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\begin{document}
\title{Hyperon semileptonic decays and quark spin content of the proton}
\author{Hyun-Chul Kim$^a$, 
Micha{\l} Prasza{\l}owicz$^b$, and Klaus Goeke$^c$}
\address{~\\
$^a$ Department of Physics, Pusan National University,\\
Pusan 609-735, Republic of Korea \\
$^b$ Institute of Physics, Jagellonian University, \\
ul. Reymonta 4, 30-059 Krak{\'o}w, Poland \\
$^c$ Institute for Theoretical Physics II, Ruhr-University Bochum, \\
D-44780 Bochum, Germany}
\date{\today}
\maketitle

\begin{abstract}
We investigate the hyperon semileptonic decays and the quark spin content 
of the proton $\Delta \Sigma$ taking into account flavor SU(3) symmetry
breaking. Symmetry breaking is implemented with the help of the
chiral quark-soliton model in an
approach, in which the dynamical parameters are fixed by the experimental
data for six hyperon semileptonic decay constants. As 
a result  we predict the unmeasured decay constants, particularly for 
$\Xi^0 \rightarrow \Sigma^+$, which will be soon measured
and examine the effect of the SU(3) symmetry breaking on the spin 
content $\Delta \Sigma $ of the proton. Unfortunately
large experimental errors of $\Xi^-$ decays propagate in our analysis
making $\Delta \Sigma$ and $\Delta s$ practically undetermined.
We conclude that statements concerning the values of these two quantities,
which are based on the exact SU(3) symmetry, are premature. We stress
that the meaningful results can be obtained only
if the experimental errors for the $\Xi$ decays are reduced. 
\end{abstract}

\pacs{PACS: 12.40.-y, 14.20.Dh}

\section{Introduction}

Since the European Muon Collaboration (EMC) measured the first moment $I_p^{%
{\rm EMC}}=0.112$ (at $Q^2$=3~(GeV/c)$^2$) of the proton spin structure
function $g^p_{1}$~\cite{EMC}, there has been a great deal of discussion
about the spin content of the proton. An  immediate and unexpected
consequence of the EMC measurement was that the quark contribution to 
the spin of the proton was very small ($\Delta \Sigma \approx 0$).
A series of following 
experiments~\cite{SMC,E143,SMC_d} confirmed the EMC measurement, 
giving, however a somewhat larger, but still small value for 
$\Delta \Sigma$.

This result is in contradiction with expectations based on the naive,
nonrelativistic quark model, supplemented by the assumption that the
contribution of strange quarks to $I_p$ was zero ($\Delta s=0$)~\cite
{EllisJaffe}. The EMC measurements required $\Delta s \ne 0$ and relatively
large. These two results: $\Delta \Sigma \approx 0$ and $\Delta s \ne 0$ are
often referred to as {\em spin crisis}. Let us shortly summarize how the
crisis arises.

Theoretical analysis of
recent measurements~\cite{EllKar} indicates that the $I_p$ is equal to:
\begin{equation}
I_p(Q^2=3~({\rm GeV}/c)^2)=0.124\pm 0.011.  \label{Ip}
\end{equation}
On the other hand the $I_p$ is related to the integrated 
polarized quark densities:
\begin{equation}
I_p=\frac 1{18}(4\Delta u+\Delta d+\Delta s)\left( 1-\frac{\alpha _{{\rm s}}}%
\pi +\ldots \right) .  \label{Ip2}
\end{equation}
Here for simplicity we neglect higher orders and higher twist contributions.
Comparing Eq.(\ref{Ip}) with Eq.(\ref{Ip2}) and assuming $\alpha _{{\rm s}%
}(Q^2=3~({\rm GeV}/c)^2)=0.4$~\cite{KarLip}, we get immediately:
\begin{equation}
\Gamma _p\equiv 4\Delta u+\Delta d+\Delta s=2.56\pm 0.23\,.  \label{Gampval}
\end{equation}
Let us quote here for completeness the experimental value for the
neutron~\cite{EllKar}:
\begin{equation}
\Gamma _n\equiv 4\Delta d+\Delta u+\Delta s=-0.928\pm 0.186\,.
\label{Gamnval}
\end{equation}
With this definition of $\Gamma _n$ the Bjorken sum rule is automatically
satisfied.

Integrated
polarized quark densities $\Delta q$ can be in principle extracted from the
hyperon semileptonic decays. It is customary to assume SU(3) symmetry to
analyze these decays. Then all decay amplitudes are given in terms of two
reduced matrix elements $F$ and $D$. For example:
\[
A_1({\rm n}\rightarrow {\rm p})=F+D\,,~~~~A_4(\Sigma ^{-}\rightarrow {\rm n}%
)=F-D\,.
\]
Here by $A_i$ we denote the ratios of axial-vector to vector coupling constants
$g_1/f_1$ for semileptonic decays as
displayed in Table I. Taking for these decays experimental values (see Table
I) one gets $F=0.46$ and $D=0.80$. The matrix elements of diagonal operators $%
\lambda _3$ and $\lambda _8$ (called $g_{{\rm A}}^{(3)}$ and $g_{{\rm A}%
}^{(8)}$ respectively), which define integrated quark densities 
$\Delta q$, can be also
expressed in terms of $F$ and $D$:
\begin{eqnarray}
g_{{\rm A}}^{(3)}\equiv \Delta u-\Delta d &=&F+D\,,  \nonumber \\
g_{{\rm A}}^{(8)}\equiv \frac 1{\sqrt{3}}\left( \Delta u+\Delta d-2\Delta
s\right) &=&\frac 1{\sqrt{3}}\left( 3F-D\right) \,.
\end{eqnarray}
Using the values of $F$ and $D$ obtained from the neutron and $\Sigma ^{-}$
decays together with Eq.(\ref{Gampval}) we get: $\Delta u=0.79$, $\Delta
d=-0.47$, and $\Delta s=-0.13$. Defining the quark content of the proton's
spin:
\begin{equation}
\Delta \Sigma =\Delta u+\Delta d+\Delta s  \label{Sigma}
\end{equation}
we obtain $\Delta \Sigma =0.19$. Had we used for $I_p$ the result of the
first EMC measurement $I_p^{{\rm EMC}}=0.112$, we would get even smaller
value: $\Delta \Sigma =0.07$.

Although quite often used, the above derivation of $\Delta \Sigma$ has, 
however, one serious flaw.
Namely we could equally well use some other decays to extract $F$ and $D$.
For example using:
\begin{eqnarray}
A_4(\Sigma^- \rightarrow {\rm n}) = F-D \, , &~~~~ & A_5(\Xi^- \rightarrow
\Lambda) = F - \frac{D}{3} \, ,  \label{ch2}
\end{eqnarray}
together with the experimental data for these decays (see Table I) and
experimental value for $\Gamma_p$, Eq.(\ref{Gampval}), 
we would get $F=0.55$ and $D=0.89$,
yielding $\Delta \Sigma=0.02$ -- almost ten times less than our previous
value. It is the breaking of the SU(3) symmetry, which is responsible for
this discrepancy. Although the symmetry breaking in hyperon decays
themselves is not that large, {\em i.e.} it amounts to no more than 
10 \%, the effect of the symmetry breaking on $\Delta \Sigma$, or 
integrated quark density $\Delta s$, is much stronger.

There are 6 measured semileptonic hyperon decays, so that the number of
combinations which one can form to extract $F$ and $D$ is 14 (actually 15,
but two conditions are linearly dependent). Taking these 14 combinations
into account and Eq.(\ref{Gampval}) we get the following values for $\Delta
u=0.75\rightarrow 0.85$, $\Delta d=-0.39\rightarrow -0.58$ and 
$\Delta s=-0.05\rightarrow -0.25 $, 
which in turn give $\Delta \Sigma =0.02 \rightarrow 0.30$. These are the
uncertainties of the {\em central values} due to the theoretical error
caused by using SU(3) symmetry to describe the hyperon decays. They are
further increased by the experimental errors of all individual decays and
the one of $\Gamma _p$. 

The authors of Ref.\cite{EhrSch} made similar
observation trying to fit the variation of $F$ and $D$ for various
decays with one parameter related to $m_{\rm s}$. Assuming further
$\Delta s=0$ they were able to fit experimental data for
$I_{p,n,deuter}$ with satisfactory accuracy.

Similarly in Refs.\cite{Lipkin,LichtLip} a simple quark model has been
proposed to describe the symmetry breaking in the hyperon decays.
It has been observed that with the increase of the symmetry breaking
parameter the value of $\Delta s$ increased, while $\Delta \Sigma$
stayed almost unchanged.

It is virtually impossible to analyze the symmetry breaking in weak
decays without resorting to some specific model \cite{KarLip}. In this
paper we will implement the symmetry breaking for the hyperon decays
using the Chiral Quark-Soliton Model ($\chi $QSM, see Ref.\cite{review}
for review).  This model has proven to give satisfactory description of
the axial-vector properties of hyperons
\cite{BloPraGo}\nocite{BPG,Wakaspin}--\cite{KimPoPraGo}.  
It describes  the baryons as solitons rotating adiabatically in flavor space.
Thus it provides a link between the matrix elements of the octet of the
axial-vector currents, responsible for hyperon decays, and the matrix elements
of the singlet axial-vector current, in our normalization equal to 
$\Delta \Sigma$.  In the present work we will study the relation between 
the semileptonic decays and integrated polarized quark distributions, 
with the help of the $\chi$QSM.
However we will use
only the collective Hamiltonian of the flavor rotational degrees 
of freedom including the corrections linear in  the strange quark mass 
$m_{\rm s}$.  
The dynamical quantities in this Hamiltonian, certain moments of inertia 
calculable within the model \cite{BloPraGo}, are not calculated but 
treated as free parameters.
By adjusting them to the experimentally known semileptonic decays
we allow for maximal phenomenological input and minimal 
model dependence.  In Ref.\cite{KimPraGo,strange} we have already
studied the magnetic moments of the octet and decuplet in this way.

As far as the symmetry breaking is concerned, our results are identical
to the ones obtained in Refs.\cite{Man} within large $N_{{\rm c}}$ QCD.
Indeed, the $\chi $QSM is a specific realization of the large $N_{{\rm c}}$
limit. The new ingredient of our analysis is the model formula for the
singlet axial-vector constant $g_{{\rm A}}^{(0)}$, which we use to calculate
quantities relevant for the polarized high energy experiments. In the
$\chi$QSM one can define two interesting limits 
\cite{DiaMog,limit,charge} in which
the soliton size is artificially changed either to zero (so called
the quark-model limit), or to $%
\infty $ (Skyrme limit). In these two limiting cases one recovers the well
known results: 1) $g_{{\rm A}}^{(0)}=1$ in the quark model limit and 
2) $g_{{\rm A}}^{(0)}/g_{{\rm A}}^{(3)}\rightarrow 0$ in the Skyrme limit. 
Also these simple
qualitative features make us believe that the model correctly 
describes physics essential for the axial-vector properties of the nucleon.

As we will see,
in the $\chi $QSM in the chiral limit we can express the singlet axial-vector
coupling through $F$ and $D$: $g_{{\rm A}}^{(0)}=9F-5D$.  We see that 
the value of $g_{{\rm A}}^{(0)}$ is very sensitive to small variations 
of $F$ and $D$, since it is a difference of the two, 
with relatively large multiplicators.  Indeed, for the 14 fits 
mentioned above (where as the input we use {\em only}
semileptonic decays plus model formula for $g_{{\rm A}}^{(0)}$) the central
value for $g_{{\rm A}}^{(0)}$ varies between $-0.25$ to approximately 1.
So despite the fact that semileptonic decays
are relatively well described by the model in the chiral limit, the singlet
axial-vector coupling is basically undetermined. This is a clear signal of the
importance of the symmetry breaking for this quantity.

One could argue that this kind of behavior is just an artifact of the
$\chi$QSM.  However, the scenario of a rotating soliton (which is by the
way used also in the Skyrme-type models) is very plausible and
cannot be {\em a priori} discarded on the basis of first principles.  The 
$\chi$QSM is a particular realization of 
this scenario and we use it as a tool to investigate the sensitivity of the
singlet axial current to the symmetry breaking effects in hyperon decays.
In fact conclusions similar to ours have been obtained in chiral
perturbation theory in Ref.\cite{SavWal}.

As a result of our present analysis we will give predictions for the
semileptonic decays not yet measured. More importantly, we will show that
the symmetry breaking effects cannot be neglected in the analysis of the
quark contribution to the spin of the proton.  In other words,
linking low energy data with high energy polarized experiments is 
meaningful only if the SU(3) breaking is taken into account.  We will
furthermore show which semileptonic decays should be measured more
accurately in order to reduce  the experimental errors for
$\Delta \Sigma$ and $\Delta s$.  

The paper is organized as follows: in the next Section we will shortly
recapitulate the formalism of the $\chi$QSM needed for the calculation of
semileptonic hyperon decays. In Section III we will discuss quantities
relevant for the polarized parton distribution. Finally in Section IV we
will draw conclusions. Formulae used to calculate hyperon decays and
axial-vector constants are collected in the Appendix.

\section{Hyperon decays in the Chrial Quark Soliton Model}

The transition matrix elements of the hadronic axial-vector current $\langle
B_2 | A_\mu^X | B_1\rangle$ can be expressed in terms of three independent
form factors:
\begin{equation}
\langle B_2| A_\mu^X |B_1\rangle\;=\;\bar{u}_{B_2} (p_2) 
\left[ \left\{g_1^{B_1\rightarrow B_2} (q^2) \gamma_\mu - 
\frac{i g_2^{B_1\rightarrow B_2} (q^2)}{M_1} \sigma_{\mu\nu} q^\nu 
+ \frac{g_3^{B_1\rightarrow B_2}
(q^2)}{M_1} q_\mu\right\}\gamma_5 \right] u_{B_1} (p_1),
\end{equation}
where the axial-vector current is defined as
\begin{equation}
A_{\mu}^X\;=\; \bar{\psi}(x) \gamma_\mu \gamma_5 \lambda_X \psi (x)
\label{Eq:current}
\end{equation}
with $X= \frac12 (1 \pm i 2)$ for strangeness conserving $\Delta S = 0$
currents and $X=\frac12 (4 \pm i 5) $ for $|\Delta S| = 1 $. Similar
expressions hold for the hadronic vector current, where the $g_i$ are
replaced by $f_i$ ($i=1,2,3$) and $\gamma_5$ by ${\bf 1}$.

The $q^2=-Q^2$ stands for the square of the momentum transfer $q=p_2-p_1$.
The form factors $g_i$ are real quantities depending only on the square of
the momentum transfer in the case of $CP$-invariant processes. We can safely
neglect $g_3$ for the reason that on account of $q_\mu $ its contribution to
the decay rate is proportional to the ratio $\frac{m_l^2}{M_1^2}\ll 1$,
where $m_l$ represents the mass of the lepton ($e$ or $\mu $) in the final
state and $M_1$ that of the baryon in the initial state .

The form factor $g_2$ is equal to 0 in the chiral limit. It gets the first
nonvanishing contribution in the linear order in $m_{\text{s}}$. The
inclusion of this effect in the discussion of the hyperon decays would
require reanalyzing the experimental data, which is beyond the scope of this
paper. However, the model calculations show that the $m_{\text{s}}$
contribution to $g_2$ enters with relatively small numerical coefficient,
which means that the numerical error due to the neglect of $g_2$ in the
full fledged analysis of the hyperon decays is small.

It is already well known how to treat hadronic matrix elements such as $%
\langle B_2|A_\mu ^X|B_1\rangle $ within the $\chi $QSM (see for example
\cite{review} and references therein.). Taking into account the $1/N_c$
rotational and $m_{{\rm s}}$ corrections, we can write the resulting
axial-vector constants $g_1^{B_1\rightarrow B_2}(0)$ in the following form%
\footnote{%
In the following we will assume that the baryons involved have $S_3 = \frac12
$.}:
\begin{eqnarray}
g_1^{(B_1\rightarrow B_2)} &=&a_1\langle B_2|D_{X3}^{(8)}|B_1\rangle
\;+\;a_2d_{pq3}\langle B_2|D_{Xp}^{(8)}\,\hat{S}_q|B_1\rangle \;+\;\frac{a_3%
}{\sqrt{3}}\langle B_2|D_{X8}^{(8)}\,\hat{S}_3|B_1\rangle  \nonumber \\
&+&m_s\left[ \frac{a_4}{\sqrt{3}}d_{pq3}\langle
B_2|D_{Xp}^{(8)}\,D_{8q}^{(8)}|B_1\rangle +a_5\langle B_2|\left(
D_{X3}^{(8)}\,D_{88}^{(8)}+D_{X8}^{(8)}\,D_{83}^{(8)}\right) |B_1\rangle
\right.  \nonumber \\
&+&\left. a_6\langle B_2|\left(
D_{X3}^{(8)}\,D_{88}^{(8)}-D_{X8}^{(8)}\,D_{83}^{(8)}\right) |B_1\rangle
\right] .  \label{Eq:g1}
\end{eqnarray}
$\hat{S}_q$ ($\hat{S}_3$) stand for the $q$-th (third)
component of the spin operator of the baryons. The $D_{ab}^{({\cal R})}$
denote the SU(3) Wigner matrices in representation ${\cal R}$.
The $a_i$ denote parameters depending on the specific dynamics of the
chiral soliton model. Their explicit form in terms of a Goldstone mean field
can be found in Ref.~\cite{BloPraGo}.  As mentioned already, in the
present approach we will not calculate this mean field but treat $a_i$ as free 
parameters to be adjusted to experimentally known
semileptonic hyperon decays.  

Because of the SU(3) symmetry breaking due to the strange quark mass $m_{%
{\rm s}}$, the collective baryon Hamiltonian is no more SU(3)-symmetric. The
octet states are mixed with the higher representations such as antidecuplet $%
\overline{{\bf 10}}$ and eikosiheptaplet ${\bf 27}$~\cite{KimPraGo}.  In 
the linear order in $m_{{\rm s}}$ the wave function of a state 
$B=(Y,I,I_3)$ of spin $S_3$ is given as:
\begin{equation}
\psi _{B,S_3}=(-)^{\frac 12-S_3}\left( \sqrt{8}\,D_{B\,S}^{(8)}+c_B^{(%
\overline{10})}\sqrt{10}\,D_{B\,S}^{(\overline{10})}+c_B^{(27)}\sqrt{27}%
\,D_{B\,S}^{(27)}\right) ,
\end{equation}
where $S=(-1,\frac 12,S_3)$. Mixing parameters $c_B^{({\cal R})}$ can be
found for example in Ref.~\cite{BloPraGo}. They are given as products of a
numerical constant $N_B^{({\cal R})}$ depending on the 
quantum numbers of the baryonic state $B$ 
and dynamical parameter $c_{{\cal R}}$ depending linearly 
on $m_{{\rm s}}$ (which we
assume to be 180~MeV) and the model parameter $I_2$, which is responsible
for the splitting between the octet and higher exotic multiplets~\cite
{antidec}.

Analogously to Eq.(\ref{Eq:g1}) one obtains in the $\chi$QSM  
diagonal axial-vector coupling constants.
In that case $X$ can take two values: $X=3$ and $X=8$. For $X=0$ (singlet
axial-vector current) we have the following expression \cite{BloPraGo,BPG}:
\begin{equation}
\frac 12\,g_B^{(0)}=\frac 12a_3+\sqrt{3}\,m_{\text{s}}\,(a_5-a_6)\;\langle
B|D_{83}^{(8)}|B\rangle .  \label{Eq:singlet}
\end{equation}

This equation is remarkable, since it provides a link
between an octet and singlet axial-vector current.  It is perhaps the most 
important model input in our analysis.  Pure QCD-arguments based the large 
$N_c$ expansion~\cite{Man} do not provide such a link.

A remark concerning constants $a_i$ is here in order. Coefficient $a_1$
contains terms which are leading and subleading in the large $N_{{\rm c}}$
expansion. The presence of the subleading terms enhances the numerical value
of $a_1$ calculated in the $\chi$QSM for the self-consistent profile and
makes model predictions {\em e.g.} for $g^{(3)}_{{\rm A}}$ remarkably close
to the experimental data \cite{WaWa,allstars}. This feature, although very
important for the model phenomenology, does not concern us here, since our
procedure is based on fitting all coefficients $a_i$ from the data.
Constants $a_2$ and $a_3$ are both subleading in $1/N_{{\rm c}}$ and come
from the anomalous part of the effective chiral action in Euclidean space.
In the Skyrme model they are related to the Wess-Zumino term. However, in
the simplest version of the Skyrme model (which is based on the
pseudo-scalar mesons only) $a_3 = 0$ identically \cite{BroEllKar}. In the
case of the $\chi$QSM $a_3 \ne 0$ and it provides a link between the SU(3)
octet of axial-vector currents and the singlet current of Eq.(\ref
{Eq:singlet}). It was shown in Ref.\cite{limit} that in the limit of the
artificially large soliton, which corresponds to the ``Skyrme limit'' of the
present model, $a_3/a_1 \rightarrow 0$ in agreement with \cite{BroEllKar}.
On the contrary, for small solitons $g_A^{(0)} \rightarrow 1$ reproducing
the result of the non-relativistic quark model.

So instead of calculating 7 dynamical parameters $a_i (i=1,\cdots,6)$ 
and $I_2$ (which enters into $c_{\overline{10}}$ and $c_{27}$) within the 
$\chi$QSM, we shall fit them from the hyperon semileptonic decays data. It is
convenient to introduce the following set of 7 new parameters:
\[
r=\frac{1}{30} \left( a_1 - \frac{1}{2} a_2 \right),\;\;\; \;\;\;s= \frac{1}{%
60} a_3, \;\;\; x = \frac{1}{540} m_{{\rm s}}\,a_4,\;\;\; y=\frac{1}{90} m_{%
{\rm s}}\,a_5,\;\;\; z=\frac{1}{30}m_{{\rm s}}\, a_6,
\]
\begin{equation}
p= \frac{1}{6}m_{{\rm s}}\, c_{\overline{10}} \left(a_1 + a_2 +\frac{1}{2}
a_3 \right),\;\;\; q=-\frac{1}{90}m_{{\rm s}}\, c_{27} \left(a_1 + 2 a_2 -
\frac{3}{2} a_3 \right) .  \label{Eq:newp}
\end{equation}

Employing this new set of parameters, we can express all possible
semileptonic decays of the octet baryons. Explicit formulae can be found in
the Appendix (see Eq.(\ref{Eq:semilep})). Let us finally note that there is
certain redundancy in Eq.(\ref{Eq:semilep}), namely by redefinition of $q$
and $x$ we can get rid of the variable $p$:
\begin{equation}
x^{\prime}=x- \frac{1}{9} p, \;\;\;\; q^{\prime}=q- \frac{1}{9} p.
\end{equation}
So there are 6 free parameters which have to be fitted from the data.

From Eq.(\ref{Eq:semilep}), we can easily find that in the chiral limit the
following eight sum rules for $\left({g_1}/{f_1}\right)$ exist:
\begin{eqnarray}
{({\rm n}\rightarrow {\rm p})} = {(\Xi^- \rightarrow \Sigma^0)}, &~~~~& {(%
{\rm n}\rightarrow {\rm p})} = {(\Sigma^{-} \rightarrow {\rm n})} +2 {%
(\Sigma^{+} \rightarrow \Lambda)},  \nonumber \\
{({\rm n}\rightarrow {\rm p})} = \frac{4}{3} {(\Sigma^{+} \rightarrow
\Lambda)} + {(\Xi^{-} \rightarrow \Lambda) }, & & {({\rm n}\rightarrow {\rm p%
})} = {(\Lambda \rightarrow {\rm p})} +\frac{2}{3} {(\Sigma^{+} \rightarrow
\Lambda)},  \nonumber \\
{({\rm n}\rightarrow {\rm p})} = 2 {(\Sigma^{+} \rightarrow \Lambda)} + {%
(\Xi^{-} \rightarrow \Xi^0)}, & & {({\rm n}\rightarrow {\rm p})} = {%
(\Sigma^{-} \rightarrow \Sigma^0)} + {(\Sigma^{+} \rightarrow \Lambda)},
\nonumber \\
{(\Sigma^{+} \rightarrow \Lambda)} = {(\Sigma^{-} \rightarrow \Lambda)}, & &
{(\Xi^0 \rightarrow \Sigma^+)} = {(\Xi^{-} \rightarrow \Sigma^0)}.
\label{srchi}
\end{eqnarray}
Only the first 4 sum rules (\ref{srchi}) contain known decays, and the
accuracy here is not worse than 10 \%.  Apparently the symmetry
breaking of SU(3) has only a small effect on the semileptonic
decays.  

With the linear $m_{{\rm s}}$ corrections turned on, 
we end up with only four sum rules: 
\begin{eqnarray}
{(\Xi^{-} \rightarrow \Sigma^0)} = {(\Xi^0 \rightarrow \Sigma^+)}, &~~~& {%
(\Sigma^{-} \rightarrow \Lambda)} = {(\Sigma^{+} \rightarrow \Lambda)},
\nonumber
\end{eqnarray}
\begin{eqnarray*}
3 {(\Lambda \rightarrow {\rm p})} - 2 {({\rm n}\rightarrow {\rm p})} +2 {%
(\Sigma^{-} \rightarrow {\rm n})} +4 {(\Sigma^{+} \rightarrow \Lambda)} & &
\\
- {(\Xi^{-} \rightarrow \Sigma^0)} +2 {(\Xi^- \rightarrow \Xi^0)} - 2 {%
(\Xi^{-} \rightarrow \Lambda) } & = & 0,
\end{eqnarray*}
\begin{equation}
3 {(\Lambda \rightarrow {\rm p})} - 2 {({\rm n}\rightarrow {\rm p})} - {%
(\Sigma^{-} \rightarrow {\rm n})} +2 {(\Sigma^{+} \rightarrow \Lambda)} -2 {%
(\Xi^{-} \rightarrow \Sigma^0)} +2 {(\Sigma^- \rightarrow \Sigma^0)}= 0.
\label{Eq:sum2}
\end{equation}

However, more experimental data are required to verify Eq.(\ref{Eq:sum2}).

\section{Linking hyperon decays with data on polarized parton distributions}

As we have demonstrated in the preceding Section, the amplitudes of the
hyperon decays are described in the $\chi$QSM by 6 free parameters.  There
are 2 {\em chiral} ones: $r$ and $s$, and 
4 proportional to $m_{{\rm s}}$: $x^\prime$, $y$, $z$, and $q^\prime$. Since 
there are 6 known hyperon decays, we can
express all model parameters as linear combinations of these decay
constants, and subsequently all quantities of interest can be expressed in
terms of the input amplitudes.  In the following we will use the experimental
values of Refs.~\cite{PDG96,BGHORS}, which
are presented in Table I.  

Before doing this, let us, however, observe that there exist two linear
combinations of the decay amplitudes which are free of the $m_{{\rm s}}$
corrections (within the model):
\begin{eqnarray}
A_1 + 2 A_6 & = & -42 r + 6 s \, ,  \nonumber \\
3 A_1 - 8 A_2 - 6 A_3 + 6 A_4 + 6 A_5 & = &90 r + 90 s \, ,  \label{rs0}
\end{eqnarray}
where $A_i$ stand for the decay constants in short-hand notation (see Table
I). Solving Eq.(\ref{rs0}) for $r$ and $s$, we obtain the {\em chiral-limit}
(i.e. with $x^\prime=y=z=q^\prime=0$) expressions for hyperon decays and
integrated
quark densities. The numerical values obtained in this way can be found in
Tables I and II.  
Reexpressing $r$ and $s$,
% $x^\prime$, $y$, $z$ and $q^\prime$ 
in terms of
the $A_i$'s allows us to write down the integrated quark densities as:
\begin{eqnarray}
\Delta u &=& {\frac{4\,{A_1}}{3}} - {\frac{16\,{A_2}}{9}} - {\frac{4\,{A_3}}{%
3}} + {\frac{4\,{A_4}}{3}} + {\frac{4\,{A_5}}{3}} + {\frac{4\,{A_6}}{3}} \, ,
\nonumber \\
\Delta d &=& {A_1} - {\frac{16\,{A_2}}{9}} - {\frac{4\,{A_3}}{3}} + {\frac{%
4\,{A_4}}{3}} + {\frac{4\,{A_5}}{3}} + {\frac{2\,{A_6}}{3}} \, ,  \nonumber
\\
\Delta s &=& {\frac{2\,{A_1}}{3}} - {\frac{10\,{A_2}}{9}} - {\frac{5\,{A_3}}{%
6}} + {\frac{5\,{A_4}}{6}} + {\frac{5\,{A_5}}{6}} + {\frac{{A_6}}{2}} \, .
\label{uds0}
\end{eqnarray}

The two least known amplitudes $A_5$ and $A_6$ are almost entirely
responsible for the errors quoted in Tables I and II. However, since the
coefficients which enter into Eq.(\ref{uds0}) are not too large, the
absolute errors are relatively small.

In Table II the yet unmeasured hyperon semileptonic decay constants 
are listed.  The $\Xi^0\rightarrow \Sigma^+$ channel is particularly
interesting, since its measurement will be soon announced
by the KTeV collaboration~\cite{KTeV}.    

Forming linear combinations of the quark densities we obtain the {\em chiral
limit} expressions for $\Gamma_{p,n}$ and $\Delta \Sigma$:
\begin{eqnarray}
\Gamma_p &=& 7\,{A_1} - 10\,{A_2} - {\frac{15\,{A_3}}{2}} + {\frac{15\,{A_4}%
}{2}} + {\frac{15\,{A_5}}{2}} + {\frac{13\,{A_6}}{2}} \, ,  \nonumber \\
\Gamma_n &=& 6\,{A_1} - 10\,{A_2} - {\frac{15\,{A_3}}{2}} + {\frac{15\,{A_4}%
}{2}} + {\frac{15\,{A_5}}{2}} + {\frac{9\,{A_6}}{2}} \, ,  \nonumber \\
\Delta \Sigma &=& 3\,{A_1} - {\frac{14\,{A_2}}{3}} - {\frac{7\,{A_3}}{2}} + {%
\frac{7\,{A_4}}{2}} + {\frac{7\,{A_5}}{2}} + {\frac{5\,{A_6}}{2}} \, .
\end{eqnarray}
The numerical values together with the error bars are listed in Table II.

The full expressions are obtained by solving the 
remaining 4 equations for $m_{{\rm s}}$ dependent parameters 
$x^\prime$, $y$, $z$ and $q^\prime$.  Also in this case
we are able to link integrated quark densities $\Delta q$ to the hyperon
decays:
\begin{eqnarray}
\Delta u &=& {\frac{8\,{A_2}}{9}} + {\frac{5\,{A_3}}{3}} + {\frac{7\,{A_4}}{3%
}} + {\frac{{A_5}}{3}} - {\frac{{A_6}}{3}}\, ,  \nonumber \\
\Delta d &=& -{A_1} + {\frac{8\,{A_2}}{9}} + {\frac{5\,{A_3}}{3}} + {\frac{%
7\,{A_4}}{3}} + {\frac{{A_5}}{3}} - {\frac{{A_6}}{3}} \, ,  \nonumber \\
\Delta s &=& {\frac{15\,{A_1}}{4}} - {\frac{101\,{A_2}}{18}} - {\frac{289\,{%
A_3}}{48}} + {\frac{13\,{A_4}}{48}} + {\frac{43\,{A_5}}{48}} + {\frac{149\,{%
A_6}}{48}} \, .
\end{eqnarray}
It is interesting to observe that the amplitudes $A_5$ and in particular $%
A_6 $ come with relatively large weight in the expression for $\Delta s$,
whereas $\Delta u$ and $\Delta d$ are much less affected by the 
relatively large experimental error of these two decays. This is 
explicitly seen in Fig.1,
where we plot the central values and error bars of $\Delta q$'s. In the same
figure we draw central values and errors of $\Delta q$'s in the {\em chiral
limit} as given by Eq.(\ref{uds0}). To guide the eye we have restored the
linear dependence on the symmetry breaking $m_{{\rm s}}$ corrections
assuming $m_{\rm s}=180$~MeV, as done in Ref.\cite{KimPraGo}.

We can first see that our results in the chiral limit correspond to typical
SU(3)-symmetric values: $F\approx 0.50$ and $D\approx 0.77$. However, the
result for individual integrated quark densities, 
where the model prediction for the
singlet current $g_{{\rm A}}^{(0)}$ plays a role, are beyond the typical
SU(3) symmetry values. Only when the chiral symmetry breaking is taken into
account the central values for $\Delta q$'s are shifted towards the
"standard" values. Unfortunately the error of $\Delta s$ becomes 7 times
larger than the one of $\Delta u$ or $\Delta d$, so that at this stage we 
are not able to make any firm conclusion concerning the value of $\Delta s$.

It is perhaps more interesting to look directly at the combinations relevant
for the polarized scattering experiments, which take the following form:
\begin{eqnarray}
\Gamma_p &=& {\frac{11\,{A_1}}{4}} - {\frac{7\,{A_2}}{6}} + {\frac{37\,{A_3}%
}{16}} + {\frac{191\,{A_4}}{16}} + {\frac{41\,{A_5}}{16}} + {\frac{23\,{A_6}%
}{16}} \, ,  \nonumber \\
\Gamma_n &=& {\frac{-{A_1}}{4}} - {\frac{7\,{A_2}}{6}} + {\frac{37\,{A_3}}{16%
}} + {\frac{191\,{A_4}}{16}} + {\frac{41\,{A_5}}{16}} + {\frac{23\,{A_6}}{16}%
}\, ,  \nonumber \\
\Delta \Sigma &=& {\frac{11\,{A_1}}{4}} - {\frac{23\,{A_2}}{6}} - {\frac{43\,%
{A_3}}{16}} + {\frac{79\,{A_4}}{16}} + {\frac{25\,{A_5}}{16}} + {\frac{39\,{%
A_6}}{16}}\, .  \label{GDSfull}
\end{eqnarray}
In Fig.2 we plot $\Gamma_{p,n}$ and $\Delta \Sigma$ both for the chiral
symmetry fit and for the full fit of Eq.(\ref{GDSfull}), together with
experimental data for the proton and neutron. Again,
to guide the eye we have restored the
linear dependence of the symmetry breaking $m_{{\rm s}}$ corrections.
We see that despite the large
uncertainty of $\Delta s$, we get reasonable values for $\Gamma_p$ and $%
\Gamma_n$. Somewhat unexpectedly we see, that $\Delta \Sigma$ is almost
independent of the chiral symmetry breaking\footnote{Similar behavior
has been observed in Ref.\cite{LichtLip}.} and stays within the range 
$0.1 \rightarrow 1.1$, if the errors of the hyperon decays are taken 
into account. $75 \% $
of the experimental error of $\Delta \Sigma$ comes from the two least known
hyperon decays $\Xi^- \rightarrow \Lambda,\, \Sigma^0 $ (corresponding to $%
A_5$ and $A_6$).

It is interesting to see how $\Delta\Sigma$ and $\Delta s$ are correlated.
To this end, instead of using two last hyperon decays $A_5$ and $A_6$ as
input, we use the experimental value for $\Gamma_p$ as given by Eq.(\ref
{Gampval}) and $\Delta\Sigma$, which we vary in the range from 0 to 1. In
Fig.3 we plot our prediction for the two amplitudes $A_5$ and $A_6$ (solid
lines), together with the experimental error bands for these two decays. It
is clearly seen from Fig.3 that the allowed region for $\Delta\Sigma$, in
which the theoretical prediction falls within the experimental error bars
amounts to $\Delta\Sigma=0.20\rightarrow 0.45$.

In Fig.4 we plot the variation of $\Delta q$'s with respect to $\Delta
\Sigma $ (with $\Gamma_p$ fixed by Eq.(\ref{Gampval})).
We see that $\Delta u$ and $\Delta d$ are relatively stable,
whereas $\Delta s$ exhibits rather strong dependence on $\Delta \Sigma$.
Within the allowed region $0.20 < \Delta \Sigma <0.45 $ strange quark
density $\Delta s$ varies between $-0.12$ and $0.30$. Interestingly, in the
central region around $\Delta\Sigma\approx 0.30$ strange quark density
vanishes in accordance with an intuitive assumption of Ellis and Jaffe~\cite
{EllisJaffe}.

Identical behavior\footnote{Note that authors of Ref.\cite{LichtLip}
use a slightly different value for $I_p$ and include higher order QCD 
corrections.} (shown in Fig.4 by a dash-dotted line) was obtained by 
Lichtenstadt and Lipkin in an analysis of 
the hyperon decays in which no model for $\Delta \Sigma$ has been
used \cite{LichtLip}. Indeed (assuming only the first order QCD corrections),
the identity:
\begin{equation}
\Delta \Sigma = \frac{1}{2} \Gamma_p
- \frac{1}{4} \left(3 g_{\rm A}^{(3)}+\sqrt{3} g_{\rm A}^{(8)} \right)\ .
\label{nomodel}                 
\end{equation} 
allows one to calculate $\Delta \Sigma$ in terms of $ g_{\rm A}^{(8)} $
(or equivalently $\Delta s$) by using $ g_{\rm A}^{(3)}=1.257$ and
$\Gamma_p$ as an additional input. In  the $\chi$QSM and also in large $N_c$
QCD one can express $ g_{\rm A}^{(8)}$ in terms of the known
hyperon semileptonic decays:
\begin{equation}
\left(3 g_{\rm A}^{(3)}+\sqrt{3} g_{\rm A}^{(8)} \right)=
\frac{1}{8}
\left(-44\,{  A_1} + 104\,{  A_2} + 123\,{  A_3} + 33\,{  A_4} - 
     9\,{  A_5} - 55\,{  A_6} \right) \ .
\label{nomodel1}
\end{equation}
Equation (\ref{nomodel1}) gives $\Delta \Sigma= 0.46\pm 0.31$, remarkably
close to the $\chi$QSM prediction in which model formula for $\Delta \Sigma$
is used. This is, in our opinion, another strong argument in support for
the model formula for $  g_{\rm A}^{(8)} $.



\section{Summary}

In this paper we studied the influence of the SU(3) symmetry breaking 
in semileptonic hyperon decays on the determination of the integrated
polarized quark densities $\Delta q$. Using the Chiral Quark Soliton Model
we have obtained a satisfactory parametrization of all available
experimental data on semileptonic decays. In this respect our analysis
is identical to the large $N_{\rm c}$ QCD analysis of Ref.\cite{Man}.
Using 6 known hyperon decays we have predicted $g_1/f_1$ for the decays
not yet measured.

The new ingredient of our analysis consists in using the model formula for
the singlet axial-vector current in order to make contact with the high energy
polarization experiments. We have argued that our model interpolates
between the quark model (the small soliton limit) and the Skyrme model
(large soliton limit) \cite{DiaMog}
reproducing the value of $\Delta \Sigma$ in these
two limiting cases \cite{limit,charge}. 
This unique feature, and also the numerical agreement with the analysis 
of Ref.\cite{LichtLip} as discussed at the end of the last Section, 
make us believe that our
approach contains all necessary physics needed to analyze the symmetry
breaking not only for the octet axial-vector currents, but also in the case of
the singlet one. 

The model contains 6 free parameters which can be fixed by 6 known hyperon
decays.  Unfortunately $g_1/f_1$ for the two known decays of $\Xi^-$
have large experimental errors, which influence our predictions for
$\Delta q$.  Our strategy was very simple: using model parametrization
we expressed $\Delta q$'s, $\Gamma_{p,n}$ and $\Delta \Sigma$ in terms
of the six known hyperon decays. Errors were added in quadrature.

First observation which should be made is that we reproduce
$\Gamma_{p,n}$ as measured in deep inelastic scattering.
We obtain $\Delta u=0.72\pm0.07$ and $\Delta
d=-0.54\pm0.07$, however, $\Delta s$ is practically undetermined
being equal to $0.33\pm0.51$. This large error is entirely due to the
experimental errors of the $\Xi^-$ decays, which also make $\Delta
\Sigma$ to lie between 0.1 and 0.9. 

There are two points which have to be stressed here. 
Our fit respects chiral symmetry in a sense that the leading order parameters
$r$ and $s$ (or equivalently $F$ and $D$) are fitted to the linear 
combinations  of the hyperon decays which are free from $m_{\rm s}$ 
corrections. Had we used this SU(3)
symmetric parametrization as given by Eq.(\ref{rs0}) we would not be able
to reproduce (as far as the central values are concerned)
$\Gamma_{p,n}$. With $m_{\rm s}$ corrections turned on we hit experimental
values for $\Gamma_{p,n}$, however, as stated above, 
the value of $\Delta \Sigma$ is practically undetermined, due the the 
experimental error of $\Xi^-$ decays. Therefore to confirm or invalidate
our analysis  it is of utmost importance to have better data for these
decays. Since we predict that $(\Xi^- \rightarrow \Sigma^0)=
(\Xi^0 \rightarrow \Sigma^+)$ the forthcoming experimental result for the
latter decay \cite{KTeV} will provide a test of our approach.

Interestingly, if we use $\Gamma_p$ and $\Delta \Sigma$ as an input
instead of the $\Xi^-$ decays, we see very strong correlation between
$\Delta \Sigma$ and $\Delta s$, whereas $\Delta u$ and $\Delta d$ are
basically $\Delta \Sigma$ independent. This behavior has been also
observed in Ref.\cite{LichtLip}. 

Our analysis shows clearly that if one wants to link the low-energy hyperon 
semileptonic decays with high-energy polarized experiments, one cannot neglect
SU(3) symmetry breaking for the former. In this respect our conclusions
agree with Refs.\cite{EhrSch,SavWal}. Similarly to Ref.\cite{EhrSch} we
see that $\Delta s=0$ is not ruled out by present experiments.
Therefore the results for $\Delta s$ and $\Delta \Sigma$ which are 
based on the exact SU(3) symmetry are in our opinion premature.
The  meaningful results for these  2 quantities can be obtained only
if the experimental errors for the $\Xi^-$ decays are reduced. 
 

\section*{Acknowledgments}

This work has partly been supported by the BMBF, the DFG and the
COSY--Project (J\" ulich).  We are grateful to M.V. Polyakov for 
fruitful discussions.
H.-Ch.K. and M.P. thank P.V.  Pobylitsa for critical comments.  M.P. 
thanks M. Karliner for his comment 
concerning Ref.\cite{LichtLip}.  H.-Ch.K. has been
supported by Pusan National University Research Grant. M.P. has been
supported by Polish grant {PB~2~P03B~010~15}.

\section*{appendix}

In this Appendix we quote the formulae used in the fits.
Semileptonic decay constants are parametrized as follows:
\begin{eqnarray}
A_1\;=\;
\left({g_1}/{f_1}\right)^{({\rm n}\rightarrow {\rm p})} & = &-14 r + 2 s -
44 x - 20 y - 4 z-4 p + 8 q,  \nonumber \\
A_2\;=\;
\left({g_1}/{f_1}\right)^{(\Sigma^{+}\rightarrow \Lambda)} & = & - 9r - 3s -
42x - 6y -3p + 15q,  \nonumber \\
A_3\;=\;
\left({g_1}/{f_1}\right)^{(\Lambda \rightarrow {\rm p})} & = & - 8r + 4s +
24x - 2z+ 2p - 6q,  \nonumber \\
A_4\;=\;
\left({g_1}/{f_1}\right)^{(\Sigma^{-} \rightarrow {\rm n})} & = & 4r + 8s
-4x - 4y + 2z+4q,  \nonumber \\
A_5\;=\;
\left({g_1}/{f_1}\right)^{(\Xi^{-} \rightarrow \Lambda) } & = & - 2r + 6s -
6x + 6y - 2 z + 6q,  \nonumber \\
A_6\;=\;
\left({g_1}/{f_1}\right)^{(\Xi^{-} \rightarrow \Sigma^0)} & = & - 14r + 2s +
22x + 10 y + 2 z +2p - 4q,  \nonumber \\
\left({g_1}/{f_1}\right)^{(\Sigma^{-} \rightarrow \Lambda)} & = & - 9 r - 3
s - 42x - 6y- 3p + 15q,  \nonumber \\
\left({g_1}/{f_1}\right)^{(\Sigma^- \rightarrow \Sigma^0)} & = &- 5r + 5s -
18x - 6y + 2 z- 2p,  \nonumber \\
\left({g_1}/{f_1}\right)^{(\Xi^- \rightarrow \Xi^0)} & = & 4r + 8s + 8x + 8y
- 4 z - 8q,  \nonumber \\
\left({g_1}/{f_1}\right)^{(\Xi^0 \rightarrow \Sigma^+)} & = & - 14r + 2 s +
22 x + 10 y + 2 z+ 2p - 4 q .  \label{Eq:semilep}
\end{eqnarray}

The U(1) and SU(3) axial-vector constants $g_A^{(0,3,8)}$ 
can be also expressed in
terms of the new set of parameters (\ref{Eq:newp}). For the singlet
axial-vector constant, we have
\begin{equation}
g_A^{(0)}=60s-18y+6z,\label{Eq:singlet1}
\end{equation}
for the triplet one\footnote{%
Triplet $g_{A}^{(3)}$'s are proportional to $I_3$, formulae in Eq.(\ref
{Eq:triplet}) correspond to the highest isospin state.}:
\begin{equation}
g_A^{(3)}=-14r+2s-44x-20y-4z-4p+8q, \label{Eq:triplet}
\end{equation}
and for the octet one, we get:
\begin{equation}
g_A^{(8)}=\sqrt{3}(-2r+6s+12x+4p+24q).  \label{Eq:octet}
\end{equation}


\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\vfill

\newpage

\begin{center}
{\Large {\bf Figure Captions}}
\end{center}

\noindent
{\bf Fig. 1}: $\Delta q$ as a function of the strange quark mass $m_s$.
While the $\Delta u$ and $\Delta d$ have less uncertainties as the $m_s$
increases, the uncertainty of $\Delta s$ becomes larger, as the $m_s$
increases. \vspace{0.8cm}

\noindent
{\bf Fig. 2}: $\Gamma_ {p,n}$ and $\Delta \Sigma$ as functions of $m_s$.
While the uncertainty of $\Gamma_ {p,n}$ decreases, as the $m_s$ increases,
the error of the $\Delta \Sigma$ remains constant. The error bars denote the
experimental data for the $\Gamma_ {p,n}$. \vspace{0.8cm}

\noindent
{\bf Fig. 3}: $A_5$ (lower line) and $A_6$ (upper line) as functions of $%
\Delta \Sigma$. \vspace{0.8cm}

\noindent
{\bf Fig. 4}: $\Delta q$'s as functions of $\Delta \Sigma$.  Dash-dotted
line below $\Delta s$ corresponds to the result of Ref.~\cite{LichtLip}.

\vfill

\newpage

\begin{center}
{\Large {\bf Figures}}
\end{center}

\vspace{1.6cm} \centerline{\epsfysize=2.7in\epsffile{ax_fig1.ps}}\vskip4pt
\noindent

\begin{center}
{\bf Figure 1}
\end{center}

\vspace{1.6cm} \centerline{\epsfysize=2.7in\epsffile{ax_fig2.ps}}\vskip4pt
\noindent

\begin{center}
{\bf Figure 2}
\end{center}

\vspace{1.6cm} \centerline{\epsfysize=2.7in\epsffile{ax_fig3.ps}}\vskip4pt
\noindent

\begin{center}
{\bf Figure 3}
\end{center}

\vspace{1.6cm} \centerline{\epsfysize=2.7in\epsffile{ax_fig4.ps}}\vskip4pt
\noindent

\begin{center}
{\bf Figure 4}
\end{center}



\newpage

\begin{table}
\caption{The parameters $r, \ldots, q^\prime$  fixed to the 
experimental data of the semileptonic decays
\protect\cite{PDG96,BGHORS} $A_1$ -- $A_6$. 
The entries for $A_1$ -- $A_6$ for the full fit (last column)
correspond to the experimental data.}
\begin{tabular}{cccc}
& & chiral limit  & with $m_{\rm s}$            \\ \hline 
& $r$ & $-0.0892 $  & $-0.0892             $      \\    
& $s$ & $ 0.0113 $  & $ 0.0113              $	   \\ 
& $x^{\prime}$ & 
      $ 0~~~~  $  & $-0.0055              $	   \\
& $y$ & $ 0~~~~  $  & $ 0.0080              $	   \\
& $z$ & $ 0~~~~  $  & $-0.0038              $	   \\
& $q^{\prime}$ & 
      $ 0~~~~  $  & $ -0.0140              $	   \\
\hline
$A_1$ & $\left({g_1}/{f_1}\right)^{n\rightarrow p}$        
      &$  1.271\pm 0.11$ & $  1.2573\pm 0.0028$\\
%
$A_2$ & $\left({g_1}/{f_1}\right)^{\Sigma^+\rightarrow \Lambda}$ 
      &$  0.769\pm 0.04$ & $0.742 \pm 0.018~~ $\\
%
$A_3$ & $\left({g_1}/{f_1}\right)^{\Lambda\rightarrow p}$        
      &$  0.758\pm 0.08$ & $  0.718 \pm 0.015~~ $\\
%
$A_4$ & $\left({g_1}/{f_1}\right)^{\Sigma^-\rightarrow n}$       
      &$ -0.267\pm0.04$ & $ -0.340 \pm 0.017~~ $\\
%
$A_5$ & $\left({g_1}/{f_1}\right)^{\Xi^-\rightarrow \Lambda}$    
      &$  0.246\pm 0.07$ & $  0.25  \pm 0.05~~~  $\\
%
$A_6$ & $\left({g_1}/{f_1}\right)^{\Xi^-\rightarrow \Sigma^0}$   
      &$  1.271\pm 0.11$ & $  1.278 \pm 0.158~~ $\\ %\hline
\end{tabular}
\vspace{1cm}
\caption{The predictions for yet unmeasured decays, 
integrated quark densities $\Delta q$ and $\Gamma_{p,n}$ and 
$\Delta\Sigma$.}
\begin{tabular}{cccc}
& & chiral limit  & with $m_{\rm s}$            \\ \hline 
      & $\left({g_1}/{f_1}\right)^{\Sigma^-\rightarrow \Lambda}$  
      &$  0.769\pm0.04$ & $  0.742\pm 0.02$ \\ 
%
      & $\left({g_1}/{f_1}\right)^{\Sigma^-\rightarrow \Sigma^0}$ 
      &$  0.502\pm 0.07$  & $  0.546\pm 0.16$ \\
%
      & $\left({g_1}/{f_1}\right)^{\Xi^-\rightarrow \Xi^0}$       
      & $-0.267\pm 0.04$  & $ -0.12\pm 0.12$ \\
%
      & $\left({g_1}/{f_1}\right)^{\Xi^0\rightarrow \Sigma^+}$    
      &$  1.271 \pm 0.11$ & $ 1.278\pm 0.16$ \\ \hline
%
     & $\Delta u$   &$ 0.98\pm 0.23$ &$ 0.72 \pm 0.07$   \\
%
     & $\Delta d$  &$-0.29\pm 0.13$ &$-0.54\pm 0.07$      \\
%
     & $\Delta s$  &$-0.02\pm 0.09$ &$0.33\pm 0.51$       \\ \hline
%
     & $\Gamma_p$  &$3.63 \pm 1.12$  & $ 2.67 \pm 0.33 $   \\
%
     & $\Gamma_n$  &$-0.19 \pm 0.84$ & $-1.10 \pm 0.33 $  \\
     & $\Delta\Sigma$ 
                   &$ 0.68 \pm 0.44$ & $ 0.51 \pm 0.41 $
\end{tabular}

\end{table}



\end{document}
































