponding measurements COMPASS:2006mhr ; COMPASS:2010wkz ;
COMPASS:2015mhb ; COMPASS:2016jwv  at $Q^{2}=3$ ${\rm GeV}^{2}$, and their
latest results show that the singlet axial charge of the proton is
$g^{(0)}_{A}=0.32\pm 0.02_{\rm stat.}\pm 0.04_{\rm syst.}\pm 0.05_{\rm
evol.}$, if the $SU(3)$ flavour symmetry is assumed, which should yield
$g^{(8)}_{A}=0.585\pm 0.025$ COMPASS:2016jwv . And very recently, the first
moment $g_{1}$ of nucleon at small $Q^{2}$ is also measured COMPASS:2017hef ;
Deur:2021klh . On the other hand, tremendous theoretical investigations on the
nucleon spin structure have also been done using different approaches, such as
the lattice QCD Aoki:1996pi ; Hagler:2007xi ; QCDSF:2011aa ; Yang:2016plb ;
Alexandrou:2017oeh ; Yamanaka:2018uud ; RQCD:2019jai , the chiral perturbation
theory Chen:2001pva ; Dorati:2007bk ; Lensky:2014dda ; Li:2015exr , the meson
cloud models Myhrer:2007cf ; Thomas:2008ga , and other phenomenological models
with higher Fock components in nucleon Zou:2005xy ; An:2005cj ;
Adamuscin:2007fk ; Bijker:2012zza ; An:2013daa . For recent reviews about the
nucleon spin structure, see Refs. Kuhn:2008sy ; Burkardt:2008jw ;
Leader:2013jra ; Wakamatsu:2014zza ; Liu:2015xha ; Deur:2018roz ; Ji:2020ena .

For the axial charges of the hyperons, experimental data is still lack up to
now. While the lattice QCD has demonstrated remarkable progress in computing
the hyperons axial transition couplings during the last two decades Lin:2007ap
; Sasaki:2008ha ; Erkol:2009ev ; Cooke:2013qqa ; Alexandrou:2014vya ;
Alexandrou:2016xok ; Savanur:2018jrb , and the corresponding calculations are
with very high accuracy. Besides, there have been also other theoretical
efforts dedicated to investigations on the hyperons axial couplings using
various approaches, including the chiral perturbation theory and
phenomenological hadronic models Lorce:2007as ; Bijker:2009up ; Jiang:2009sf ;
Choi:2010ty ; Ledwig:2014rfa ; Liu:2018jiu ; Jun:2020lfx ; Flores-
Mendieta:2021wzh .

Recently, the extended chiral constituent quark model (E$\chi$CQM), within
which higher Fock components in the baryons are included, has been applied to
the sea content of the octet baryons An:2012kj , the sea flavor asymmetry of
the proton $\bar{d}-\bar{u}=0.118\pm 0.012$ NuSea:2001idv ; SeaQuest:2021zxb
could be very well reproduced by taking the model parameters to be the
empirical values. Later, we have applied the work An:2012kj  to the baryon
sigma terms An:2014aea ; Duan:2016rkr , the intrinsic light and strange quark-
antiquark pair in the proton and nonperturbative strangeness suppression
An:2017flb , and the orbital angular momentum of the proton An:2019tld . And
in a very recent work, the axial charges of the proton has been investigated
employing the E$\chi$CQM Wang:2021ild . The numerical results obtained in all
the above referred works are in good agreements with predictions by other
theoretical approaches. Consequently, we extend the work Wang:2021ild  to
estimations on the hyperons axial charges in present work.

The present paper is organized as follows. In Sec. II, we give the framework
which includes the extended chiral constituent quark model and the formalism
for the octet baryons axial charges in corresponding model, the explicit
numerical results are presented in Sec. III. Finally, a brief summary is given
in Sec. IV.

##  II Framework

In this work, we investigate the axial charges of the ground state octet
baryons employing the E$\chi$CQM, which has been developed to investigate the
intrinsic sea content of the octet baryons in An:2012kj . Accordingly, in this
section, we will briefly introduce the E$\chi$CQM in Sec. II.1, and present
the formalism for calculations of octet baryons’ axial charges in Sec. II.2.

###  II.1 E$\chi$CQM

In the E$\chi$CQM developed in An:2012kj , the wave functions of the ground
state octet baryons can be expressed as:

$|B\rangle=\frac{1}{\sqrt{\mathcal{N}}}\left(|qqq\rangle+\sum_{i}C_{i}^{q}|qqq(q\bar{q}),i\rangle\right),$
(1)

where the first term represents the wave function for the three-quark
component of the octet baryons, and the second term denotes the wave functions
for the compact five-quark components, with the sum over $i$ runs over all the
possible five-quark configurations with a $q\bar{q}$ ($d\bar{d}$, $u\bar{u}$,
$s\bar{s}$, …) 111In this work, we do not take $c\bar{c}$ and $b\bar{b}$ into
account, since the probabilities for them are much smaller than other five-
quark components inside the octet baryons in the low energy scale. pair which
may form higher Fock components in the octet baryons,
$C_{i}^{q}/\sqrt{\mathcal{N}}$ are just the corresponding probability
amplitudes for the five-quark components with $\mathcal{N}$ being a
normalization constant.

In the present case, we consider the ground states of baryon octet, whose
parities are positive, so that the orbital quantum number $L$ must be an odd
number $1$, $3$, …, $2n+1$. On the other hand, the total spin $S$ of a five-
quark system arising from the quark spin can only be $\frac{1}{2}$,
$\frac{3}{2}$, or $\frac{5}{2}$, therefore, $L$ cannot be higher than $3$ to
combine with $S$ to form spin $J_{B}=L\oplus S$ for the ground state octet
baryons. As discussed explicitly in An:2012kj , only the five-quark
configurations with $L$ = 1 and the $n_{r}=0$ can form possible Fock
components in the ground state octet baryons with considerable probability
amplitudes.

Accordingly, there are $17$ possible five-quark configurations with different
orbital, flavor, spin and colour symmetries in each of the octet baryons,
which are shown explicitly in Table 1, where the symbols $[\dots]_{\nu}$ are
the Young tableaux of corresponding wave functions with $\nu=\chi$, $F$, $S$
and $C$ representing orbital, flavor, spin and color, respectively. Finally,
$[\dots]_{FS}[\dots]_{F}[\dots]_{S}$ is just a shorthand for corresponding
flavor-spin decomposition.

As we can see in Table 1, the $17$ five-quark configurations can be
categorized into two groups according to the spin wave functions of the four-
quark subsystem, namely, spin symmetry $[22]_{S}$ and $[31]_{S}$, those should
lead to the total spin of the four-quark system $S_{4}=0$ and $1$,
respectively. For the configurations with spin symmetry $[22]_{S}$, explicit
wave functions can be expressed as

$\displaystyle|B,\uparrow\rangle_{5q}$ $\displaystyle=$
$\displaystyle\sum_{ijkln}\sum_{ab}\sum_{m\bar{s}_{z}}C^{\frac{1}{2},\uparrow}_{1,m;\frac{1}{2},\bar{s}_{z}}C^{[1^{4}]}_{[31]_{\chi
FS}^{k};[211]_{C}^{\bar{k}}}C^{[31]_{\chi
FS}^{k}}_{[O]_{\chi}^{i};[FS]_{FS}^{j}}$ (2) $\displaystyle
C^{[FS]_{FS}^{j}}_{[F]_{F}^{l};[22]_{S}^{n}}C_{a,b}^{[2^{3}]_{C}}|[211]_{C}^{\bar{k}}(a)\rangle|[11]_{C,\bar{q}}(b)\rangle|I,I_{3}\rangle^{[F]_{F}^{l}}$
$\displaystyle|1,m\rangle^{[O]_{\chi}^{i}}|[22]_{S}^{n}\rangle|\bar{\chi},\bar{s}_{z}\rangle\phi(\\{\vec{r}_{q}\\}),$

where the coefficients $C^{[\cdots]}_{[\cdots][\cdots]}$ represent the CG
coefficients of the $S_{4}$ permutation group,
$|[211]^{C}_{\bar{k}}(a)\rangle$ and $|[11]^{C,\bar{q}}(b)\rangle$ are the
color wave functions for the four-quark subsystem and the antiquark,
combination of which should lead to the color singlet $[2^{3}]_{C}$.

While combination of the four-quark spin $[31]_{S}$ which leads to $S_{4}=1$
and the orbital angular momentum of the five-quark system $L=1$ should result
in $J=L\oplus S_{4}=0$ or $1$, both of the two cases of $J$ are legal in
present framework, since couplings of both the values of $J$ and spin of the
antiquark $s_{\bar{q}}=1/2$ could lead to $J_{B}=1/2$ for the octet baryons.
Hereafter, we denote these two cases as Set I and Set II, respectively. And
wave functions of the five-quark components octet baryons in these two cases
are

$\displaystyle|B,\uparrow\rangle_{5q}$ $\displaystyle=$
$\displaystyle\\!\\!\sum_{ijkln}\sum_{ab}\sum_{ms_{z}}C^{00}_{1,m;1,s_{z}}C^{[1^{4}]}_{[31]_{\chi
FS}^{k};[211]_{C}^{\bar{k}}}C^{[31]_{\chi
FS}^{k}}_{[O]^{\chi}_{i};[FS]_{FS}^{j}}$ (3)
$\displaystyle\\!\\!C^{[FS]_{FS}^{j}}_{[F]_{F}^{l};[31]_{S}^{n}}C_{a,b}^{[2^{3}]_{C}}|[211]_{C}^{\bar{k}}(a)\rangle|[11]_{C,\bar{q}}(b)\rangle|I,I_{3}\rangle^{[F]_{F}^{l}}$
$\displaystyle\\!\\!|1,m\rangle^{[O]_{\chi}^{i}}|[31]_{S}^{n},s_{z}\rangle|\bar{\chi},\bar{s}_{z}\rangle\phi(\\{\vec{r}_{q}\\}).$

$\displaystyle|B,\uparrow\rangle_{5q}$ $\displaystyle=$
$\displaystyle\\!\\!\sum_{ijkln}\sum_{ab}\sum_{J_{z}\bar{s}_{z}}\sum_{ms_{z}}C^{\frac{1}{2},\frac{1}{2}}_{1,J_{z};\frac{1}{2},\bar{s}_{z}}C^{1,J_{z}}_{1,m;1,s_{z}}C^{[1^{4}]}_{[31]_{\chi
FS}^{k};[211]_{C}^{\bar{k}}}$ (4) $\displaystyle\\!\\!C^{[31]_{\chi
FS}^{k}}_{[O]_{\chi}^{i};[FS]_{FS}^{j}}C^{[FS]_{FS}^{j}}_{[F]_{F}^{l};[31]_{S}^{n}}C_{a,b}^{[2^{3}]_{C}}|[211]_{C}^{\bar{k}}(a)\rangle|[11]_{C,\bar{q}}(b)\rangle$
$\displaystyle\\!\\!|I,I_{3}\rangle^{[F]_{F}^{l}}|1,m\rangle^{[O]^{\chi}_{i}}|[31]^{S}_{n},s_{z}\rangle|\bar{\chi},\bar{s}_{z}\rangle\phi(\\{\vec{r}_{q}\\})\,,$

respectively.

One should note that $|I,I_{3}\rangle^{[F]_{F}^{l}}$ in Eqs. (2)-(4) just
represents coupling of the flavour wave functions for the four-quark subsystem
and the antiquark to form appropriate isospin quantum number. Explicit flavor
decompositions of the five-quark configurations in the octet baryons are given
in Appendix A.

Table 1: The orbital-flavor-spin configurations for five-quark configurations those may exist as higher Fock components in ground octet baryons. $i$ | $1$ | $2$ | $3$ | $4$ | $5$
---|---|---|---|---|---
Config. | $[31]_{\chi}[4]_{FS}[22]_{F}[22]_{S}$ | $[31]_{\chi}[31]_{FS}[211]_{F}[22]_{S}$ | $[31]_{\chi}[31]_{FS}[31]_{F_{1}}[22]_{S}$ | $[31]_{\chi}[31]_{FS}[31]_{F_{2}}[22]_{S}$ | $[4]_{\chi}[31]_{FS}[211]_{F}[22]_{S}$
$i$ | $6$ | $7$ | $8$ | $9$ | $10$
Config. | $[4]_{\chi}[31]_{FS}[31]_{F_{1}}[22]_{S}$ | $[4]_{\chi}[31]_{FS}[31]_{F_{2}}[22]_{S}$ | $[31]_{\chi}[4]_{FS}[31]_{F_{1}}[31]_{S}$ | $[31]_{\chi}[4]_{FS}[31]_{F_{2}}[31]_{S}$ | $[31]_{\chi}[31]_{FS}[211]_{F}[31]_{S}$
$i$ | $11$ | $12$ | $13$ | $14$ | $15$
Config. | $[31]_{\chi}[31]_{FS}[22]_{F}[31]_{S}$ | $[31]_{\chi}[31]_{FS}[31]_{F_{1}}[31]_{S}$ | $[31]_{\chi}[31]_{FS}[31]_{F_{2}}[31]_{S}$ | $[4]_{\chi}[31]_{FS}[211]_{F}[31]_{S}$ | $[4]_{\chi}[31]_{FS}[22]_{F}[31]_{S}$
$i$ | $16$ | $17$ |  |  |
Config. | $[4]_{\chi}[31]_{FS}[31]_{F_{1}}[31]_{S}$ | $[4]_{\chi}[31]_{FS}[31]_{F_{2}}[31]_{S}$ |  |  |

The coefficients $C_{i}^{q}$ in Eq. (1), in the framework of the E$\chi$CQM
developed in Ref. An:2012kj , can be related to the coupling between the
three-quark and the corresponding five-quark components, which reads

$C_{i}^{q}=\frac{\langle qqq(q\bar{q}),i|\hat{T}|qqq\rangle}{M_{B}-E_{i}}\,,$
(5)

here $M_{B}$ denote the physical mass of the baryon $B$
ParticleDataGroup:2020ssz , and $E_{i}$ is the energy of the $i$-th
$qqq(q\bar{q})$ five-quark component. The transition coupling operator
$\hat{T}$ depends on the quark-antiquark creation mechanism, in present work,
we take the widely accepted ${}^{3}P_{0}$ mechanism following Ref. An:2012kj ,
where the explicit form of $\hat{T}$ is

$\displaystyle\hat{T}$ $\displaystyle=$
$\displaystyle-\gamma\sum_{j=1,4}\mathcal{F}_{j,5}^{00}\mathcal{C}_{j,5}^{00}\mathcal{C}_{OFSC}\sum_{m}\langle
1,m;1,-m|00\rangle\times$ (6)
$\displaystyle\chi_{j,5}^{1,m}\mathcal{Y}_{j,5}^{1,-m}(\vec{p}_{j}-\vec{p}_{5})b^{\dagger}(\vec{p}_{j})d^{\dagger}(\vec{p}_{5})\,,$

where $\gamma$ is an dimensionless transition coupling constant,
$\mathcal{F}_{j,5}^{00}\text{ and }\mathcal{C}_{j,5}^{00}$ are the flavor and
color singlet of the created quark-antiquark pair $q_{j}\bar{q}_{5}$,
$\chi_{j,5}^{1,m}\text{ and }\mathcal{Y}_{j,5}^{1,-m}$ are the total spin
$S_{q\bar{q}}=1$ and relative orbital $P-$ state of the created quark-
antiquark system, the operator $\mathcal{C}_{OFSC}$ is to calculate the
overlap factor between the residual three-quark configuration in the five-
quark component and the valence three-quark component, finally,
$b^{\dagger}(\vec{p}_{j}),d^{\dagger}(\vec{p}_{5})$ are the quark and
antiquark creation operators.

Finally, to calculate the energy for a given five-quark configuration $E_{i}$
in Eq. (5), we employ the chiral constituent quark model, in which the quark-
quark hyperfine interaction is flavor-dependent Glozman:1995fu , as follow

$\displaystyle H_{hyp}$ $\displaystyle=$
$\displaystyle-\sum_{i<j}\vec{\sigma}_{i}\cdot\vec{\sigma}_{j}\Bigg{[}\sum_{a=1}^{3}V_{\pi}(r_{ij})\lambda^{a}_{i}\lambda^{a}_{j}$
(7)
$\displaystyle+\sum_{a=4}^{7}V_{K}(r_{ij})\lambda^{a}_{i}\lambda^{a}_{j}+V_{\eta}(r_{ij})\lambda^{8}_{i}\lambda^{8}_{j}\Bigg{]}\,,$

numerical values for all the exchange coupling strength constants $V_{M}$ used
in present work are taken to be the empirical ones Glozman:1995fu . Thus, the
energies $E_{i}$ for the 17 five-quark configurations in Table 1 should be

$E_{i}=E_{0}+\langle H_{hyp}\rangle+\delta_{q\bar{q}},$ (8)

where $E_{0}$ is a degenerated energy for the 17 five-quark configurations
shown in Table 1. The parameter $E_{0}$ is dependent on the constituent quark
masses, the kinetic quark energies, and also the energies of the quark
confinement interactions and the quark potentials. Here we take $E_{0}=2127$
MeV, $\delta_{u\bar{u}}=\delta_{d\bar{d}}=0$, and $\delta_{s\bar{s}}=240$ MeV
as used in Ref. An:2012kj .

###  II.2 Formalism for the axial charges of octet baryons

In this section, we present the formalism for the axial charges $g_{A}^{(0)}$,
$g_{A}^{(3)}$ and $g_{A}^{(8)}$ of octet baryons within the framework of
E$\chi$CQM shown in Sec. II.1.

The quark spin contribution $\Delta q$, which is related to the flavour-
dependent axial vector current operator, is defined as

$\displaystyle\langle
B,s_{z}|\int\mathrm{d}x\bar{q}\gamma^{\mu}\gamma^{5}q|B,s_{z}\rangle$
$\displaystyle=s^{\mu}\cdot\Delta q\,,$ (9)

where $s^{\mu}$ is the baryon states polarization vector, and $\Delta q$ is

$\Delta
q=(q^{\uparrow}+\bar{q}^{\uparrow})-(q^{\downarrow}+\bar{q}^{\downarrow})\,.$
(10)

Combinations of different flavor $\Delta f$ with $f=u,d,s$ lead to flavor-
singlet, isovector, and SU(3) octet axial charges of octet baryons, as follows

$\displaystyle g^{(0)}_{A}$ $\displaystyle=$ $\displaystyle\Delta u+\Delta
d+\Delta s.$ (11) $\displaystyle g^{(3)}_{A}$ $\displaystyle=$
$\displaystyle\Delta u-\Delta d,$ (12) $\displaystyle g^{(8)}_{A}$
$\displaystyle=$ $\displaystyle\Delta u+\Delta d-2\Delta s.$ (13)

In the non-relativistic approximation, $\Delta f$ can be gotten directly
through the formula:

$\Delta f=\langle
Bs_{z}|\sum_{j}\,\hat{\sigma}^{z}_{j}\delta_{jf}|Bs_{z}\rangle,$ (14)

where $|Bs_{z}\rangle$ is the wave function Eq. (1) of octet baryons,
$\hat{\sigma}^{z}_{j}$ is Pauli operator acting on $j$-th quark, and
$\delta_{jf}$ is a flavor-dependent operator, defined as

$\displaystyle\delta_{jf}=\begin{cases}1&\text{if the favour of $j$-th quark
is $f$}\\\ 0&\text{if the favour of $j$-th quark is not $f$}\end{cases}$ (15)

Consequently, for each of the octet baryons, one can get

$\displaystyle\\!\\!\\!\\!\\!\Delta f=\frac{1}{\mathcal{N}}\langle
qqq,s_{z}|\sum_{j=1,3}\,\hat{\sigma}^{z}_{j}\delta_{jf}|qqq,s_{z}\rangle+\sum_{i}\frac{(C_{i}^{q})^{2}}{\mathcal{N}}$
$\displaystyle\\!\\!\\!\\!\\!\langle
qqq(q\bar{q}),i,s_{z}|\sum_{j=1,5}\,\hat{\sigma}^{z}_{j}\delta_{jf}|qqq(q\bar{q}),i,s_{z}\rangle,$
(16)

where the non-diagonal terms are neglected.

For simplicity, we denote the matrix elements for the three-quark components
and an explicit given five-quark component in Eq. (16) as

$\displaystyle\Delta f^{3q}$ $\displaystyle=$ $\displaystyle\langle
qqq,s_{z}|\sum_{j=1,3}\,\hat{\sigma}^{z}_{j}\delta_{jf}|qqq,s_{z}\rangle\,,$
(17) $\displaystyle\Delta f_{i}^{5q}$ $\displaystyle=$ $\displaystyle\langle
qqq(q\bar{q}),i,s_{z}|\sum_{j=1}^{5}\,\hat{\sigma}^{z}_{j}\delta_{jf}|qqq(q\bar{q}),i,s_{z}\rangle.$
(18)

Accordingly, one can easily obtain the explicit expression of the matrix
elements results for $\Delta u$, $\Delta d$ and $\Delta s$ of octet baryons as
following

$\Delta u=\frac{1}{\mathcal{N}}\Delta
u^{3q}+\sum_{i,q}\frac{(C_{i}^{q})^{2}}{\mathcal{N}}\Delta u_{i}^{5q},$ (19)
$\Delta d=\frac{1}{\mathcal{N}}\Delta
d^{3q}+\sum_{i,q}\frac{(C_{i}^{q})^{2}}{\mathcal{N}}\Delta d_{i}^{5q},$ (20)
$\Delta s=\frac{1}{\mathcal{N}}\Delta
s^{3q}+\sum_{i,q}\frac{(C_{i}^{q})^{2}}{\mathcal{N}}\Delta s_{i}^{5q}.$ (21)

Finally, considering the $SU(2)$ isospin symmetry, one can get the relations
of the spin contributions of quarks with different flavour to spin of the
proton and neutron as

$\displaystyle\Delta{u}(p)=\Delta{d}(n),\Delta{d}(p)=\Delta{u}(n),\Delta{s}(p)=\Delta{s}(n)\,,$
(22)

and the resulted relations of the axial charges of proton and neutron as

$\displaystyle
g^{(0)}_{A}(p)=g^{(0)}_{A}(n),g^{(3)}_{A}(p)=-g^{(3)}_{A}(n)\,,$
$\displaystyle g^{(8)}_{A}(p)=g^{(8)}_{A}(n)\,.$ (23)

For the spin of quarks with different flavor and the axial charges of the
other octet baryons, analogous relationship can be obtained.

##  III The numerical results and discussions

In this section, we present our numerical results of the calculations on the
quark spin contributions and the axial charges of the octet baryons in the
framework shown in Sec. II.

Before going to the final numerical results, firstly, we have to discuss the
model parameters in the E$\chi$CQM. In ref. An:2012kj , the intrinsic sea
contents of the octet baryons have been studied in the E$\chi$CQM by taking
the empirical values for the model parameters, only except for the newly
introduced $V$ which should indicate the transition coupling strength for the
processes $qqq\leftrightarrow qqq(q\bar{q})$, arising from calculations on the
matrix elements of the operator (6). The parameter $V$ is determined by
fitting the sea asymmetry in proton $\bar{d}-\bar{u}=0.118\pm 0.012$
NuSea:2001idv ; SeaQuest:2021zxb , which yields

$V=570\pm 46~{}~{}{\rm MeV}\,.$ (24)

As we have discussed in Sec. II.1, the spin symmetry of the fourquark
subsystem in the configurations listed in Table 1 should lead to two sets for
the wave functions of the five-quark components in the octet baryons, namely,
Eqs. (3) and (4), respectively. In Ref. An:2012kj , only the former case was
considered because of the possible lower energy, while in the very recent work
Wang:2021ild , both of the two sets of wave functions have been employed to
investigate the axial charges of the proton. Keeping all the other model
parameters being the empirical values, the parameter $V$ should be

$V=697\pm{80}~{}~{}{\rm MeV}\,,$ (25)

to fit the intrinsic sea asymmetry of the proton when the set of wave
functions (4) is used.

In present work, we will consider both Set I and Set II to study the axial
charges of the octet baryons following work Wang:2021ild . And the resulting
numerical results for the probabilities of the five-quark Fock components, the
quark spin contributions to the octet baryons, and the flavor-dependent axial
charges of the octet baryons are presented in the following three subsections,
respectively.

###  III.1 Probabilities of the intrinsic five-quark Fock components in the
octet baryons

Employing the values for all the model parameters mentioned above, explicit
calculations on the transition couplings between three- and five-quark
components, and energies for the corresponding five-quark components by taking
the wave functions of Set I and Set II lead to the numerical results for the
probabilities of the intrinsic five-quark components in the octet baryons,
those are shown in Tables AI\- AVIII.

As discussed in Ref. Wang:2021ild , taking the wave functions of Set II, one
can obtain larger probabilities for the five-quark components in the proton
than those resulted by using the wave functions of Set I. While the results of
Set II are very close to the predictions obtained by the BHPS model
Chang:2011vx . It’s very similar for the probabilities of the five-quark in
the other octet baryons, as one can see in the tables.

One may note that the most significant difference of the results obtained in
Sets I and II, is that the probabilities of the five-quark configuration with
$i=8$ in all the octet baryons obtained in Set II are about two times of those
obtained in Set I. In fact, one can also find similar amplifications for the
probabilities of the five-quark configurations with $i=9-17$ in those the spin
symmetry of the four-quark subsystem is $[31]_{S}$. These amplifications are
mainly caused by the larger matrix elements $\langle\hat{T}\rangle$ for the
transition couplings $qqq\leftrightarrow qqq(q\bar{q})$ via the ${}^{3}P_{0}$
mechanism obtained using the wave function (4) in Set II than using (3) in Set
I.

Straightforwardly, for Set I, one can get the total probabilities of the five-
quark components in the octet baryons

$\displaystyle\mathcal{P}^{N}_{\textrm{I}}=0.373,~{}~{}\mathcal{P}^{\Sigma}_{\textrm{I}}=0.338\,,$
$\displaystyle\mathcal{P}^{\Xi}_{\textrm{I}}=0.303,~{}~{}\mathcal{P}^{\Lambda}_{\textrm{I}}=0.330\,,$
(26)

with uncertainties $\sim 10\%$ for all the values caused by the experimental
errors of the data for $\bar{d}-\bar{u}$ in proton, and

$\displaystyle\mathcal{P}^{N}_{\textrm{II}}=0.548,~{}~{}\mathcal{P}^{\Sigma}_{\textrm{II}}=0.519\,,$
$\displaystyle\mathcal{P}^{\Xi}_{\textrm{II}}=0.467,~{}~{}\mathcal{P}^{\Lambda}_{\textrm{II}}=0.504\,,$
(27)

with uncertainties $\sim 10\%$.

An examination of the obtained probabilities is to estimate the pion- and
strangeness-baryon $\sigma$ terms of the octet baryons. In the model developed
in Ref. An:2014aea , the corresponding $\sigma$ terms could be related to the
probabilities of the five-quark components with light and strangeness quark-
antiquark pairs $\mathcal{P}_{q\bar{q}}$ as

$\displaystyle\sigma_{\pi N}$ $\displaystyle=$
$\displaystyle\frac{3+2(\mathcal{P}^{N}_{u\bar{u}}+\mathcal{P}^{N}_{d\bar{d}})}{3+2(\mathcal{P}^{N}_{u\bar{u}}+\mathcal{P}^{N}_{d\bar{d}}-2\mathcal{P}^{N}_{s\bar{s}})}\hat{\sigma},$
(28) $\displaystyle\sigma_{sN}$ $\displaystyle=$
$\displaystyle\frac{m_{s}}{m_{l}}\frac{2\mathcal{P}_{N}^{s\bar{s}}}{3+2(\mathcal{P}_{N}^{u\bar{u}}+\mathcal{P}_{N}^{d\bar{d}})}\sigma_{\pi
N}\,,$ (29)

for nucleon, and

$\displaystyle\sigma_{\pi Y}$ $\displaystyle=$
$\displaystyle\frac{2+2(\mathcal{P}^{Y}_{u\bar{u}}+\mathcal{P}^{Y}_{d\bar{d}})}{3+2(\mathcal{P}^{N}_{u\bar{u}}+\mathcal{P}^{N}_{d\bar{d}})}\sigma_{\pi
N},$ (30) $\displaystyle\sigma_{sY}$ $\displaystyle=$
$\displaystyle\frac{1+2\mathcal{P}^{Y}_{s\bar{s}}}{2\mathcal{P}^{N}_{s\bar{s}}}\sigma_{sN}$
(31) $\displaystyle=$
$\displaystyle\frac{m_{s}}{m_{l}}\frac{1+2\mathcal{P}^{Y}_{s\bar{s}}}{3+2(\mathcal{P}^{N}_{u\bar{u}}+\mathcal{P}^{N}_{d\bar{d}})}\sigma_{\pi
N}\,,$

for the hyperons $\Sigma$ and $\Lambda$, and

$\displaystyle\sigma_{\pi\Xi}$ $\displaystyle=$
$\displaystyle\frac{1+2(\mathcal{P}^{\Xi}_{u\bar{u}}+\mathcal{P}^{\Xi}_{d\bar{d}})}{3+2(\mathcal{P}^{N}_{u\bar{u}}+\mathcal{P}^{N}_{d\bar{d}})}\sigma_{\pi
N},$ (32) $\displaystyle\sigma_{s\Xi}$ $\displaystyle=$
$\displaystyle\frac{2+2\mathcal{P}^{\Xi}_{s\bar{s}}}{2\mathcal{P}^{N}_{s\bar{s}}}\sigma_{sN}$
(33) $\displaystyle=$
$\displaystyle\frac{m_{s}}{m_{l}}\frac{2+2\mathcal{P}^{\Xi}_{s\bar{s}}}{3+2(\mathcal{P}^{N}_{u\bar{u}}+\mathcal{P}^{N}_{d\bar{d}})}\sigma_{\pi
N}\,,$

for the hyperon $\Xi$, where the nonsinglet component was taken to be
$\hat{\sigma}=33\pm 5$ MeV, as extracted within the chiral perturbation theory
Borasoy:1996bx  and the ratio of the light and strange quark masses was taken
to be $\frac{m_{s}}{m_{l}}=27.5\pm 1$ as shown in PDG
ParticleDataGroup:2020ssz .

As shown in An:2014aea , the numerical results of the obtained employing the
wave functions of Set I in present work are in general consistent with the
predictions of chiral perturbation theory and lattice QCD. While by the
obtained probabilities of the five-quark components shown in Tables AI\- AVIII
employing the wave functions of Set II, one can get

$\displaystyle\sigma_{\pi N}=36\pm 5~{}(\rm MeV),~{}~{}~{}~{}\sigma_{sN}=46\pm
8~{}(\rm MeV)\,,$ (34) $\displaystyle\sigma_{\pi\Sigma}=26\pm 4~{}(\rm
MeV),~{}\sigma_{s\Sigma}=302\pm 57~{}(\rm MeV)\,,$ (35)
$\displaystyle\sigma_{\pi\Xi}=16\pm 3~{}(\rm MeV),~{}\sigma_{s\Xi}=535\pm
102~{}(\rm MeV)\,,$ (36) $\displaystyle\sigma_{\pi\Lambda}=27\pm 4~{}(\rm
MeV),~{}\sigma_{s\Lambda}=286\pm 55~{}(\rm MeV)\,,$ (37)

accordingly, all the results are very close to those obtained in Ref.
An:2014aea .

###  III.2 The quark spin contributions to the octet baryons

Table 2: Numerical results for $\Delta{f}$ $(f=u,d,s)$ of the octet baryons, compared to the predictions by lattice QCD QCDSF:2011aa  and chiral effective field theory Li:2015exr  shown in the last two rows, respectively. |  | $p$ | $n$ | $\Sigma^{+}$ | $\Sigma^{0}$ | $\Sigma^{-}$ | $\Xi^{0}$ | $\Xi^{-}$ | $\Lambda$
---|---|---|---|---|---|---|---|---|---
$\Delta{u}$ | Set I | $0.883\pm 0.005$ | $-0.213\pm 0.003$ | $0.922\pm 0.005$ | $0.473\pm 0.004$ | $0.023\pm 0.003$ | $-0.215\pm 0.002$ | $0.030\pm 0.006$ | $0.026\pm 0.003$
Set II | $0.710\pm 0.012$ | $-0.225\pm 0.008$ | $0.762\pm 0.014$ | $0.348\pm 0.004$ | $-0.066\pm 0.007$ | $-0.194\pm 0.002$ | $-0.059\pm 0.007$ | $-0.020\pm 0.002$
LQCD QCDSF:2011aa  | $0.794(21)(2)$ |  |  |  |  |  |  |
$\chi$PT Li:2015exr  | $0.90^{+0.03}_{-0.04}$ |  |  |  |  |  |  |
$\Delta{d}$ | Set I | $-0.213\pm 0.003$ | $0.883\pm 0.005$ | $0.023\pm 0.003$ | $0.473\pm 0.004$ | $0.922\pm 0.005$ | $0.030\pm 0.006$ | $-0.215\pm 0.002$ | $0.026\pm 0.003$
Set II | $-0.225\pm 0.008$ | $0.710\pm 0.012$ | $-0.066\pm 0.007$ | $0.348\pm 0.004$ | $0.762\pm 0.014$ | $-0.059\pm 0.007$ | $-0.194\pm 0.002$ | $-0.020\pm 0.002$
LQCD QCDSF:2011aa  | $-0.289(16)(1)$ |  |  |  |  |  |  |
$\chi$PT Li:2015exr  | $-0.38^{+0.03}_{-0.03}$ |  |  |  |  |  |  |
$\Delta{s}$ | Set I | $0.015\pm 0.002$ | $0.015\pm 0.002$ | $-0.205\pm 0.002$ | $-0.205\pm 0.002$ | $-0.205\pm 0.002$ | $0.929\pm 0.001$ | $0.929\pm 0.001$ | $0.674\pm 0.001$
Set II | $-0.020\pm 0.003$ | $-0.020\pm 0.003$ | $-0.180\pm 0.003$ | $-0.180\pm 0.003$ | $-0.180\pm 0.003$ | $0.798\pm 0.011$ | $0.798\pm 0.011$ | $0.551\pm 0.006$
LQCD QCDSF:2011aa  | $-0.023(10)(1)$ |  |  |  |  |  |  |
$\chi$PT Li:2015exr  | $-0.007^{+0.004}_{-0.007}$ |  |  |  |  |  |  |

In this section, we present our numerical results for the quark spin
contributions to the octet baryons. Employing the wave functions of Sets I and
II, explicit calculations on the matrix elements (18) lead to numerical
results for $\Delta f_{i}^{5q}$ shown in Tables AI\- AVIII in Appendix B.

Analogously to the results in Ref. Wang:2021ild , as we can see in the tables
in Appendix B, the numerical results for $\Delta f_{i}^{5q}$ with $i=1-7$ of
Set I are the same with those of Set II, this is because of that the wave
functions of the five-quark configurations $i=1-7$ in Sets I and II are the
same one as shown in Eq. (2). And only spin of the antiquarks contribute to
$\Delta f_{i}^{5q}$ of these five-quark configurations, since spin of the
fourquark subsystem is $0$. For $\Delta f_{i}^{5q}$ with $i=8-17$, both the
quarks and antiquark should contribute, and the different wave functions of
the five-quark configurations in Eqs. (3) and (4) in Sets I and II lead to
different results.

Straightforwardly, we can obtain the numerical results for the quark spin
contributions of the octet baryons $\Delta f$ by the obtained $\Delta
f^{5q}_{i}$ and $\mathcal{P}_{q\bar{q}}$ using Eqs. (19)-(21), the results are
shown in Table 2.

In most of the theoretical studies on the quark spin contributions of baryons,
only nucleon has been considered. As discussed in Ref. Wang:2021ild , the
numerical results of $\Delta f$ of the proton obtained using the E$\chi$CQM
are in general consistent with the predictions by lattice QCD theory.
Therefore, here we only compare the present numerical results to those
predicted in Refs. QCDSF:2011aa  and Li:2015exr , a more comprehensive
comparison of the numerical results for $\Delta f$ of the nucleon obtained in
present model to other theoretical predictions can be found in Wang:2021ild .

It’s very interesting that $\Delta u$ and $\Delta d$ of the $\Lambda$ hyperon
in present model are small but nonzero values. As we know, in the traditional
constituent quark model in which the baryons are assumed to be composed of
three quarks, the two light quarks in $\Lambda$ are completely antisymmetric,
therefore, the $\Lambda$ spin arising from the light quarks spin vanishes.
While in present work, contributions of the five-quark components are taken
into account, as we can see in Table AVIII, several of the five-quark
components have nonzero contributions to $\Delta u$ and $\Delta d$.
Accordingly, experimental investigations on the contributions of light quarks
to $\Lambda$ spin may be an examination of the present model.

On the other hand, the magnetic moments, spins and orbital angular momenta of
the octet baryons have been studied in an unquenched quark model in Ref.
Bijker:2009up . In that work, the spin of proton carried by different flavor
of quarks were obtained to be $\Delta u=1.098$, $\Delta d=-0.417$ and $\Delta
s=-0.005$, and for the $\Lambda$ hyperon, the values $\Delta u=-0.055$,
$\Delta d=-0.055$ and $\Delta s=0.961$ were predicted.

###  III.3 The flavor-dependent axial charges of the octet baryons

The singlet axial charge $g^{(0)}_{A}$, isovector axial charge $g^{(3)}_{A}$
and $SU(3)$ octet axial charge $g^{(8)}_{A}$ of the octet baryons can be
directly obtained by using Eqs. (11)-(13) with the values of $\Delta f$ shown
in Table 2, the numerical results are shown in Table 3 in Columns Set I and
Set II for the two sets of wave functions discussed in Sec. II. As we can see
in the table, the obtained numerical results of $g^{(0)}_{A}$ of the $\Sigma$
and $\Xi$ hyperons are larger than that of the nucleon, which should indicate
larger contributions of the quark spin to hyperons spin than those to nucleon
spin.

In Ref. Wang:2021ild , the axial charges of the proton obtained using the
E$\chi$CQM have been explicitly compared to the experimental data
COMPASS:2010wkz  and predictions of lattice QCD Aoki:1996pi ; Hagler:2007xi ;
QCDSF:2011aa ; Yang:2016plb ; Alexandrou:2017oeh ; Yamanaka:2018uud ;
RQCD:2019jai , and the chiral perturbation theory Chen:2001pva ; Dorati:2007bk
; Lensky:2014dda ; Li:2015exr . For the singlet axial charge $g^{(0)}_{A}$,
which should indicate the proton spin arising from the quarks spin, it was
shown that the results in E$\chi$CQM were consistent with predictions of other
theoretical approaches, but larger than the experimental data. In fact, it may
be not convenient for us to directly compare $g^{(0)}_{A}$ estimated in the
static quark model with the COMPASS data, since $g^{(0)}_{A}$ is often
measured at $Q^{2}=3$ ${\rm GeV}^{2}$, and the increasing $Q^{2}$ may result
in decreased $g^{(0)}_{A}$ Jaffe:1987sx . Therefore, one may conclude that the
estimation of the axial charges of the proton in E$\chi$CQM are consistent
with the data and predictions by lattice QCD and chiral perturbation theory.

Consequently, in present work, we only compare our numerical results for the
axial charges of the other octet baryons with predictions by other theoretical
approaches, as we can see in Table 3.

Table 3: Numerical results of $g^{(0)}_{A}$, $g^{(3)}_{A}$, $g^{(8)}_{A}$ of octet baryons, compared to the predictions by other theoretical approaches. | Baryon | Set I | Set II |  LQCD Lin:2007ap  |  LQCD Alexandrou:2016xok  |  $\chi$PT Jiang:2009sf  |  $\chi$EFT Li:2015exr  |  RCQM Choi:2010ty  |  PCQM Liu:2018jiu
---|---|---|---|---|---|---|---|---|---
$g^{(0)}_{A}$ | $N$ | $0.685\pm 0.007$ | $0.465\pm 0.023$ |  |  |  | $0.51^{+0.07}_{-0.08}$ |  |
$\Sigma$ | $0.740\pm 0.010$ | $0.516\pm 0.024$ |  |  |  |  |  |
$\Xi$ | $0.744\pm 0.009$ | $0.545\pm 0.020$ |  |  |  |  |  |
$\Lambda$ | $0.726\pm 0.008$ | $0.511\pm 0.010$ |  |  |  |  |  |
$g^{(3)}_{A}$ | $N$ | $1.096\pm 0.005$ | $0.935\pm 0.019$ | $1.18(4)(6)$ |  | $1.18$ | $1.27$ | $1.65$ | $1.263$
$\Sigma$ | $0.899\pm 0.008$ | $0.828\pm 0.021$ | $0.900(42)(54)$ | $0.762(22)$ | $1.03$ |  | $0.93$ | $0.896$
$\Xi$ | $-0.245\pm 0.008$ | $-0.135\pm 0.009$ | $-0.277(15)(19)$ | $-0.248(9)$ | $-0.23$ |  | $-0.32$ | $-0.275$
| $\Lambda$ | $0$ | $0$ |  | $0.085(15)$ |  | $0$ |  |
$g^{(8)}_{A}$ | $N$ | $0.640\pm 0.010$ | $0.525\pm 0.026$ |  |  |  | $0.53^{+0.06}_{-0.06}$ |  |
$\Sigma$ | $1.355\pm 0.011$ | $1.056\pm 0.026$ |  |  |  |  |  |
$\Xi$ | $-2.043\pm 0.010$ | $-1.849\pm 0.031$ |  |  |  |  |  |
| $\Lambda$ | $-1.296\pm 0.009$ | $-1.142\pm 0.016$ |  |  |  |  |  |

In Ref. Lin:2007ap , the first lattice calculations on the axial charge
$g^{(3)}_{A}$ of the $\Sigma$ and $\Xi$ hyperons was performed using
$2+1$-flavor lattices by Lin and Orginos. And in 2018, Savanur and Lin
presented the first chiral-continuum–finite-volume extrapolation of the
hyperons axial couplings $g_{\Sigma\Sigma}$ and $g_{\Xi\Xi}$ from
$N_{f}=2+1+1$ lattice QCD in Savanur:2018jrb , where the axial charges
$g^{(3)}_{A}$ of $\Sigma$ and $\Xi$ with respect to the nucleon axial charges
were predicted to be $g^{(3)}_{A}/g_{A}=0.702(12)(4)$ for $\Sigma$ and
$g^{(3)}_{A}/g_{A}=-0.213(5)(1)$ for $\Xi$, respectively.

In Ref. Erkol:2009ev , the strangeness-conserving $NN$, $\Sigma\Sigma$,
$\Xi\Xi$, $\Lambda\Sigma$ and the strangeness-changing $\Lambda N$, $\Sigma
N$, $\Lambda\Xi$, $\Sigma\Xi$ axial charges was studied in lattice QCD with
$N_{f}=2$. The resulting axial charges of the $\Sigma$ and $\Xi$ fall in the
same range as the present obtained numerical results.

And in a recent work Alexandrou:2016xok , the axial couplings of the low lying
baryons were evaluated using a total of five ensembles of dynamical twisted
mass fermion gauge configurations. As we can see in Table 3, the present
numerical results for the axial charges $g^{(3)}_{A}$ and $g^{(8)}_{A}$ of the
$\Sigma$, $\Xi$ and $\Lambda$ hyperons are very close to those depicted in the
figures in Ref. Alexandrou:2016xok , and in Table 3, we only show their
predicted values for $g^{(3)}_{A}$ of the $\Sigma$ and $\Xi$ hyperons.

The axial charges of the octet baryons have also been investigated using
chiral perturbation theory Li:2015exr ; Jiang:2009sf . In Li:2015exr , all the
three flavor-dependent axial charges of the nucleon has been studied, as shown
in the column $\chi$EFT in Table 3. In Ref. Jiang:2009sf , $g^{(3)}_{A}$ of
the nucleon was predicted to be $1.18$, while that for the hyperons $\Sigma$
and $\Xi$ were predicted to be $1.03$ and $-0.23$, respectively, as we can see
in the column $\chi$PT in Table 3.

In addition, we also compare the present obtained results with those obtained
using the relativistic constituent $qqq$ quark model Choi:2010ty  and
perturbative chiral quark model Liu:2018jiu , which are shown in the columns
RCQM and PCQM in Table 3, respectively.

##  IV Summary

To summarize, we investigate the intrinsic five-quark components and the
flavor-dependent axial charges of the octet baryons within the framework of
the extended chiral constituent quark model, in which wave functions of the
higher five-quark Fock components in baryons are taken into account.

The probabilities of the five-quark components in the nucleon are in
agreements with other theoretical predictions. And with the obtained
probabilities of the five-quark components in the wave functions of the octet
baryons, we get the pion- and strangeness-baryon sigma terms consistent with
predictions by lattice QCD.

Finally, the present obtained numerical results show that the singlet axial
charges of the octet baryons, which should indicate total baryons spin arising
from the spin of the quarks, fall in the range $0.45-0.75$ in present model,
it’s in consistent with the predictions by lattice QCD, chiral perturbation
theory, as well the other theoretical approaches. The present obtained axial
charges $g^{(3)}_{A}$ and $g^{(8)}_{A}$ of the octet baryons are also
comparable to those predicted by lattice QCD and chiral perturbation theory.
It’s also very interesting that the quark spin $\Delta u$ and $\Delta d$ of
$\Lambda$ arising from the five-quark components are of small but nonzero
values.

###### Acknowledgements.

This work is partly supported by the Chongqing Natural Science Foundation
under Project No. cstc2021jcyj-msxmX0078, and No. cstc2019jcyj-msxmX0409, and
the National Natural Science Foundation of China under Grant Nos. 12075288,
12075133, 11735003, 11961141012 and 11835015. It is also supported by the
Youth Innovation Promotion Association CAS, Taishan Scholar Project of
Shandong Province (Grant No.tsqn202103062), the Higher Educational Youth
Innovation Science and Technology Program Shandong Province (Grant No.
2020KJJ004).

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##  Appendix A Flavor decompositions of five-quark components in the octet
baryons

In each five-quark configuration, $[v]_{F}$ is the flavour wave function of
four-quark subsystem, one can get the flavour wave function of the five-quark
system by combining $[v]_{F}$ and antiquark flavour wave function
$|\bar{q}\rangle$. Here we give the explicit flavor decompositions for all the
possible five-quark configurations in the octet baryons.

###  A.1 Nucleon

For the proton, whose isospin wave function is
$|\frac{1}{2},\frac{1}{2}\rangle_{I}$, the configurations with
$[v]_{F}=[31]_{F_{1}}$ rules out the strangeness component. Then the four-
quark subsystem should couple with the antiquark as

$\displaystyle|\frac{1}{2},\frac{1}{2}\rangle_{I}^{[31]_{F_{1}}}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{2}{3}}|u^{3}d_{[31]_{F_{1}}}\rangle\otimes|\bar{u}\rangle+$
(38)
$\displaystyle\sqrt{\frac{1}{3}}|u^{2}d^{2}_{[31]_{F_{1}}}\rangle\otimes|\bar{d}\rangle,$

For the configurations with $[v]_{F}=[31]_{F_{2}}$, the quark-antiquark pair
can only be $s\bar{s}$, and the corresponding isospin wave function of the
proton is

$|\frac{1}{2},\frac{1}{2}\rangle_{I}^{[31]_{F_{2}}}=|u^{2}ds_{[31]_{F_{2}}}\rangle\otimes|\bar{s}\rangle,$
(39)

For the configurations with $[v]_{F}=[22]_{F}$, the quark-antiquark pair can
be $d\bar{d}$ and $s\bar{s}$. To take into account the $SU(3)$ flavour
symmetry breaking effects, we treat these two kinds of $qqq(q\bar{q})$
configurations separately. And the flavour decompositions are

$\displaystyle|\frac{1}{2},\frac{1}{2}\rangle_{I}^{[22]_{F}}$ $\displaystyle=$
$\displaystyle|u^{2}d^{2}_{[22]_{F}}\rangle\otimes|\bar{d}\rangle,$ (40)
$\displaystyle|\frac{1}{2},\frac{1}{2}\rangle_{I}^{[22]_{F}}$ $\displaystyle=$
$\displaystyle|u^{2}ds_{[22]_{F}}\rangle\otimes|\bar{s}\rangle\,,$ (41)

At last, for the configurations with $[v]_{F}=[211]_{F}$, which limits the
quark-antiquark pair to be $s\bar{s}$, the flavour decomposition is

$|\frac{1}{2},\frac{1}{2}\rangle_{I}^{[211]_{F}}=|u^{2}ds_{[211]_{F}}\rangle\otimes|\bar{s}\rangle.$
(42)

Analogously, one can get the following flavour decompositions for the five-
quark configurations in neutron according to the isospin quark number:

$\displaystyle|\frac{1}{2},-\frac{1}{2}\rangle_{I}^{[31]_{F_{1}}}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{1}{3}}|u^{2}d^{2}_{[31]_{F_{1}}}\rangle\otimes|\bar{u}\rangle+$
(43)
$\displaystyle\sqrt{\frac{2}{3}}|ud^{3}_{[31]_{F_{1}}}\rangle\otimes|\bar{d}\rangle\,,$
$\displaystyle|\frac{1}{2},-\frac{1}{2}\rangle_{I}^{[31]_{F_{2}}}$
$\displaystyle=$
$\displaystyle|ud^{2}s_{[31]_{F_{2}}}\rangle\otimes|\bar{s}\rangle\,,$ (44)
$\displaystyle|\frac{1}{2},-\frac{1}{2}\rangle_{I}^{[22]_{F}}$
$\displaystyle=$
$\displaystyle-|u^{2}d^{2}_{[22]^{F}}\rangle\otimes|\bar{u}\rangle\,,$ (45)
$\displaystyle|\frac{1}{2},-\frac{1}{2}\rangle_{I}^{[22]_{F}}$
$\displaystyle=$
$\displaystyle|ud^{2}s_{[22]^{F}}\rangle\otimes|\bar{s}\rangle\,,$ (46)
$\displaystyle|\frac{1}{2},-\frac{1}{2}\rangle_{I}^{[211]_{F}}$
$\displaystyle=$
$\displaystyle|ud^{2}s_{[211]_{F}}\rangle\otimes|\bar{s}\rangle\,.$ (47)

###  A.2 $\Sigma$ baryons

For the $\Sigma^{+}$ baryon, the isospin wave function is $|1,1\rangle_{I}$.
In the five-quark configurations with $[v]_{F}=[31]_{F_{1}}$, both the light
and strange quark-antiquark pairs survive. The corresponding flavour
decompositions are

$\displaystyle|1,1\rangle_{I}^{[31]_{F_{1}}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{3}{4}}|u^{3}s_{[31]_{F_{1}}}\rangle\otimes|\bar{u}\rangle+$
(48)
$\displaystyle\sqrt{\frac{1}{4}}|u^{2}ds_{[31]_{F_{1}}}\rangle\otimes|\bar{d}\rangle\,,$
$\displaystyle|1,1\rangle_{I}^{[31]_{F_{1}}}$ $\displaystyle=$
$\displaystyle|u^{2}s^{2}_{[31]_{F_{1}}}\rangle\otimes|\bar{s}\rangle\,,$ (49)

respectively.

For the configurations with $[v]_{F}=[31]_{F_{2}}$, in present case, flavour
decomposition of the $uuss\bar{s}$ configuration is the same as the
configurations with $[v]_{F}=[31]_{F_{1}}$. And the flavour decomposition of
the five-quark configurations with light quark antiquark pair should be

$|1,1\rangle_{I}^{[31]_{F_{2}}}=|u^{2}ds_{[31]_{F_{2}}}\rangle\otimes|\bar{d}\rangle\,.$
(50)

For the configurations with $[v]_{F}=[22]_{F}$, both light and strangeness
five-quark components exist, and the flavor decompositions are

$\displaystyle|1,1\rangle_{I}^{[22]_{F}}$ $\displaystyle=$
$\displaystyle-|u^{2}ds_{[22]^{F}}\rangle\otimes|\bar{d}\rangle,$ (51)
$\displaystyle|1,1\rangle_{I}^{[22]_{F}}$ $\displaystyle=$
$\displaystyle|u^{2}s^{2}_{[22]^{F}}\rangle\otimes|\bar{s}\rangle,$ (52)

Finally, the configurations with $[v]_{F}=[211]_{F}$ rules out the strangeness
five-quark component in $\Sigma^{+}$, only $uuds\bar{d}$ component exists. And
the flavour decomposition reads

$|1,1\rangle_{I}^{[211]_{F}}=-|u^{2}ds_{[211]_{F}}\rangle\otimes|\bar{d}\rangle\,.$
(53)

Considering the isospin $SU(2)$ symmetry, one can obtain the flavour
decomposition

$\displaystyle|1,0\rangle_{I}^{[31]_{F_{1}}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}}|u^{2}ds_{[31]_{F_{1}}}\rangle\otimes|\bar{u}\rangle-$
(54)
$\displaystyle\sqrt{\frac{1}{2}}|ud^{2}s_{[31]_{F_{1}}}\rangle\otimes|\bar{d}\rangle\,,$
$\displaystyle|1,0\rangle_{I}^{[31]_{F_{1}}}$ $\displaystyle=$
$\displaystyle|uds^{2}_{[31]_{F_{1}}}\rangle\otimes|\bar{s}\rangle\,,$ (55)
$\displaystyle|1,0\rangle_{I}^{[31]_{F_{2}}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}}|u^{2}ds_{[31]_{F_{2}}}\rangle\otimes|\bar{u}\rangle-$
(56)
$\displaystyle\sqrt{\frac{1}{2}}|ud^{2}s_{[31]_{F_{2}}}\rangle\otimes|\bar{d}\rangle\,,$
$\displaystyle|1,0\rangle_{I}^{[22]_{F}}$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{1}{2}}|u^{2}ds_{[22]_{F}}\rangle\otimes|\bar{u}\rangle+$
(57)
$\displaystyle\sqrt{\frac{1}{2}}|ud^{2}s_{[22]_{F}}\rangle\otimes|\bar{d}\rangle\,,$
$\displaystyle|1,0\rangle_{I}^{[22]_{F}}$ $\displaystyle=$
$\displaystyle|uds^{2}_{[22]_{F}}\rangle\otimes|\bar{s}\rangle\,,$ (58)
$\displaystyle|1,0\rangle_{I}^{[211]_{F_{2}}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}}|u^{2}ds_{[211]_{F}}\rangle\otimes|\bar{u}\rangle-$
(59)
$\displaystyle\sqrt{\frac{1}{2}}|ud^{2}s_{[211]_{F}}\rangle\otimes|\bar{d}\rangle\,,$

for the five-quark components in the $\Sigma^{0}$ baryon, and

$\displaystyle|1,-1\rangle_{I}^{[31]_{F_{1}}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{4}}|ud^{2}s_{[31]_{F_{1}}}\rangle\otimes|\bar{u}\rangle+$
(60)
$\displaystyle\sqrt{\frac{3}{4}}|d^{3}s_{[31]_{F_{1}}}\rangle\otimes|\bar{d}\rangle\,,$
$\displaystyle|1,-1\rangle_{I}^{[31]_{F_{1}}}$ $\displaystyle=$
$\displaystyle|d^{2}s^{2}_{[31]_{F_{1}}}\rangle\otimes|\bar{s}\rangle\,,$ (61)
$\displaystyle|1,-1\rangle_{I}^{[31]_{F_{2}}}$ $\displaystyle=$
$\displaystyle|ud^{2}s_{[31]_{F_{2}}}\rangle\otimes|\bar{u}\rangle\,,$ (62)
$\displaystyle|1,-1\rangle_{I}^{[22]_{F}}$ $\displaystyle=$
$\displaystyle-|ud^{2}s_{[22]^{F}}\rangle\otimes|\bar{u}\rangle\,,$ (63)
$\displaystyle|1,-1\rangle_{I}^{[22]_{F}}$ $\displaystyle=$
$\displaystyle|d^{2}s^{2}_{[22]^{F}}\rangle\otimes|\bar{s}\rangle\,,$ (64)
$\displaystyle|1,-1\rangle_{I}^{[211]_{F}}$ $\displaystyle=$
$\displaystyle|ud^{2}s_{[211]_{F}}\rangle\otimes|\bar{u}\rangle\,,$ (65)

for the five-quark components in the $\Sigma^{-}$ baryon, respectively.

###  A.3 $\Xi$ baryon

The isospin wave function of the $\Xi^{+}$ baryon is $|1/2,1/2\rangle_{I}$.
Accordingly, for the configurations with $[v]_{F}=[31]_{F_{1}}$, both the
five-quark components with light and strange quark-antiquark pairs can exist.
Accordingly, the flavour decompositions are

$\displaystyle|\frac{1}{2},\frac{1}{2}\rangle_{I}^{[31]_{F_{1}}}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{2}{3}}|u^{2}s^{2}_{[31]_{F_{1}}}\rangle\otimes|\bar{u}\rangle+$
(66)
$\displaystyle\sqrt{\frac{1}{3}}|uds^{2}_{[31]_{F_{1}}}\rangle\otimes|\bar{d}\rangle\,,$
$\displaystyle|\frac{1}{2},\frac{1}{2}\rangle_{I}^{[31]_{F_{1}}}$
$\displaystyle=$
$\displaystyle|us^{3}_{[31]_{F_{1}}}\rangle\otimes|\bar{s}\rangle,$ (67)

respectively.

For the configurations with $[v]_{F}=[31]_{F_{2}}$, the $usss\bar{s}$
component is the same as the one with $[31]_{F_{1}}$. So we only have to
consider the light five-quark components, which should be

$|\frac{1}{2},\frac{1}{2}\rangle_{I}^{[31]_{F_{2}}}=|uds^{2}_{[31]_{F_{2}}}\rangle\otimes|\bar{d}\rangle\,.$
(68)

The flavour symmetry $[v]_{F}=[22]_{F}$ rules out the five-quark component in
$\Xi^{+}$ with strange quark-antiquark pair, and the flavour decomposition for
the $uss(q\bar{q})$ component is

$\displaystyle|\frac{1}{2},\frac{1}{2}\rangle_{I}^{[22]_{F}}$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{2}{3}}|u^{2}s^{2}_{[22]_{F}}\rangle\otimes|\bar{u}\rangle-$
(69)
$\displaystyle\sqrt{\frac{1}{3}}|uds^{2}_{[22]_{F}}\rangle\otimes|\bar{d}\rangle\,.$

Finally, for the configurations with $[v]_{F}=[211]_{F}$, only $udss\bar{d}$
component exists in $\Xi^{+}$. And

$|\frac{1}{2},\frac{1}{2}\rangle_{I}^{[211]_{F}}=-|uds^{2}_{[211]_{F}}\rangle\otimes|\bar{d}\rangle\,.$
(70)

The flavour decompositions of the five-quark components in the $\Xi^{-}$
baryon can be obtained by considering the $SU(2)$ isospin symmetry as

$\displaystyle|\frac{1}{2},-\frac{1}{2}\rangle_{I}^{[31]_{F_{1}}}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{1}{3}}|uds^{2}_{[31]_{F_{1}}}\rangle\otimes|\bar{u}\rangle+$
(71)
$\displaystyle\sqrt{\frac{2}{3}}|d^{2}s^{2}_{[31]_{F_{1}}}\rangle\otimes|\bar{d}\rangle\,,$
$\displaystyle|\frac{1}{2},-\frac{1}{2}\rangle_{I}^{[31]_{F_{1}}}$
$\displaystyle=$
$\displaystyle|ds^{3}_{[31]_{F_{1}}}\rangle\otimes|\bar{s}\rangle\,,$ (72)
$\displaystyle|\frac{1}{2},-\frac{1}{2}\rangle_{I}^{[31]_{F_{2}}}$
$\displaystyle=$
$\displaystyle|uds^{2}_{[31]_{F_{2}}}\rangle\otimes|\bar{u}\rangle\,,$ (73)
$\displaystyle|\frac{1}{2},-\frac{1}{2}\rangle_{I}^{[22]_{F}}$
$\displaystyle=$
$\displaystyle-\sqrt{\frac{1}{3}}|uds^{2}_{[22]_{F}}\rangle\otimes|\bar{u}\rangle-$
(74)
$\displaystyle\sqrt{\frac{2}{3}}|d^{2}s^{2}_{[22]_{F}}\rangle\otimes|\bar{d}\rangle\,,$
$\displaystyle|\frac{1}{2},-\frac{1}{2}\rangle_{I}^{[211]_{F}}$
$\displaystyle=$
$\displaystyle|uds^{2}_{[211]_{F}}\rangle\otimes|\bar{u}\rangle\,.$ (75)

###  A.4 $\Lambda$ baryon

In the $\Lambda$ baryon, isospin zero rules out the five-quark configurations
with $[v]_{F}=[31]_{F_{1}}$. For the configurations with
$[v]_{F}=[31]_{F_{2}}$, the light five-quark components should be

$\displaystyle|0,0\rangle_{I}^{[31]_{F_{2}}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}}|u^{2}ds_{[31]_{F_{2}}}\rangle\otimes|\bar{u}\rangle+$
(76)
$\displaystyle\sqrt{\frac{1}{2}}|ud^{2}s_{[31]_{F_{2}}}\rangle\otimes|\bar{d}\rangle\,.$

Additionally, the strangeness five-quark component can also exist in
$\Lambda$, as

$|0,0\rangle_{I}^{[31]_{F_{2}}}=|uds^{2}_{[31]_{F_{2}}}\rangle\otimes|\bar{s}\rangle\,.$
(77)

For the five-quark configurations with $[v]_{F}=[22]_{F}$, only the ones with
light quark-antiquark pairs exist in $\Lambda$ baryon, and the flavour
decomposition is

$\displaystyle|0,0\rangle_{I}^{[22]_{F}}$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{1}{2}}|u^{2}ds_{[22]_{F}}\rangle\otimes|\bar{u}\rangle-$
(78)
$\displaystyle\sqrt{\frac{1}{2}}|ud^{2}s_{[22]_{F}}\rangle\otimes|\bar{d}\rangle.$

Finally, for the configurations with $[v]_{F}=[211]_{F}$, the light five-quark
components should be

$\displaystyle|0,0\rangle_{I}^{[211]_{F_{2}}}$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{1}{2}}|u^{2}ds_{[211]_{F}}\rangle\otimes|\bar{u}\rangle-$
(79)
$\displaystyle\sqrt{\frac{1}{2}}|ud^{2}s_{[211]_{F}}\rangle\otimes|\bar{d}\rangle\,,$

and the strangeness component is

$|0,0\rangle_{I}^{[211]_{F}}=|uds^{2}_{[211]_{F}}\rangle\otimes|\bar{s}\rangle\,.$
(80)

##  Appendix B Numerical results of each five-quark components
$\mathcal{P}{q\bar{q}}$ and $\Delta{f^{q}_{i}}$$(f=u,d,s)$ of octet baryons.

In present appendix, we show the numerical results on the matrix elements of
$\Delta{f^{q}_{i}}$$(f=u,d,s)$ for the octet baryons, as shown in Tables AI\-
AVIII.

Table AI: The numerical results of the probabilities $\mathcal{P}{q\bar{q}}$ and the matrix elements of the quark spin $\Delta f_{i}^{5q}$ of all the 17 five-quark configurations of proton in Set I and Set II. Note that we have denoted the five-quark configurations $uudq\bar{q}$ with light quark-antiquark pairs as $l\bar{l}$, and those with strange quark-antiquark pairs as $s\bar{s}$. |  | Set I | Set II
---|---|---|---
Con. $i$ | $q\bar{q}$ | $\mathcal{P}{q\bar{q}}$ | $\Delta u_{i}^{5q}$ | $\Delta d_{i}^{5q}$ | $\Delta s_{i}^{5q}$ | $\mathcal{P}{q\bar{q}}$ | $\Delta u_{i}^{5q}$ | $\Delta d_{i}^{5q}$ | $\Delta s_{i}^{5q}$
$1$ | $l\bar{l}$ | $0.146\pm 0.015$ | $0$ | $-1/3$ | $0$ | $0.157\pm 0.016$ | $0$ | $-1/3$ | $0$
$s\bar{s}$ | $0.010\pm 0.001$ | $0$ | $0$ | $-1/3$ | $0.011\pm 0.002$ | $0$ | $0$ | $-1/3$
$2$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0.004\pm 0.001$ | $0$ | $0$ | $-1/3$ | $0.004\pm 0.001$ | $0$ | $0$ | $-1/3$
$3$ | $l\bar{l}$ | $0.016\pm 0.002$ | $-2/9$ | $-1/9$ | $0$ | $0.018\pm 0.002$ | $-2/9$ | $-1/9$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$4$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0.003\pm 0.001$ | $0$ | $0$ | $-1/3$ | $0.003\pm 0.001$ | $0$ | $0$ | $-1/3$
$5$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0.009\pm 0.001$ | $0$ | $0$ | $-1/3$ | $0.009\pm 0.001$ | $0$ | $0$ | $-1/3$
$6$ | $l\bar{l}$ | $0.041\pm 0.004$ | $-2/9$ | $-1/9$ | $0$ | $0.045\pm 0.005$ | $-2/9$ | $-1/9$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$7$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0.007\pm 0.001$ | $0$ | $0$ | $-1/3$ | $0.007\pm 0.001$ | $0$ | $0$ | $-1/3$
$8$ | $l\bar{l}$ | $0.073\pm 0.007$ | $2/3$ | $1/3$ | $0$ | $0.157\pm 0.016$ | $4/9$ | $-1/9$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$9$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0.006\pm 0.001$ | $0$ | $0$ | $1$ | $0.014\pm 0.002$ | $4/9$ | $-1/9$ | $0$
$10$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0.003\pm 0.001$ | $0$ | $0$ | $1$ | $0.007\pm 0.001$ | $1/2$ | $1/12$ | $-1/4$
$11$ | $l\bar{l}$ | $0.006\pm 0.001$ | $0$ | $1$ | $0$ | $0.013\pm 0.001$ | $1/3$ | $0$ | $0$
$s\bar{s}$ | $0.002\pm 0.000$ | $0$ | $0$ | $1$ | $0.004\pm 0.001$ | $1/3$ | $1/6$ | $-1/6$
$12$ | $l\bar{l}$ | $0.005\pm 0.001$ | $2/3$ | $1/3$ | $0$ | $0.010\pm 0.001$ | $1/3$ | $0$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$13$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0.001\pm 0.000$ | $0$ | $0$ | $1$ | $0.002\pm 0.001$ | $7/18$ | $1/36$ | $-1/12$
$14$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0.008\pm 0.001$ | $0$ | $0$ | $1$ | $0.017\pm 0.002$ | $1/2$ | $1/12$ | $-1/4$
$15$ | $l\bar{l}$ | $0.015\pm 0.002$ | $0$ | $1$ | $0$ | $0.032\pm 0.003$ | $1/3$ | $0$ | $0$
$s\bar{s}$ | $0.004\pm 0.001$ | $0$ | $0$ | $1$ | $0.009\pm 0.001$ | $1/3$ | $1/6$ | $-1/6$
$16$ | $l\bar{l}$ | $0.012\pm 0.001$ | $2/3$ | $1/3$ | $0$ | $0.025\pm 0.002$ | $1/3$ | $0$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$17$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0.002\pm 0.000$ | $0$ | $0$ | $1$ | $0.004\pm 0.001$ | $7/18$ | $1/36$ | $-1/12$
Table AII: The numerical results of the probabilities $\mathcal{P}{q\bar{q}}$ and the matrix elements of the quark spin $\Delta f_{i}^{5q}$ of the neutron, conventions are the same as in Table AI. |  | Set I | Set II
---|---|---|---
Con. $i$ | $q\bar{q}$ | $\mathcal{P}{q\bar{q}}$ | $\Delta u_{i}^{5q}$ | $\Delta d_{i}^{5q}$ | $\Delta s_{i}^{5q}$ | $\mathcal{P}{q\bar{q}}$ | $\Delta u_{i}^{5q}$ | $\Delta d_{i}^{5q}$ | $\Delta s_{i}^{5q}$
$1$ | $l\bar{l}$ | $0.146\pm 0.015$ | $-1/3$ | $0$ | $0$ | $0.157\pm 0.016$ | $-1/3$ | $0$ | $0$
$s\bar{s}$ | $0.010\pm 0.001$ | $0$ | $0$ | $-1/3$ | $0.011\pm 0.002$ | $0$ | $0$ | $-1/3$
$2$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0.004\pm 0.001$ | $0$ | $0$ | $-1/3$ | $0.004\pm 0.001$ | $0$ | $0$ | $-1/3$
$3$ | $l\bar{l}$ | $0.016\pm 0.002$ | $-1/9$ | $-2/9$ | $0$ | $0.018\pm 0.002$ | $-1/9$ | $-2/9$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$4$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0.003\pm 0.001$ | $0$ | $0$ | $-1/3$ | $0.003\pm 0.001$ | $0$ | $0$ | $-1/3$
$5$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0.009\pm 0.001$ | $0$ | $0$ | $-1/3$ | $0.009\pm 0.001$ | $0$ | $0$ | $-1/3$
$6$ | $l\bar{l}$ | $0.041\pm 0.004$ | $-1/9$ | $-2/9$ | $0$ | $0.045\pm 0.005$ | $-1/9$ | $-2/9$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$7$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0.007\pm 0.001$ | $0$ | $0$ | $-1/3$ | $0.007\pm 0.001$ | $0$ | $0$ | $-1/3$
$8$ | $l\bar{l}$ | $0.073\pm 0.007$ | $1/3$ | $2/3$ | $0$ | $0.157\pm 0.016$ | $-1/9$ | $4/9$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$9$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0.006\pm 0.001$ | $0$ | $0$ | $1$ | $0.014\pm 0.002$ | $-1/9$ | $4/9$ | $0$
$10$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0.003\pm 0.001$ | $0$ | $0$ | $1$ | $0.007\pm 0.001$ | $1/12$ | $1/2$ | $-1/4$
$11$ | $l\bar{l}$ | $0.006\pm 0.001$ | $1$ | $0$ | $0$ | $0.013\pm 0.001$ | $0$ | $1/3$ | $0$
$s\bar{s}$ | $0.002\pm 0.000$ | $0$ | $0$ | $1$ | $0.004\pm 0.001$ | $1/6$ | $1/3$ | $-1/6$
$12$ | $l\bar{l}$ | $0.005\pm 0.001$ | $1/3$ | $2/3$ | $0$ | $0.010\pm 0.001$ | $0$ | $1/3$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$13$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0.001\pm 0.000$ | $0$ | $0$ | $1$ | $0.002\pm 0.001$ | $1/36$ | $7/18$ | $-1/12$
$14$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0.008\pm 0.001$ | $0$ | $0$ | $1$ | $0.017\pm 0.002$ | $1/12$ | $1/2$ | $-1/4$
$15$ | $l\bar{l}$ | $0.015\pm 0.002$ | $1$ | $0$ | $0$ | $0.032\pm 0.003$ | $0$ | $1/3$ | $0$
$s\bar{s}$ | $0.004\pm 0.001$ | $0$ | $0$ | $1$ | $0.009\pm 0.001$ | $1/6$ | $1/3$ | $-1/6$
$16$ | $l\bar{l}$ | $0.012\pm 0.001$ | $1/3$ | $2/3$ | $0$ | $0.025\pm 0.002$ | $0$ | $1/3$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$17$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0.002\pm 0.000$ | $0$ | $0$ | $1$ | $0.004\pm 0.001$ | $1/36$ | $7/18$ | $-1/12$
Table AIII: The numerical results of the probabilities $\mathcal{P}{q\bar{q}}$ and the matrix elements of the quark spin $\Delta f_{i}^{5q}$ of $\Sigma^{+}$, conventions are the same as in Table AI. |  | Set I | Set II
---|---|---|---
Con. $i$ | $q\bar{q}$ | $\mathcal{P}{q\bar{q}}$ | $\Delta u_{i}^{5q}$ | $\Delta d_{i}^{5q}$ | $\Delta s_{i}^{5q}$ | $\mathcal{P}{q\bar{q}}$ | $\Delta u_{i}^{5q}$ | $\Delta d_{i}^{5q}$ | $\Delta s_{i}^{5q}$
$1$ | $l\bar{l}$ | $0.066\pm 0.007$ | $0$ | $-1/3$ | $0$ | $0.071\pm 0.008$ | $0$ | $-1/3$ | $0$
$s\bar{s}$ | $0.020\pm 0.002$ | $0$ | $0$ | $-1/3$ | $0.021\pm 0.003$ | $0$ | $0$ | $-1/3$
$2$ | $l\bar{l}$ | $0.010\pm 0.001$ | $0$ | $-1/3$ | $0$ | $0.010\pm 0.001$ | $0$ | $-1/3$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$3$ | $l\bar{l}$ | $0.017\pm 0.002$ | $-1/4$ | $-1/12$ | $0$ | $0.018\pm 0.002$ | $-1/4$ | $-1/12$ | $0$
$s\bar{s}$ | $0.004\pm 0.001$ | $0$ | $0$ | $-1/3$ | $0.004\pm 0.001$ | $0$ | $0$ | $-1/3$
$4$ | $l\bar{l}$ | $0.001\pm 0.000$ | $0$ | $-1/3$ | $0$ | $0.001\pm 0.000$ | $0$ | $-1/3$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$5$ | $l\bar{l}$ | $0.022\pm 0.002$ | $0$ | $-1/3$ | $0$ | $0.023\pm 0.002$ | $0$ | $-1/3$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$6$ | $l\bar{l}$ | $0.043\pm 0.005$ | $-1/4$ | $-1/12$ | $0$ | $0.046\pm 0.005$ | $-1/4$ | $-1/12$ | $0$
$s\bar{s}$ | $0.010\pm 0.001$ | $0$ | $0$ | $-1/3$ | $0.011\pm 0.001$ | $0$ | $0$ | $-1/3$
$7$ | $l\bar{l}$ | $0.002\pm 0.001$ | $0$ | $-1/3$ | $0$ | $0.002\pm 0.000$ | $0$ | $-1/3$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$8$ | $l\bar{l}$ | $0.054\pm 0.006$ | $3/4$ | $1/4$ | $0$ | $0.117\pm 0.013$ | $37/72$ | $-1/72$ | $-12/72$
$s\bar{s}$ | $0.008\pm 0.001$ | $0$ | $0$ | $1$ | $0.018\pm 0.002$ | $1/3$ | $0$ | $0$
$9$ | $l\bar{l}$ | $0.003\pm 0.001$ | $0$ | $1$ | $0$ | $0.006\pm 0.001$ | $4/9$ | $-4/9$ | $1/3$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$10$ | $l\bar{l}$ | $0.008\pm 0.001$ | $0$ | $1$ | $0$ | $0.017\pm 0.002$ | $1/2$ | $-1/4$ | $1/12$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$11$ | $l\bar{l}$ | $0.004\pm 0.000$ | $0$ | $1$ | $0$ | $0.009\pm 0.001$ | $1/3$ | $-1/6$ | $1/6$
$s\bar{s}$ | $0.004\pm 0.001$ | $0$ | $0$ | $1$ | $0.010\pm 0.002$ | $1/3$ | $0$ | $0$
$12$ | $l\bar{l}$ | $0.005\pm 0.001$ | $3/4$ | $1/4$ | $0$ | $0.011\pm 0.002$ | $13/36$ | $-1/36$ | $0$
$s\bar{s}$ | $0.001\pm 0.000$ | $0$ | $0$ | $1$ | $0.002\pm 0.001$ | $1/3$ | $0$ | $0$
$13$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$14$ | $l\bar{l}$ | $0.019\pm 0.002$ | $0$ | $1$ | $0$ | $0.040\pm 0.004$ | $1/2$ | $-1/4$ | $1/12$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$15$ | $l\bar{l}$ | $0.010\pm 0.001$ | $0$ | $1$ | $0$ | $0.021\pm 0.002$ | $1/3$ | $-1/6$ | $1/6$
$s\bar{s}$ | $0.011\pm 0.001$ | $0$ | $0$ | $1$ | $0.024\pm 0.003$ | $1/3$ | $0$ | $0$
$16$ | $l\bar{l}$ | $0.013\pm 0.002$ | $3/4$ | $1/4$ | $0$ | $0.029\pm 0.003$ | $13/36$ | $-1/36$ | $0$
$s\bar{s}$ | $0.003\pm 0.000$ | $0$ | $0$ | $1$ | $0.007\pm 0.001$ | $1/3$ | $0$ | $0$
$17$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0.001\pm 0.000$ | $7/18$ | $-1/12$ | $1/36$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
Table AIV: The numerical results of the probabilities $\mathcal{P}{q\bar{q}}$ and the matrix elements of the quark spin $\Delta f_{i}^{5q}$ of $\Sigma^{0}$, conventions are the same as in Table AI. |  | Set I | Set II
---|---|---|---
Con. $i$ | $q\bar{q}$ | $\mathcal{P}{q\bar{q}}$ | $\Delta u_{i}^{5q}$ | $\Delta d_{i}^{5q}$ | $\Delta s_{i}^{5q}$ | $\mathcal{P}{q\bar{q}}$ | $\Delta u_{i}^{5q}$ | $\Delta d_{i}^{5q}$ | $\Delta s_{i}^{5q}$
$1$ | $l\bar{l}$ | $0.066\pm 0.007$ | $-1/6$ | $-1/6$ | $0$ | $0.071\pm 0.008$ | $-1/6$ | $-1/6$ | $0$
$s\bar{s}$ | $0.020\pm 0.002$ | $0$ | $0$ | $-1/3$ | $0.021\pm 0.003$ | $0$ | $0$ | $-1/3$
$2$ | $l\bar{l}$ | $0.010\pm 0.001$ | $-1/6$ | $-1/6$ | $0$ | $0.010\pm 0.001$ | $-1/6$ | $-1/6$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$3$ | $l\bar{l}$ | $0.017\pm 0.002$ | $-1/6$ | $-1/6$ | $0$ | $0.018\pm 0.002$ | $-1/6$ | $-1/6$ | $0$
$s\bar{s}$ | $0.004\pm 0.001$ | $0$ | $0$ | $-1/3$ | $0.004\pm 0.001$ | $0$ | $0$ | $-1/3$
$4$ | $l\bar{l}$ | $0.001\pm 0.000$ | $-1/6$ | $-1/6$ | $0$ | $0.001\pm 0.000$ | $-1/6$ | $-1/6$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$5$ | $l\bar{l}$ | $0.022\pm 0.002$ | $-1/6$ | $-1/6$ | $0$ | $0.023\pm 0.002$ | $-1/6$ | $-1/6$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$6$ | $l\bar{l}$ | $0.043\pm 0.005$ | $-1/6$ | $-1/6$ | $0$ | $0.046\pm 0.005$ | $-1/6$ | $-1/6$ | $0$
$s\bar{s}$ | $0.010\pm 0.001$ | $0$ | $0$ | $-1/3$ | $0.011\pm 0.001$ | $0$ | $0$ | $-1/3$
$7$ | $l\bar{l}$ | $0.002\pm 0.001$ | $-1/6$ | $-1/6$ | $0$ | $0.002\pm 0.000$ | $-1/6$ | $-1/6$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$8$ | $l\bar{l}$ | $0.054\pm 0.006$ | $1/2$ | $1/2$ | $0$ | $0.117\pm 0.013$ | $1/4$ | $1/4$ | $-1/6$
$s\bar{s}$ | $0.008\pm 0.001$ | $0$ | $0$ | $1$ | $0.018\pm 0.002$ | $1/6$ | $1/6$ | $0$
$9$ | $l\bar{l}$ | $0.003\pm 0.001$ | $1/2$ | $1/2$ | $0$ | $0.006\pm 0.001$ | $0$ | $0$ | $1/3$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$10$ | $l\bar{l}$ | $0.008\pm 0.001$ | $1/2$ | $1/2$ | $0$ | $0.017\pm 0.002$ | $1/8$ | $1/8$ | $1/12$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$11$ | $l\bar{l}$ | $0.004\pm 0.000$ | $1/2$ | $1/2$ | $0$ | $0.009\pm 0.001$ | $1/12$ | $1/12$ | $1/6$
$s\bar{s}$ | $0.004\pm 0.001$ | $0$ | $0$ | $1$ | $0.010\pm 0.002$ | $1/6$ | $1/6$ | $0$
$12$ | $l\bar{l}$ | $0.005\pm 0.001$ | $1/2$ | $1/2$ | $0$ | $0.011\pm 0.002$ | $1/6$ | $1/6$ | $0$
$s\bar{s}$ | $0.001\pm 0.000$ | $0$ | $0$ | $1$ | $0.002\pm 0.001$ | $1/6$ | $1/6$ | $0$
$13$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$14$ | $l\bar{l}$ | $0.019\pm 0.002$ | $1/2$ | $1/2$ | $0$ | $0.040\pm 0.004$ | $1/8$ | $1/8$ | $1/12$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$15$ | $l\bar{l}$ | $0.010\pm 0.001$ | $1/2$ | $1/2$ | $0$ | $0.021\pm 0.002$ | $1/12$ | $1/12$ | $1/6$
$s\bar{s}$ | $0.011\pm 0.001$ | $0$ | $0$ | $1$ | $0.024\pm 0.003$ | $1/6$ | $1/6$ | $0$
$16$ | $l\bar{l}$ | $0.013\pm 0.002$ | $1/2$ | $1/2$ | $0$ | $0.029\pm 0.003$ | $1/6$ | $1/6$ | $0$
$s\bar{s}$ | $0.003\pm 0.000$ | $0$ | $0$ | $1$ | $0.007\pm 0.001$ | $1/6$ | $1/6$ | $0$
$17$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0.001\pm 0.000$ | $1/24$ | $1/24$ | $1/4$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
Table AV: The numerical results of the probabilities $\mathcal{P}{q\bar{q}}$ and the matrix elements of the quark spin $\Delta f_{i}^{5q}$ of $\Sigma^{-}$, conventions are the same as in Table AI. |  | Set I | Set II
---|---|---|---
Con. $i$ | $q\bar{q}$ | $\mathcal{P}{q\bar{q}}$ | $\Delta u_{i}^{5q}$ | $\Delta d_{i}^{5q}$ | $\Delta s_{i}^{5q}$ | $\mathcal{P}{q\bar{q}}$ | $\Delta u_{i}^{5q}$ | $\Delta d_{i}^{5q}$ | $\Delta s_{i}^{5q}$
$1$ | $l\bar{l}$ | $0.066\pm 0.007$ | $-1/3$ | $0$ | $0$ | $0.071\pm 0.008$ | $-1/3$ | $0$ | $0$
$s\bar{s}$ | $0.020\pm 0.002$ | $0$ | $0$ | $-1/3$ | $0.021\pm 0.003$ | $0$ | $0$ | $-1/3$
$2$ | $l\bar{l}$ | $0.010\pm 0.001$ | $-1/3$ | $0$ | $0$ | $0.010\pm 0.001$ | $-1/3$ | $0$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$3$ | $l\bar{l}$ | $0.017\pm 0.002$ | $-1/12$ | $-1/4$ | $0$ | $0.018\pm 0.002$ | $-1/12$ | $-1/4$ | $0$
$s\bar{s}$ | $0.004\pm 0.001$ | $0$ | $0$ | $-1/3$ | $0.004\pm 0.001$ | $0$ | $0$ | $-1/3$
$4$ | $l\bar{l}$ | $0.001\pm 0.000$ | $-1/3$ | $0$ | $0$ | $0.001\pm 0.000$ | $-1/3$ | $0$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$5$ | $l\bar{l}$ | $0.022\pm 0.002$ | $-1/3$ | $0$ | $0$ | $0.023\pm 0.002$ | $-1/3$ | $0$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$6$ | $l\bar{l}$ | $0.043\pm 0.005$ | $-1/12$ | $-1/4$ | $0$ | $0.046\pm 0.005$ | $-1/12$ | $-1/4$ | $0$
$s\bar{s}$ | $0.010\pm 0.001$ | $0$ | $0$ | $-1/3$ | $0.011\pm 0.001$ | $0$ | $0$ | $-1/3$
$7$ | $l\bar{l}$ | $0.002\pm 0.001$ | $-1/3$ | $0$ | $0$ | $0.002\pm 0.000$ | $-1/3$ | $0$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$8$ | $l\bar{l}$ | $0.054\pm 0.006$ | $1/4$ | $3/4$ | $0$ | $0.117\pm 0.013$ | $-1/72$ | $37/72$ | $-12/72$
$s\bar{s}$ | $0.008\pm 0.001$ | $0$ | $0$ | $1$ | $0.018\pm 0.002$ | $0$ | $1/3$ | $0$
$9$ | $l\bar{l}$ | $0.003\pm 0.001$ | $1$ | $0$ | $0$ | $0.006\pm 0.001$ | $-4/9$ | $4/9$ | $1/3$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$10$ | $l\bar{l}$ | $0.008\pm 0.001$ | $1$ | $0$ | $0$ | $0.017\pm 0.002$ | $-1/4$ | $1/2$ | $1/12$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$11$ | $l\bar{l}$ | $0.004\pm 0.000$ | $1$ | $0$ | $0$ | $0.009\pm 0.001$ | $-1/6$ | $1/3$ | $1/6$
$s\bar{s}$ | $0.004\pm 0.001$ | $0$ | $0$ | $1$ | $0.010\pm 0.002$ | $0$ | $1/3$ | $0$
$12$ | $l\bar{l}$ | $0.005\pm 0.001$ | $1/4$ | $3/4$ | $0$ | $0.011\pm 0.002$ | $-1/36$ | $13/36$ | $0$
$s\bar{s}$ | $0.001\pm 0.000$ | $0$ | $0$ | $1$ | $0.002\pm 0.001$ | $0$ | $1/3$ | $0$
$13$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$14$ | $l\bar{l}$ | $0.019\pm 0.002$ | $1$ | $0$ | $0$ | $0.040\pm 0.004$ | $-1/4$ | $1/2$ | $1/12$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$15$ | $l\bar{l}$ | $0.010\pm 0.001$ | $1$ | $0$ | $0$ | $0.021\pm 0.002$ | $-1/6$ | $1/3$ | $1/6$
$s\bar{s}$ | $0.011\pm 0.001$ | $0$ | $0$ | $1$ | $0.024\pm 0.003$ | $0$ | $1/3$ | $0$
$16$ | $l\bar{l}$ | $0.013\pm 0.002$ | $1/4$ | $3/4$ | $0$ | $0.029\pm 0.003$ | $-1/36$ | $13/36$ | $0$
$s\bar{s}$ | $0.003\pm 0.000$ | $0$ | $0$ | $1$ | $0.007\pm 0.001$ | $0$ | $1/3$ | $0$
$17$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0.001\pm 0.000$ | $-1/12$ | $7/18$ | $1/36$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
Table AVI: The numerical results of the probabilities $\mathcal{P}{q\bar{q}}$ and the matrix elements of the quark spin $\Delta f_{i}^{5q}$ of $\Xi^{0}$, conventions are the same as in Table AI. |  | Set I | Set II
---|---|---|---
Con. $i$ | $q\bar{q}$ | $\mathcal{P}{q\bar{q}}$ | $\Delta u_{i}^{5q}$ | $\Delta d_{i}^{5q}$ | $\Delta s_{i}^{5q}$ | $\mathcal{P}{q\bar{q}}$ | $\Delta u_{i}^{5q}$ | $\Delta d_{i}^{5q}$ | $\Delta s_{i}^{5q}$
$1$ | $l\bar{l}$ | $0.083\pm 0.009$ | $-2/9$ | $-1/9$ | $0$ | $0.094\pm 0.011$ | $-2/9$ | $-1/9$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$2$ | $l\bar{l}$ | $0.010\pm 0.001$ | $0$ | $-1/3$ | $0$ | $0.011\pm 0.001$ | $0$ | $-1/3$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$3$ | $l\bar{l}$ | $0.011\pm 0.002$ | $-2/9$ | $-1/9$ | $0$ | $0.012\pm 0.001$ | $-2/9$ | $-1/9$ | $0$
$s\bar{s}$ | $0.009\pm 0.001$ | $0$ | $0$ | $-1/3$ | $0.010\pm 0.002$ | $0$ | $0$ | $-1/3$
$4$ | $l\bar{l}$ | $0.002\pm 0.000$ | $0$ | $-1/3$ | $0$ | $0.002\pm 0.001$ | $0$ | $-1/3$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$5$ | $l\bar{l}$ | $0.022\pm 0.002$ | $0$ | $-1/3$ | $0$ | $0.025\pm 0.003$ | $0$ | $-1/3$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$6$ | $l\bar{l}$ | $0.027\pm 0.003$ | $-2/9$ | $-1/9$ | $0$ | $0.030\pm 0.004$ | $-2/9$ | $-1/9$ | $0$
$s\bar{s}$ | $0.023\pm 0.002$ | $0$ | $0$ | $-1/3$ | $0.026\pm 0.003$ | $0$ | $0$ | $-1/3$
$7$ | $l\bar{l}$ | $0.005\pm 0.001$ | $0$ | $-1/3$ | $0$ | $0.006\pm 0.001$ | $0$ | $-1/3$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$8$ | $l\bar{l}$ | $0.028\pm 0.003$ | $2/3$ | $1/3$ | $0$ | $0.064\pm 0.007$ | $1/18$ | $-1/18$ | $1/3$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$9$ | $l\bar{l}$ | $0.006\pm 0.001$ | $0$ | $1$ | $0$ | $0.014\pm 0.002$ | $0$ | $-1/3$ | $2/3$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$10$ | $l\bar{l}$ | $0.008\pm 0.001$ | $0$ | $1$ | $0$ | $0.018\pm 0.002$ | $1/12$ | $-1/4$ | $1/2$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$11$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$12$ | $l\bar{l}$ | $0.003\pm 0.001$ | $2/3$ | $1/3$ | $0$ | $0.006\pm 0.001$ | $1/18$ | $-1/18$ | $1/3$
$s\bar{s}$ | $0.003\pm 0.001$ | $0$ | $0$ | $1$ | $0.006\pm 0.001$ | $0$ | $0$ | $1/3$
$13$ | $l\bar{l}$ | $0.001\pm 0.001$ | $0$ | $1$ | $0$ | $0.001\pm 0.000$ | $1/12$ | $-1/4$ | $1/2$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$14$ | $l\bar{l}$ | $0.018\pm 0.002$ | $0$ | $1$ | $0$ | $0.041\pm 0.005$ | $1/12$ | $-1/4$ | $1/2$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$15$ | $l\bar{l}$ | $0.028\pm 0.003$ | $2/3$ | $1/3$ | $0$ | $0.065\pm 0.008$ | $1/18$ | $-1/18$ | $1/3$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$16$ | $l\bar{l}$ | $0.008\pm 0.001$ | $2/3$ | $1/3$ | $0$ | $0.018\pm 0.002$ | $1/18$ | $-1/18$ | $1/3$
$s\bar{s}$ | $0.007\pm 0.001$ | $0$ | $0$ | $1$ | $0.015\pm 0.002$ | $0$ | $0$ | $1/3$
$17$ | $l\bar{l}$ | $0.001\pm.0.001$ | $0$ | $1$ | $0$ | $0.003\pm 0.001$ | $1/12$ | $-1/4$ | $1/2$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
Table AVII: The numerical results of the probabilities $\mathcal{P}{q\bar{q}}$ and the matrix elements of the quark spin $\Delta f_{i}^{5q}$ of $\Xi^{-}$, conventions are the same as in Table AI. |  | Set I | Set II
---|---|---|---
Con. $i$ | $q\bar{q}$ | $\mathcal{P}{q\bar{q}}$ | $\Delta u_{i}^{5q}$ | $\Delta d_{i}^{5q}$ | $\Delta s_{i}^{5q}$ | $\mathcal{P}{q\bar{q}}$ | $\Delta u_{i}^{5q}$ | $\Delta d_{i}^{5q}$ | $\Delta s_{i}^{5q}$
$1$ | $l\bar{l}$ | $0.083\pm 0.009$ | $-1/9$ | $-2/9$ | $0$ | $0.094\pm 0.011$ | $-1/9$ | $-2/9$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$2$ | $l\bar{l}$ | $0.010\pm 0.001$ | $-1/3$ | $0$ | $0$ | $0.011\pm 0.001$ | $-1/3$ | $0$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$3$ | $l\bar{l}$ | $0.011\pm 0.002$ | $-1/9$ | $-2/9$ | $0$ | $0.012\pm 0.001$ | $-1/9$ | $-2/9$ | $0$
$s\bar{s}$ | $0.009\pm 0.001$ | $0$ | $0$ | $-1/3$ | $0.010\pm 0.002$ | $0$ | $0$ | $-1/3$
$4$ | $l\bar{l}$ | $0.002\pm 0.000$ | $-1/3$ | $0$ | $0$ | $0.002\pm 0.001$ | $-1/3$ | $0$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$5$ | $l\bar{l}$ | $0.022\pm 0.002$ | $-1/3$ | $0$ | $0$ | $0.025\pm 0.003$ | $-1/3$ | $0$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$6$ | $l\bar{l}$ | $0.027\pm 0.003$ | $-1/9$ | $-2/9$ | $0$ | $0.030\pm 0.004$ | $-1/9$ | $-2/9$ | $0$
$s\bar{s}$ | $0.023\pm 0.002$ | $0$ | $0$ | $-1/3$ | $0.026\pm 0.003$ | $0$ | $0$ | $-1/3$
$7$ | $l\bar{l}$ | $0.005\pm 0.001$ | $-1/3$ | $0$ | $0$ | $0.006\pm 0.001$ | $-1/3$ | $0$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$8$ | $l\bar{l}$ | $0.028\pm 0.003$ | $1/3$ | $2/3$ | $0$ | $0.064\pm 0.007$ | $-1/18$ | $1/18$ | $1/3$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$9$ | $l\bar{l}$ | $0.006\pm 0.001$ | $1$ | $0$ | $0$ | $0.014\pm 0.002$ | $-1/3$ | $0$ | $2/3$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$10$ | $l\bar{l}$ | $0.008\pm 0.001$ | $1$ | $0$ | $0$ | $0.018\pm 0.002$ | $-1/4$ | $1/12$ | $1/2$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$11$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$12$ | $l\bar{l}$ | $0.003\pm 0.001$ | $1/3$ | $2/3$ | $0$ | $0.006\pm 0.001$ | $-1/18$ | $1/18$ | $1/3$
$s\bar{s}$ | $0.003\pm 0.001$ | $0$ | $0$ | $1$ | $0.006\pm 0.001$ | $0$ | $0$ | $1/3$
$13$ | $l\bar{l}$ | $0.001\pm 0.001$ | $1$ | $0$ | $0$ | $0.001\pm 0.000$ | $-1/4$ | $1/12$ | $1/2$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$14$ | $l\bar{l}$ | $0.018\pm 0.002$ | $1$ | $0$ | $0$ | $0.041\pm 0.005$ | $-1/4$ | $1/12$ | $1/2$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$15$ | $l\bar{l}$ | $0.028\pm 0.003$ | $1/3$ | $2/3$ | $0$ | $0.065\pm 0.008$ | $-1/18$ | $1/18$ | $1/3$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$16$ | $l\bar{l}$ | $0.008\pm 0.001$ | $1/3$ | $2/3$ | $0$ | $0.018\pm 0.002$ | $-1/18$ | $1/18$ | $1/3$
$s\bar{s}$ | $0.007\pm 0.001$ | $0$ | $0$ | $1$ | $0.015\pm 0.002$ | $0$ | $0$ | $1/3$
$17$ | $l\bar{l}$ | $0.001\pm.0.001$ | $1$ | $0$ | $0$ | $0.003\pm 0.001$ | $-1/4$ | $1/12$ | $1/2$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
Table AVIII: The numerical results of the probabilities $\mathcal{P}{q\bar{q}}$ and the matrix elements of the quark spin $\Delta f_{i}^{5q}$ of $\Lambda$, conventions are the same as in Table AI. |  | Set I | Set II
---|---|---|---
Con. $i$ | $q\bar{q}$ | $\mathcal{P}{q\bar{q}}$ | $\Delta u_{i}^{5q}$ | $\Delta d_{i}^{5q}$ | $\Delta s_{i}^{5q}$ | $\mathcal{P}{q\bar{q}}$ | $\Delta u_{i}^{5q}$ | $\Delta d_{i}^{5q}$ | $\Delta s_{i}^{5q}$
$1$ | $l\bar{l}$ | $0.115\pm 0.013$ | $-1/6$ | $-1/6$ | $0$ | $0.128\pm 0.014$ | $-1/6$ | $-1/6$ | $0$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$2$ | $l\bar{l}$ | $0.003\pm 0.001$ | $-1/6$ | $-1/6$ | $0$ | $0.003\pm 0.000$ | $-1/6$ | $-1/6$ | $0$
$s\bar{s}$ | $0.003\pm 0.001$ | $0$ | $0$ | $-1/3$ | $0.003\pm 0.000$ | $0$ | $0$ | $-1/3$
$3$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$4$ | $l\bar{l}$ | $0.015\pm 0.001$ | $-1/6$ | $-1/6$ | $0$ | $0.016\pm 0.002$ | $-1/6$ | $-1/6$ | $0$
$s\bar{s}$ | $0.006\pm 0.001$ | $0$ | $0$ | $-1/3$ | $0.007\pm 0.001$ | $0$ | $0$ | $-1/3$
$5$ | $l\bar{l}$ | $0.006\pm 0.001$ | $-1/6$ | $-1/6$ | $0$ | $0.007\pm 0.001$ | $-1/6$ | $-1/6$ | $0$
$s\bar{s}$ | $0.006\pm 0.001$ | $0$ | $0$ | $-1/3$ | $0.007\pm 0..001$ | $0$ | $0$ | $-1/3$
$6$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$7$ | $l\bar{l}$ | $0.037\pm 0.004$ | $-1/6$ | $-1/6$ | $0$ | $0.041\pm 0.004$ | $-1/6$ | $-1/6$ | $0$
$s\bar{s}$ | $0.015\pm 0.002$ | $0$ | $0$ | $-1/3$ | $0.017\pm 0.002$ | $0$ | $0$ | $-1/3$
$8$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$9$ | $l\bar{l}$ | $0.053\pm 0.006$ | $1/2$ | $1/2$ | $0$ | $0.117\pm 0.013$ | $0$ | $0$ | $1/3$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$10$ | $l\bar{l}$ | $0.003\pm 0.001$ | $1/2$ | $1/2$ | $0$ | $0.006\pm 0.001$ | $1/8$ | $1/8$ | $1/12$
$s\bar{s}$ | $0.002\pm 0.001$ | $0$ | $0$ | $1$ | $0.005\pm 0.001$ | $1/12$ | $1/12$ | $1/6$
$11$ | $l\bar{l}$ | $0.010\pm 0.001$ | $1/2$ | $1/2$ | $0$ | $0.022\pm 0.002$ | $1/12$ | $1/12$ | $1/6$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$12$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$13$ | $l\bar{l}$ | $0.004\pm 0.001$ | $1/2$ | $1/2$ | $0$ | $0.009\pm 0.001$ | $1/24$ | $1/24$ | $1/4$
$s\bar{s}$ | $0.002\pm 0.000$ | $0$ | $0$ | $1$ | $0.004\pm 0.001$ | $1/12$ | $1/12$ | $1/6$
$14$ | $l\bar{l}$ | $0.005\pm 0.001$ | $1/2$ | $1/2$ | $0$ | $0.011\pm 0.002$ | $1/8$ | $1/8$ | $1/12$
$s\bar{s}$ | $0.006\pm 0.000$ | $0$ | $0$ | $1$ | $0.013\pm 0.001$ | $1/12$ | $1/12$ | $1/6$
$15$ | $l\bar{l}$ | $0.025\pm 0.002$ | $1/2$ | $1/2$ | $0$ | $0.055\pm 0.006$ | $1/12$ | $1/12$ | $1/6$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$16$ | $l\bar{l}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$s\bar{s}$ | $0$ | $-$ | $-$ | $-$ | $0$ | $-$ | $-$ | $-$
$17$ | $l\bar{l}$ | $0.010\pm 0.001$ | $1/2$ | $1/2$ | $0$ | $0.023\pm 0.002$ | $1/24$ | $1/24$ | $1/4$
$s\bar{s}$ | $0.004\pm 0.001$ | $0$ | $0$ | $1$ | $0.010\pm 0.001$ | $1/12$ | $1/12$ | $1/6$