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\preprint{\vbox{\hbox{PNU-NTG-03/2002} \hbox{RUB-TP2-11-02}}}
\title{Strange and singlet form factors of the nucleon: Predictions
for G0, A4, and HAPPEX-II experiments}

\author{Antonio Silva$^{(1,2)}$
\footnote{E-mail address: Antonio.Silva@tp2.ruhr-uni-bochum.de},
Hyun-Chul Kim$^{(3)}$
\footnote{E-mail address: hchkim@pnu.edu},
and Klaus Goeke$^{(1)}$
\footnote{E-mail address: Klaus.Goeke@tp2.ruhr-uni-bochum.de}}

\affiliation{(1) Institut f\"ur Theoretische  Physik  II, \\  
Ruhr-Universit\" at Bochum, \\
 D--44780 Bochum, Germany  \\
(2) Departamento de F\'\i sica and Centro de F\'\i sica Computacional,\\
Universidade de Coimbra,\\
P-3000 Coimbra, Portugal\\
(3) Department of Physics,
Pusan National University,\\
609-735 Pusan, Republic of Korea }
\date{August 2002}

\begin{abstract}
We investigate the strange and flavor-singlet electric and magnetic 
form factors of the nucleon within the framework of the SU(3) chiral 
quark-soliton model.  Isospin symmetry is assumed and the
symmetry-conserving SU(3) quantization is employed, rotational and
strange quark mass corrections being included.  
For the experiments G0, A4, and HAPPEX-II we
predict the quantities $G^{0}_E + \beta G^{0}_M$ and $G^{\rm s}_E + 
\beta G^{\rm s}_M$.  The dependence of the results on the parameters 
of the model and the treatment of the Yukawa asymptotic behavior of the
soliton are investigated.  
\end{abstract}

\pacs{12.40.-y, 14.20.Dh\\
Key words: Strange vector form factors, singlet vector form factors,
chiral quark-soliton model.}
\maketitle
%\tableofcontents

\section{Introduction}
It is of great importance to understand the strangeness content of the
nucleon, since it gives a clue about its internal quark structure.
In particular, the deviation from the valence quark
picture and the polarization of the quark sea is investigated.  
A great deal of experimental and theoretical effort has
been put into the study of the strangeness in the nucleon in various
channels: The spin content of the nucleon~\cite{EMC,SMC1,SMC2,E142,E143}, 
the $\pi N$ sigma term $\Sigma_{\pi N}$~\cite{Gasser:1991ce}, 
and the strange vector form factors~\cite{Kaplan:1988ku,Jaffe:1989mj}.  
In particular, the strange vector form factors have been a hot issue recently,
as their first measurement was achieved by the SAMPLE
collaboration~\cite{Mueller:1997mt,Spayde:2000qg,SAMPLE00s} at
MIT/Bates, parity-violating electron scattering being used.
The most recent result by the SAMPLE collaboration~\cite{SAMPLE00s} for the 
strange magnetic form factor finds ($Q^2$ in (GeV/$c$)$^2$) 
\begin{equation} 
        G_M^{\rm s} (Q^2=0.1) = ( + 0.14 \pm 0.29 \, ({\rm stat.}) 
\pm 0.31 \, ({\rm syst.})) \;\; \mbox{n.m.}\, .  
\label{eq:samplenew}
\end{equation}
%
It is extracted from knowledge of both the neutral weak magnetic 
form factor $G_M^Z$ measured in parity-violating elastic $e$-$p$
scattering and the electromagnetic form factors $G_{M}^{p\gamma}$,  
$G_{M}^{n\gamma}$ by using the relation (assuming isospin invariance)
\begin{equation}
G_M^Z =  \left(1-4\sin^2\theta_W\right)  G_M^{p\gamma} 
- G_M^{n\gamma} - G_M^{s}\, ,       
\label{eq:GMZ}
\end{equation} 
where $\theta_W$ is the Weinberg mixing angle determined  
experimentally~\cite{PDG00}: $\sin^2\theta_W=0.23147$.

The HAPPEX collaboration at TJNAF also announced the measurement of 
the strange vector form factors~\cite{Aniol:2000at}.  The asymmetry
$A_{\rm th}$ can be obtained from the parity-violating polarized
electron scattering, 
from which the singlet form factors are extracted:
\begin{equation}
 \frac{(G_E^{0} + 0.392 G_M^{0})}{(G_M^{p\gamma }/\mu_p)} (Q^2=0.477)
        =  1.527\pm 0.048\pm 0.027\pm 0.011.
\label{eq:happex0}
\end{equation}
With the help of the available data for the electromagnetic form
factors via the following relation
\begin{equation}
G_{E,M}^{\rm s}  =  
G_{E,M}^0  -  G_{E,M}^{p\gamma}  -  G_{E,M}^{n\gamma},
\label{eq:GEMs}
\end{equation}
they arrive at the strange form factors:
\begin{equation}
 (G_E^{\rm s} + 0.392 G_M^{\rm s})(Q^2=0.477)  =  0.025\pm 0.020 \pm 0.014,
\label{eq:happexs}
\end{equation}
where the first error is experimental and the second one is from the 
uncertainties in electromagnetic form factors.  
 
There has been a great deal of theoretical effort in order to predict
the strange vector form factors~\cite{Theory} and each approach emphasizes     
different aspects.  Beck and Holstein reviewed most of theoretical works in
Ref.~\cite{Beck:2001dz}.    

A proper description of the strange form factors of the nucleon should
be based on QCD.  Since, however, these form factors basically reflect
the excitation of $s\bar{s}$ pairs, it is very difficult to use
lattice gauge techniques because they are still hampered by technical
problems, in particular, with light quarks.  Thus, appropriate models
are required, which are based on QCD and treat the relevant degrees of
freedom in an approximate way.  One of those is the chiral
quark-soliton model ($\chi$QSM).  It is an effective quark theory of
the instanton-degrees of freedom of the QCD vacuum and results in a
Lagrangian for valence and sea quarks interacting via a static
self-consistent Goldstone background field.  It has successfully been
applied to electromagnetic and axial-vector form
factors~\cite{Christov:1996vm} and to forward and
generalized parton distributions~\cite{Diakonov:1996sr,Petrov:1998kf,
Goeke:2001tz}.    

Recently, we have investigated some aspects 
of the SAMPLE, HAPPEX, and A4 experiments within the framework of this
$\chi$QSM~\cite{Silva:2001st}.  
Results have shown a fairly good agreement with experimental
data of the SAMPLE and HAPPEX.  In this work, we want to extend the 
former investigation, to document the relevant part of the formalism and to
present the results pertinent to future experiments: G0 experiments 
being conducted at TJNAF will measure the combination of the strange
vector form factors at seven different values of the momentum transfer
$Q^2$ with two different angles, i.e. the forward angle $\theta=10^\circ$
and the backward angle $108^\circ$.  With these measurements, the
strange electric and magnetic form factors can be separately obtained.    
The A4 experiment at MAMI will soon bring out the new data at 
$Q^2=0.227\ {\rm GeV}^2$ with $\theta=35^\circ$.  The planned HAPPEX II
experiment will measure the combination of the strange vector form
factors at $Q^2=0.11\ {\rm GeV}^2$, which is the same momentum
transfer as the SAMPLE experiment, with the forward angle 
$\theta=6^\circ$ to extract the separated strange electric and
magnetic form factors with the SAMPLE data combined.  Thus, in the
present work, we will continue our previous work~\cite{Silva:2001st}
and will concentrate on predicting the above-given future experiments,
in particular, G0 experiment.

The outline of the paper is as follows: Section II explains briefly
the general formalism for obtaining the strange vector form factors 
of the baryons within the framework of the $\chi$QSM.  Section III
gives the explicit expressions of the flavor-singlet and
strange vector form factors of the nucleon.  In Section IV we
investigate the dependence of the strange and singlet form factors on
the parameters of the model (basically $M$ is the only one), we present
corresponding results and predictions for G0, A4, and HAPPEX-II and discuss
them.  Section V summarizes the  present work and draws the conclusion.
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{General Formalism}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section we briefly review the formalism of the $\chi$QSM.   
Details of the model~\cite{Diakonov:1988ty} can be found in 
ref.~\cite{Christov:1996vm}.
We start with the low-energy partition function in Euclidean space
given by the functional integral over pseudoscalar meson ($\pi^a$) and 
quark fields($\psi$)~\cite{Diakonov:1988ty}:
\begin{eqnarray}
{\mathcal{Z}} & = & \int {\mathcal{D}}
\psi {\mathcal{D}}\psi ^{\dagger }{\mathcal{D}}\pi ^{a}\exp \left[ 
\int d^{4}x\psi ^{\dagger \alpha }_{f}\left( i\rlap {/}{\partial }
+iMe^{i\gamma _{5}\lambda ^{a}\pi ^{a}}\right) _{fg}\psi ^{\alpha }_{g}\right],
\nonumber \\  
& = & \int {\mathcal{D}}\pi ^{a}\exp {(-S_{\rm eff}[\pi ])},\label{Eq:action} 
\end{eqnarray}
 where $S_{\rm eff}$ is the effective action 
\begin{equation}
S_{\rm eff}[\pi ]\; =\; -{\rm Tr} {\rm ln}D.
\end{equation}
${\rm Tr}$ denotes the functional trace.   
$D$ represents the Dirac differential operator 
\begin{equation}
D\; =\; i\rlap {/}{\partial }+i\hat{m}+iMU^{\gamma5 }
\end{equation}
 with the pseudoscalar chiral field 
\begin{equation}
U^{\gamma _{5}}\; =\; \exp {(i\pi ^{a}\lambda ^{a}\gamma _{5})}
\; =\; \frac{1+\gamma _{5}}{2}U+\frac{1-\gamma _{5}}{2}U^{\dagger }.
\end{equation}
$\hat{m}$ is the matrix of the current quark mass given by 
\begin{equation}
\label{Eq:mass}
\hat{m}\; =\; \mbox {diag}(m_{\rm u},m_{\rm d},m_{\rm s})\; =\; 
m_{0}{\textbf {1}} \; +\; m_{8}\lambda _{8},
\end{equation}
where $\lambda ^{a}$ designate the usual Gell-Mann matrices normalized
as ${\rm tr}(\lambda ^{a}\lambda ^{b})=2\delta ^{ab}$.  Here, we have
assumed isospin symmetry ($m_{\rm u}=m_{\rm d}$).  $M$ stands 
for the dynamical quark mass arising from the spontaneous chiral
symmetry breaking, which is in general momentum-dependent. We regard 
$M$ as a constant and introduce the proper-time regularization for 
convenience.  The $m_{0}$ and $m_{8}$ in Eq.~(\ref{Eq:mass}) are
defined, respectively, by 
\begin{equation}
m_{0}\; =\; \frac{m_{\rm u}+m_{\rm d}+m_{\rm s}}{3},\; \; \; \; \; m_{8}
\; =\; \frac{m_{\rm u}+m_{\rm d}-2m_{\rm s}}{2\sqrt{3}}.
\end{equation}
The operator $D$ is expressed in Euclidean space in terms of the 
Euclidean time derivative $\partial _{4}$ and the Dirac 
one--particle Hamiltonian $H(U^{\gamma _{5}})$
\begin{equation}
\label{Eq:Dirac}
D\; =\; i\partial _{4}\; +\; iH(U^{\gamma _{5}})+i\gamma _{4}\hat{m}
-i\gamma _{4}\bar{m}{\textbf {1}}
\end{equation}
with 
\begin{equation}
\label{Eq:hamil}
H(U^{\gamma_{5}})\; =\; \gamma _{4}\gamma _{k}\partial _{k}
\; +\; \gamma _{4}MU^{\gamma _{5}}\; +\; \gamma _{4}\bar{m}{\textbf {1}}.
\end{equation}
$\bar{m}$ is introduced in such a way that it produces a correct 
Yukawa-type asymptotic behavior of the profile function. The model is
allowed only to have the same asymptote for the pion- and kaon-field 
because both originate from the same solitonic profile functions. In 
order to estimate the systematic error caused by this shortcoming we 
perform two calculations with different values of $\bar{m}$ 
such that one shows a pion tail and the other a kaon
tail.  The effect of adding the $\bar{m}$ term in the single quark 
Hamiltonian given in Eq.~(\ref{Eq:hamil}) is compensated approximately 
by subtracting this term in the collective part of the single quark 
propagator. 

The $U$ is assumed to have a structure corresponding to the so-called 
trivial embedding of the SU(2)-hedgehog into SU(3): 
\begin{equation}
\label{Eq:imbed}
U\; =\; \left( \begin{array}{cc}
U_{0} & 0\\
0 & 1
\end{array} \right) ,
\end{equation}
 with 
\begin{equation}
\label{Eq:profile}
U_{0}\; =\; \exp {[i{\bf n}\cdot {\bm \tau }P(r)]}.
\end{equation}
 The partition function ${\mathcal{Z}}$ of Eq.(\ref{Eq:action}) is 
simplified by the stationary phase approximation, which is justified 
in the large $N_{c}$ limit of ${\mathcal{Z}}$.  One ends up with one 
stationary profile function $P(r)$ which is determined numerically 
by solving the Euler-Lagrange equation corresponding to 
$\frac{\delta S_{\rm eff}}{\delta P(r)}=0$.  This yields a
selfconsistent classical field $U_{0}$ and a set of single quark energies
and corresponding states $E_{n}$ and $\Psi _{n}$ resulting from
solving $H\Psi_{n} = E_{n} \Psi_{n}$.  Note that the $E_{n}$ and $\Psi
_{n}$ do not constitute the nucleon  $|N\rangle$ yet because the
collective spin and and isospin  quantum numbers are missing. Those are
obtained by the semiclassical  quantization procedure, which consists in
performing the functional integral over rotations and translations of the
selfconsistent field $U_0 (x)$. 

The baryon state is expressed in terms of 
the Ioffe-type baryon current: $J_{N}(x)$ in Euclidean space 
($x_{0}=-ix_{4}$): 
\begin{eqnarray}
| {N(p)}\rangle & = & \lim _{y_{4}\rightarrow -\infty }
e^{p_{4}y_{4}}{\mathcal{N}}^{*}(p_{1})\int d^{3}y
e^{i{\bm p}\cdot {\bm y}}J^{\dagger }_{N}(y)| {0}\rangle\nonumber \\
\langle N(p')| & = & \lim _{x_{4}\rightarrow +\infty }
e^{-p_{4}x_{4}}{\mathcal{N}}(p')\int d^{3}x
e^{-i{\bm p}'\cdot {\bm x}}\langle 0 | J_{N}(x).
\end{eqnarray}
 The nucleon current $J^{\dagger }_{N}(J_{N})$ plays a role of creating
(annihilating) baryons. ${\mathcal{N}}^{*}({\mathcal{N}})$ are the 
normalization factors depending on the initial (final) momenta.  The 
baryon current consists of $N_{c}$ quarks: 
\begin{equation}
J_{N}(x)\; =\; \frac{1}{N_{c}!}\epsilon ^{c_{1}c_{2}\cdots c_{N_{c}}}
\Gamma ^{\alpha _{1}\alpha _{2}\cdots 
\alpha _{N_{c}}}_{(TT_{3}Y)(JJ_{3}Y_{R})}\psi _{\alpha _{1}c_{1}}(x)
\cdots \psi _{\alpha _{N_{c}}c_{N_{c}}}(x).
\end{equation}
$\alpha _{1}\cdots \alpha _{N_{c}}$ and $c_{1}\cdots c_{N_{c}}$ denote
respectively spin-isospin and color indices. 
$\Gamma ^{\{\alpha \}}_{(TT_{3}Y)(JJ_{3}Y_{R})}$
are matrices with the quantum numbers $(TT_{3}Y)(JJ_{3}Y_{R})$. 
The right hypercharge will be constrained by the baryon number.  
The creation baryon current is written as 
\begin{equation}
J^{\dagger }_{N}(y)\; =\; \frac{1}{N_{c}!}\epsilon ^{c_{1}c_{2}
\cdots c_{N_{c}}}\Gamma ^{\alpha _{1}\alpha _{2}\cdots 
\alpha _{N_{c}}* }_{(TT_{3}Y)(JJ_{3}Y_{R})}\left( -i\psi ^{\dagger }
\gamma _{4}\right) _{\alpha _{N_{c}}c_{N_{c}}}(x)\cdots 
\left( -i\psi ^{\dagger }\gamma _{4}\right) _{\alpha _{1}c_{1}}(x).
\end{equation}

The matrix elements of the strange vector current in Euclidean space is then
written as follows: 
\begin{eqnarray}
\langle N'(p')|-is^{\dagger }\gamma _{\mu }s|N(p)\rangle  
& = & \lim _{y_{4}\rightarrow -\infty 
\atop x_{4}\rightarrow +\infty }{\mathcal{N}}^{*}(p){\mathcal{N}}
(p')e^{p_{4}y_{4}-p_{4}x_{4}}\nonumber \\
 & \times  & \int d^{3}yd^{3}xe^{i{\bm p}\cdot {\bm y}-i{\bm p}'
\cdot {\bm x}}\langle 0 | J_{N}(x)(-i\psi^\dagger)\gamma_\mu
\hat{Q}_{\rm s} 
\psi J^{\dagger }_{N}(y)| {0}\rangle,
\label{Eq:mat1}
\end{eqnarray}
where $\hat{Q}_{\rm s}={\rm diag}(0,0,1)$ is called the {\em
strangeness operator}: We employ the non-standard sign 
convention used by Jaffe~\cite{Jaffe:1989mj} for the strange current.  
The matrix elements in Eq.(\ref{Eq:mat1}) 
can be related to the following correlator: 
\begin{equation}
\label{Eq:correl}
\lim _{y_{4}\rightarrow -\infty \atop x_{4}\rightarrow +\infty }
\langle 0 | J_{N}(x)(-i\psi^\dagger)\gamma_\mu \hat{Q}_{\rm s} 
\psi J^{\dagger }_{N}(y)| {0}\rangle\; =\; 
\lim _{y_{4}\rightarrow -\infty 
\atop x_{4}\rightarrow +\infty }{\mathcal{K}}
\end{equation}
with 
\begin{equation}
{\mathcal{K}}\; =\; \frac{1}{\mathcal{Z}}\int D\psi 
D\psi ^{\dagger }DUJ_{N}(x)\left( -i\psi ^{\dagger }\right)
 \gamma_\mu \hat{Q}_{\rm s} \psi J^{\dagger }_{N}(y)\exp 
\left[ \int d^{4}x\psi ^{\dagger }
\left( i\rlap {/}\partial +iMU^{\gamma ^{5}}+i\hat{m}\right) \psi \right] .
\end{equation}
In order to calculate the nucleonic correlation function, we introduce the 
corresponding generating functional: 
\begin{eqnarray}
{\mathcal{W}}[\eta ^{\dagger },\eta ,s_{\mu }] 
& = & \frac{1}{\mathcal{Z}}\int D\psi D\psi ^{\dagger }
DU\exp \left[ \int d^{4}x\left( \psi ^{\dagger } D\psi +i\eta ^{\dagger }\psi 
+i\psi ^{\dagger }\eta +\psi ^{\dagger }is_{\mu }\gamma_\mu
\hat{Q}_{\rm s}\psi \right) \right] .
\end{eqnarray}
 Having integrated the quark fields out, we obtain 
\begin{eqnarray}
{\mathcal{W}}[\eta ^{\dagger },\eta ,s_{\mu }] 
& = & \frac{1}{\mathcal{Z}}\int DU\det \left[ D+is_{\mu }
\gamma_\mu \hat{Q}_{\rm s} \right] \nonumber \\
 & \times  & \exp \left[ -\int d^{4}xd^{4}y
\eta ^{\dagger }_{\alpha }(x) \left \langle x,\alpha\left|
\frac{1}{D+is_{\mu }\gamma_\mu \hat{Q}_{\rm s}}
\right| y,\beta\right\rangle \eta _{\beta }(y)\right] ,
\end{eqnarray}
from which we can calculate the correlation function~(\ref{Eq:correl}) as
follows: 
\begin{eqnarray}
{\cal K} & = & \Gamma ^{\alpha _{1}\alpha _{2}\cdots 
\alpha _{N_{c}}}_{(\frac{1}{2}T_{3}Y)(\frac{1}{2}J_{3}Y_{R})}
\Gamma ^{\beta _{1}\beta _{2}\cdots \beta _{N_{c}}* }_{(
\frac{1}{2}T_{3}Y)(\frac{1}{2}J_{3}Y_{R})}\nonumber \\
 & \times  & \frac{\delta }{\delta \eta ^{\dagger }_{
\alpha _{1}}({\bm x},x_{0})}\frac{\delta }{\delta 
\eta ^{\dagger }_{\alpha _{2}}({\bm x},x_{0})}\cdots 
\frac{\delta }{\delta \eta ^{\dagger }_{\alpha _{N_{c}}}({\bm x},x_{0})}
\frac{\delta }{\delta s_{\mu }(0)}\left. {\mathcal{W}}[\eta ^{\dagger },
\eta ,s_{\mu }]\right| _{\eta ^{\dagger },\eta ,s_{\mu }=0}\nonumber \\
 & \times  & \frac{\stackrel{\leftarrow }{\delta }}{\delta 
\eta _{\beta _{1}}({\bm y},y_{0})}\frac{\stackrel{\leftarrow }{\delta }}{
\delta \eta _{\beta _{2}}({\bm y},y_{0})}\cdots 
\frac{\stackrel{\leftarrow }{\delta }}{\delta 
\eta _{\beta _{N_{c}}}({\bm y},y_{0})}.
\end{eqnarray}
Having carried out the functional derivatives, we have two terms, {\em i.e.}
valence and sea parts: 
\begin{equation}
{\mathcal{K}}  =  {\cal K}_{\rm val} + {\cal K}_{\rm sea},
\end{equation}
where 
\begin{eqnarray}
{\mathcal{K}}_{{\rm val}} & = & \frac{1}{\mathcal{Z}}\int 
DU\Gamma ^{\{\alpha \}}_{T'T'_{3}Y'}
\Gamma ^{\{\beta * \}}_{TT_{3}Y}N_{c}e^{-S_{{\rm eff}}}
\prod ^{N_{c}}_{k=2}{\mathcal{G}}_{\alpha _{k}\beta _{k}}(x|y)
\left[ {\mathcal{G}}_{\alpha _{1}\alpha }(x|0)(-1)\left(\gamma_\mu
\gamma _{4}\hat{Q}_{\rm s}\right) _{\alpha \beta }{\mathcal{G}}_{\beta 
\beta _{1}}(0|y)\right], \nonumber \\
{\mathcal{K}}_{{\rm sea}} & = & \frac{1}{\mathcal{Z}}
\int DU\Gamma ^{\{\alpha \}}_{T'T'_{3}Y'}
\Gamma ^{\{\beta * \}}_{TT_{3}Y}N_{c}e^{-S_{{\rm eff}}}
\prod ^{N_{c}}_{k=1}{\mathcal{G}}_{\alpha _{k}\beta _{k}}(x|y)
\left[ (-1){{\rm tr}}\left(\gamma_\mu\gamma _{4}
\hat{Q}_{\rm s} {\mathcal{G}}(0|0)\right) \right]. 
\label{Eq:correl1}
\end{eqnarray}
Here, ${\mathcal{G}}(x|y)$ denotes the quark propagator:
\begin{eqnarray}
{\mathcal{G}}(x|y) & = & \left\langle x \left | \frac{1}{i\rlap {/}\partial 
+iMU^{\gamma ^{5}}+i\hat{m}}i\gamma _{4}\right | {y}\right \rangle\nonumber \\
 & = & \left\langle x \left|\frac{1}{\partial _{4 }+H(U^{\gamma_5}) +
\gamma _{4}\hat{m}-\gamma _{4}\bar{m}{\textbf {1}}}\right | {y}\right\rangle.
\end{eqnarray}

Since the hedgehog solution given in Eq.(\ref{Eq:imbed}) is not invariant 
under the rotational and translational symmetries of the effective action, 
we need to restore these continuous symmetries, {\em i.e.} to consider 
quantum fluctuations around the classical solution, so that
the quantum numbers of the baryon are well determined.  
The quantum fluctuations in the direction of the zero modes play a 
particular role, since they are not at all small.  
There are three translational zero modes and seven rotational ones 
for the SU(3) soliton.    

Explicitly, the zero modes are taken into account by adiabatically 
rotating and translating the classical hedgehog configuration.
The rotational zero modes can be considered as follows:
\begin{equation}
U({\bm x},t)\; =\; R(t)U_c({\bm x})R^{\dagger }(t),
\end{equation}
where $R(t)$ denotes the unitary time-dependent SU(3) orientation matrix
of the soliton and $U_c({\bm x})$ is the stationary meson configuration.
The translational zero modes can be taken into account as follows:
\begin{equation}
U({\bm x},t)\; =\; U_c({\bm x}-{\bm Z}(t))\;=\;
T_{\bm Z(t)} U_c ({\bm x}) T^\dagger_{\bm Z(t)},
\end{equation}
where $T_{\bm Z(t)}$ is the unitary translational operator which 
causes a translation ${\bm Z}$.  Explicitly, they can be written as
\begin{eqnarray}
R(t) &=& \exp \left(i\Omega (t) \right)\nonumber \cr 
T_{\bm Z(t)} &=& \exp\left(i{\bm P}\cdot {\bm Z}(t)\right)
\end{eqnarray}
with the angular velocity in flavor space and the momentum operator:
\begin{equation}
\Omega (t) \;=\; \frac12 \Omega^a(t) \lambda^a,\;\;\;
{\bm P} = -i \nabla.
\end{equation}
Thus, we write 
\begin{eqnarray}
U({\bm x},t)& =& R(t)T_{\bm Z(t)}U_c({\bm x})T^\dagger_{\bm Z(t)}
R^{\dagger }(t)\nonumber \\
&=& R(t)U_c({\bm x}-{\bm Z}(t))R^{\dagger }(t).
\end{eqnarray}
The transformed Dirac operator becomes
\begin{eqnarray}
D(U) &=& i\rlap {/}\partial +iMU^{\gamma ^{5}}+i\hat{m}\nonumber \\
&=& R(t)T_{\bm Z(t)}\left(i\rlap {/}\partial +iMU_{c}^{\gamma ^{5}} 
+ i \gamma_4 R^\dagger \dot{R} -i\dot{{\bm Z}}\cdot {\bm \nabla } + 
iR^\dagger(t) \hat{m}R(t) \right)T^\dagger_{\bm Z(t)} R^\dagger(t),
\end{eqnarray}
where $\dot{\bm Z}$ denotes the velocity of the translational motion:
\begin{equation}
\dot{\bm Z} \;=\; \frac{d{\bm Z}}{dt}.
\end{equation}
The corresponding effective action can now be expressed by 
\begin{equation}
S_{{\rm eff}} \; = \; -N_{c}{{\rm Sp}}\log\left( i\rlap {/}\partial +
iMU^{\gamma ^{5}}_c+i\gamma _{4}R^{\dagger } \dot{R}+iR^{\dagger }\hat{m}R-
i\dot{{\bm Z}}\cdot {\bm \nabla }\right) .
\label{Eq:efS}
\end{equation}
Expanding Eq.(\ref{Eq:efS}) in powers of angular and translational velocities
($\sim1/N_c$), we end up with the effective action for the collective 
coordinates:
\begin{equation}
S_{\rm eff}\; \simeq \; -N_{c}{{\rm Sp}}\log\left( i\rlap {/}\partial 
+iMU^{\gamma ^{5}}_c \right) +S_{{\rm rot}}[R]+S_{{\rm trans}}[{\bm Z}]
+ S_{\rm br} [R^\dagger \hat{m} R]
\end{equation}
with 
\begin{eqnarray}
S_{{\rm rot}}[R] & = & \frac{1}{2}I_{ab}\int ^{T}_{0}dt
\Omega ^{a}\Omega ^{b} + \frac{iN_c}{2\sqrt{3}} 
\int ^{T}_{0}dt  \Omega^8 (t),
\nonumber \\
S_{{\rm trans}}[{\bm Z}] & = & \int ^{T}_{0}dt
\frac{M_{{\rm cl}}\dot{{\bm Z}}^{2}}{2}.
\end{eqnarray}
The $S_{\rm br}$ is the current quark mass corrections taken into account
to order ${\cal O} (m^2_{s})$.  The collective Hamiltonian can be easily
constructed from the collective actions and corresponding colllective
wave functions~\cite{Blotzetal}.
    
With the zero modes considered, the correlation function ${\cal K}$ in 
Eq.(\ref{Eq:correl1}) becomes:
\begin{eqnarray}
{\mathcal{K}}_{{\rm val}} & = & \frac{1}{\mathcal{Z}}\int 
DR D{\bm Z} \Gamma ^{\{\alpha \}}_{T'T'_{3}Y'}
\Gamma ^{\{\beta * \}}_{TT_{3}Y}N_{c}e^{-S_{{\rm eff}}}\cr
&\times&
\prod ^{N_{c}}_{k=2}{\tilde{\mathcal{G}}}_{\alpha _{k}\beta _{k}}(x|y)
\left[ \tilde{{\mathcal{G}}}_{\alpha _{1}\alpha }(x|0)(-1)\left(
\gamma_\mu\gamma_{4}\hat{Q}_{\rm s}
\right)_{\alpha \beta }{\tilde{\mathcal{G}}}_{\beta 
\beta _{1}}(0|y)\right], \nonumber \\
{\mathcal{K}}_{{\rm sea}} & = & \frac{1}{\mathcal{Z}}
\int DR D{\bm Z} \Gamma ^{\{\alpha \}}_{T'T'_{3}Y'}
\Gamma ^{\{\beta * \}}_{TT_{3}Y}N_{c}e^{-S_{{\rm eff}}}\cr
&\times&
\prod ^{N_{c}}_{k=1}\tilde{{\mathcal{G}}}_{\alpha _{k}\beta _{k}}(x|y)
\left[ (-1){{\rm tr}}\left(\gamma_\mu\gamma_{4}\hat{Q}_{\rm s}
\tilde{{\mathcal{G}}}(0|0)
\gamma _{4}\right) \right], 
\label{Eq:correl2}
\end{eqnarray}
where
\begin{eqnarray}
&& \tilde{{\mathcal{G}}}_{\alpha _{k}\beta _{k}}(x|y) \nonumber \\
&=&\left\langle x,\alpha \left|\frac{1}{\partial _{4 }+H(U)+
\gamma _{4}\hat{m}-\gamma _{4}\bar{m}{\textbf {1}}}\right| 
{y,\beta }\right\rangle\\
 & = & \left\langle x_{4},{\bm x}-{\bm Z},\alpha \left|R(x_{4})
\frac{1}{\partial _{4 }+H(U_c)+i\Omega +R^{\dagger }\gamma _{4}
\hat{m}R-\gamma _{4}\bar{m}{\textbf {1}}}R^{\dagger }(y_{4})
\right| {y_{4},{\bm y}-{\bm Z},\beta }\right\rangle.
\label{Eq:prop2}
\end{eqnarray}
We take into consideration the rotational $1/N_c$ corrections and linear
quark mass corrections, so that we expand the propagator given
by Eq.(\ref{Eq:prop2}) to linear order in powers of $i\Omega$ and 
$R^{\dagger }\gamma _{4}\hat{m}R-\gamma _{4}\bar{m}{\textbf {1}}$:
\begin{eqnarray}
&& \tilde{{\mathcal{G}}}_{\alpha _{k}\beta _{k}}(x|y) \nonumber \\
 & \simeq & \langle x_{4},{\bm x}-{\bm Z},\alpha |R(x_{4}) {\mathcal{G}}^{(0)}
R^{\dagger }(y_{4})| {y_{4},{\bm y}-{\bm Z},\beta }\rangle \nonumber \\
 & - & \langle x_{4},{\bm x}-{\bm Z},\alpha |R(x_{4}){\mathcal{G}}^{(0)}i
\Omega {\mathcal{G}}^{(0)}R^{\dagger }(y_{4})
| {y_{4},{\bm y}-{\bm Z},\beta }\rangle
\nonumber \\
& - & \langle x_{4},{\bm x}-{\bm Z},\alpha |R(x_{4}) {\mathcal{G}}^{(0)}
\left( R^{\dagger }\gamma _{4}\hat{m}R-\gamma _{4}\bar{m}{\textbf {1}}\right)
 {\mathcal{G}}^{(0)}R^{\dagger }(y_{4})| {y_{4},{\bm y}-{\bm Z},\beta }
\rangle. 
\end{eqnarray}
The functional integral over the zero modes quantize the soliton to have
the right quantum numbers of the baryon.  Then the angular velocity 
$R^\dagger \dot{R}=i\Omega$ ($\sim 1/N_c$) is related to the right 
generator (spin operator) after the quantization:
\begin{equation}
{\cal R}_a \;=\;-\Omega_b I_{ba} + \frac{N_c}{2\sqrt{3}}\delta_{8a}
- 2 K_{ab} m_3 D^{(8)}_{3a}(R) - 2K_{ab} m_8 D^{(8)}_{8b}(R),
\end{equation}
where $I_{ab}$ and $K_{ab}$ denote the moments of inertia~\cite{Blotzetal}.
The right generator does not commute with the rotational matrix $R$,
which requires that one has to take care of the ordering of those collective
operators.  It can be understood that the functional integral over the
collective operators will be identified as the time-ordering of the collective
operators:
\begin{eqnarray}
&& \int_{R(T_1)}^{R(T_2)} DR {\cal O}(R(t_1))\cdots {\cal O}(R(t_n))
e^{-S_{\rm eff}}\nonumber \\
&=& \langle R(T_2 ),T_2 | {\cal T}
[\hat{\cal O}(R(t_1))\cdots \hat{\cal O}(R(t_n))] |R(t_1), T_1 \rangle.
\end{eqnarray}
While this time-ordering makes the $\chi$QSM describe the observables
quantitatively in SU(2), it has a flaw in SU(3) in subleading order of 
$1/N_c$ because it violates the Gell-Mann--Nishijima
relation~\cite{Watabe:1998vi}.   This is deeply related to the non-locality in
time inherited in the $\chi$QSM and the large $N_c$ limit. Therefore, we need
to modify the quantization scheme in such a way that it satisfies the
commutation rules of the generators even in subleading order of 
$1/N_c$ . See ref.~\cite{Praszalowicz:1998jm} for details.   

The octet baryon collective wave functions are eigenfunctions of the
collective Hamiltonian corresponding to $S_{\rm rot}$.  Since we take 
into account the $m_{\rm s}$ corrections, these wave functions are not in
pure octet states but in mixed states with higher representations:
\begin{equation} \Psi_{N} (R) \;=\; \Psi_N^{(8)} (R) + 
m_8 c_{\overline{10}}^N \Psi_{N}^{({\overline{10}})} (R) + 
m_8 c_N^{27} \Psi_N^{(27)} (R),
\end{equation}
where
\begin{equation}
\label{aaB}
c_{\overline{10}}^N=-c_{\overline{10}}\sqrt{5},
\;\;\; c_{27}^N=-c_{27}3\sqrt{2}.
\end{equation}
The coefficients $c_{\overline{10}}$ and $c_{27}$ are defined as 
\begin{equation}
c_{\overline{10}}=\frac{I_2}{15}
\left( \sigma -\frac{K_1}{I_1}\right) ,\;\;\;
c_{27}=\frac{I_2}{25}\left( \sigma +\frac{K_1}{3I_1}-\frac{4K_2}{3I_2}
\right) ,
\end{equation}
where $I_i$ and $K_i$ ($i=1,2$) are moments of inertia~\cite{Blotzetal}.
The constant $\sigma$ is related to the SU(2) $\pi N$ sigma term
$\Sigma_{\rm SU(2)}=\frac{3\sigma}{2(m_u+m_d)}$.
The collective wave function can be explicitly expressed in terms of the
SU(3) Wigner $D^{\cal R}$ function:
\begin{equation}
\Psi_{N}^{({\cal R})} \;=\; (-1)^{S_3 - 1/2} \sqrt{{\rm dim}({\cal R})}
\left[D_{(YTT_3)(-1SS_3)}^{({\cal R})}\right]^*.
\end{equation}
Thus, there are two sources for the SU(3) symmetry breaking terms within
the $\chi$QSM: One from the action, the other from the baryon wave functions.

\section{Strange and flavor-singlet form factors in the $\chi$QSM}

The strange and singlet vector form factors of the baryons are expressed
in the quark matrix elements as follows: 
\begin{equation}
\langle N (p')|J^{{\rm s},(0)}_{\mu }| N(p)\rangle \; =\; \bar{u}_{N}(p')
\left[ \gamma _{\mu }F^{{\rm s},0}_{1}(q^{2})+i\sigma _{\mu \nu }
\frac{q^{\nu }}{2M_{N}}F^{{\rm s},0}_{2}(q^{2})\right] u_{N}(p),
\label{Eq:ff1}
\end{equation}
 where \( q^{2} \) is the square of the four momentum transfer 
\( q^{2}=-Q^{2} \) with \( Q^{2}>0 \). \( M_{N} \) and \( u_{N}(p) \) 
stand for the nucleon mass and its spinor, respectively. The strange quark 
current $J^{\rm s}_{\mu }$ can be expressed in terms of the
flavor-singlet and flavor-octet currents: 
\begin{equation}
\label{Eq:scur}
J^{\rm s}_{\mu } \;=\; -i\psi^\dagger \gamma _{\mu }\hat{Q}_{s} \psi
\; =\; \frac{1}{N_{c}}J^{(0)}_{\mu }-\frac{1}{\sqrt{3}}J^{(8)}_{\mu },
\end{equation}
where $J^{(0)}_{\mu }$ and $J^{(8)}_{\mu }$ are the flavor-singlet
and flavor-octet currents, respectively: 
\begin{eqnarray}
J^{(0)}_{\mu } & = & -i\psi^\dagger\gamma _{\mu }\psi
\nonumber \\
J^{(8)}_{\mu } & = & -i\psi^\dagger \gamma _{\mu }\lambda _{8}\psi.
\label{Eq:bar}
\end{eqnarray}
$N_{c}=3$ denotes the number of colors of the quark, 
which is correctly introduced to make it sure that the baryon number must
be equal to unity.  
The baryon and hypercharge currents are equal to the singlet and 
octet currents, respectively. 

The strange (singlet) Dirac form factors 
$F^{{\rm s},0}_{1}$ and $F^{{\rm s},0}_{2}$ can be
written in terms of the strange (singlet) Sachs form factors, 
$G^{{\rm s},0}_{E}(Q^{2})$ and $G^{{\rm s},0}_{M}(Q^{2})$: 
\begin{eqnarray}
G^{{\rm s},0}_{E}(Q^{2}) & = & F^{{\rm s},0}_{1}(Q^{2})
-\frac{Q^{2}}{4M^{2}_{N}}F^{{\rm s},0}_{2}(Q^{2})\nonumber \\
G_{M}^{{\rm s},0}(Q^{2}) & = & F^{{\rm s},0}_{1}(Q^{2})
+F^{{\rm s},0}_{2}(Q^{2}).
\end{eqnarray}
 In the non--relativistic limit($Q^{2}\ll M^{2}_{N}$, which however is not
used in the present paper), the Sachs-type form factors $G^{{\rm
s},0}_{E}(Q^{2})$ and $G^{{\rm s},0}_{M}(Q^{2})$ are  related to the time and
space components of the strange current, respectively:  \begin{eqnarray}
G^{{\rm s},0}_{E}(Q^{2})&=&\langle N'(p')|J^{{\rm s},(0)}_{0}(0)|N(p)\rangle 
\nonumber \\
G^{{\rm s},0}_{M}(Q^{2}) & = &i M_N \epsilon_{ilk} \frac{q_l}{6q^2} 
{\rm tr}\left(\langle p',\lambda'| J^{{\rm s},(0)}_{i} |p,\lambda\rangle 
\sigma_k \right).
\label{Eq:gm} 
\end{eqnarray}
where $\sigma _{j}$ stand for Pauli spin matrices. 
The $|\lambda \rangle$ is the corresponding spin state of the 
baryon.  

Having carried out a lengthy calculation of Eq.(\ref{Eq:correl2}) following
strictly Ref.\cite{Praszalowicz:1998jm}, we obtain the expressions 
for the strange vector form factors and flavor-singlet form factors 
for the nucleon:
\begin{eqnarray}
G_{E,N}^{{\rm s},0} ({\bm Q}^2) &=& 
G_{E,N}^{{\rm s},0,m_s^{0}} ({\bm Q}^2) + 
G_{E,N}^{{\rm s},0,m_{\rm s}^{1},{\rm op}} ({\bm Q}^2) +
G_{E,N}^{{\rm s},0,m_{\rm s}^{1},{\rm wf}} ({\bm Q}^2),\nonumber \\
G_{M,N}^{{\rm s},0}({\bm Q}^2)&=& 
G_{M,N}^{{\rm s},0,m_s^{0}} ({\bm Q}^2) + 
G_{M,N}^{{\rm s},0,m_{\rm s}^{1},{\rm op}} ({\bm Q}^2) +
G_{M,N}^{{\rm s},0,m_{\rm s}^{1},{\rm wf}} ({\bm Q}^2),
\label{Eq:final}
\end{eqnarray}
where $G_{E,N}^{{\rm s},m_{\rm s}^{0}} ({\bm Q}^2) (G_{M,N}^{0,
m_{\rm s}^{0}} ({\bm Q}^2))$ 
stands for the SU(3) symmetric part of the strange (flavor-singlet) 
electric and magnetic form factors, whereas the symmetry breaking parts 
$G_{E,N}^{{\rm s},m_{\rm s}^{1},{\rm op}} ({\bm Q}^2)
(G_{M,N}^{0,m_{\rm s}^{1},{\rm op}} ({\bm Q}^2))$ and 
$G_{E,N}^{{\rm s},m_{\rm s}^{1},{\rm wf}} ({\bm Q}^2)
(G_{M,N}^{0,m_{\rm s}^{1},{\rm wf}} ({\bm Q}^2))$ correspond to 
the symmetry breaking in the operator and in the baryon wave functions,
respectively.  The final expressions for the strange vector form
factors are written explicitly as follows below. There the coefficients
like $c_{\overline{10}}$ are known from the SU(3) algebra, the $I_1$ etc. are
moments of ineria whose expressions can be found in ref.
\cite{Christov:1996vm}. The other quantities like ${\cal I}_2 ({\bm Q}^2)$ are
explicitely given in appendix A. 


\begin{eqnarray}
G_{E,N}^{{\rm s},m_{\rm s}^{0}} ({\bm Q}^2) &=& 
\frac1{10}\left(7 {\cal B}({\bm Q}^2)
- \frac{{\cal I}_1 ({\bm Q}^2)}{I_1}-6\frac{{\cal I}_2 ({\bm
Q}^2)}{I_2} \right) \label{es0B8} \\
&&  \nonumber \\
G_{E,N}^{{\rm s},m_{\rm s}^{1},{\rm op}} ({\bm Q}^2) &=&\frac 1{15}\left(
(m_0 - \bar{m})13
+m_8\frac{5}{\sqrt{3}}\right) {\cal C}({\bm Q}^2) \nonumber \\
&+&m_8 \frac{12}{15\sqrt{3}}\left(I_1 {\cal K}_1 ({\bm Q}^2 ) - 
K_1{\cal I}_1({\bm Q}^2)\right) \nonumber \\
&+& m_8\frac{12}{5\sqrt{3}} \left(I_2 {\cal K}_2 ({\bm Q}^2 ) - 
K_2{\cal I}_2({\bm Q}^2)\right) \label{es1opB8} \\
&&  \nonumber \\
G_{E,N}^{{\rm s},m_s^{1},{\rm wf}} ({\bm Q}^2) &=&
-m_8 \left(c_{\overline{10}} + \frac{6\sqrt{3}}{5I_1} \, 
c_{27}\right)\,{\cal B}({\bm Q}^2 )  
\nonumber \\  
&-& m_8\frac{1}{5I_1} \left(5 \,c_{\overline{10}} - 6
  \,c_{27}\right)\, {\cal I}_1 ({\bm Q}^2)
- m_8 \frac {24}{5\sqrt{3} I_2}c_{27} {\cal I}_2 ({\bm Q}^2), \\
G_{M,N}^{{\rm s},m_{\rm s}^{0}} ({\bm Q}^2) &=&\frac{M_N}{3|{\bm
    Q}|}\left\{-\frac 8{30} \left({\cal Q}_0 ({\bm Q}^2 ) + 
\frac{1}{I_1} {\cal Q}_1 ({\bm Q}^2 )  + \frac{1}{6I_2} {\cal X}_2({\bm Q}^2) 
\right)\,S_3 \right.\cr
&-&\left.\frac{1}{15I_1}{\cal X}_1 ({\bm Q}^2)\,S_3 \right\}  \\
G_{M,N}^{{\rm s},m_{\rm s}^{1},{\rm op}} ({\bm Q}^2) &=& 
\frac{M_N}{3|{\bm Q}|}\left\{-m_8\frac 4{135} \left(6{\cal M}_2 ({\bm Q}^2) - 2
    \frac{K_2}{I_2} {\cal X}_2 ({\bm Q}^2) \right)\,S_3\right. \cr
&-& m_8\frac1{9}\left({\cal M}_0 ({\bm Q}^2) 
+ {\cal M}_1 ({\bm Q}^2)
-\frac13\frac{K_1}{I_1}{\cal X}_1 ({\bm Q}^2)\right)\,S_3 \cr
&-& m_8\frac 1{15} \left({\cal M}_0 ({\bm Q}^2) - {\cal M}_1 ({\bm Q}^2)
+\frac13\frac{K_1}{I_1}{\cal X}_1 ({\bm Q}^2)\right)\,S_3\cr
&+& \left.(m_0 - \bar{m})\frac 8{15} {\cal M}_0 ({\bm Q}^2)\,
S_3\right\},  \label{ms1opB8} \\
G_{M,N}^{{\rm s},m_s^{1},{\rm wf}} ({\bm Q}^2) &=&\frac{M_N}{3|{\bm
    Q}|}\left\{-m_8\frac 8{45}\,c_{27}\,
\left({\cal Q}_0 ({\bm Q}^2 ) + 
\frac{1}{I_1} {\cal Q}_1 ({\bm Q}^2 )   \right.\right.\cr
&+& \left.\left. \frac{2}{I_2} {\cal X}_2
({\bm Q}^2)-\frac{3}{2I_1} {\cal X}_1 ({\bm Q}^2) \right)\right\} \,S_3,
\label{Eq:semform}
\end{eqnarray}
while the flavor-singlet vector form factor can be written by:
\begin{eqnarray}
G_{E,N}^{0,m_{\rm s}^{0}} ({\bm Q}^2) &=& 
{\cal B}({\bm Q}^2), \label{e0B8} \\
&&  \nonumber \\
G_{E,N}^{{0},m_{\rm s}^{1},{\rm op}} ({\bm Q}^2) &=& \left(
2 (m_0 - \bar{m})
+m_8\frac{3}{10\sqrt{3}}\right) {\cal C}({\bm Q}^2), \label{e1opB8} \\ 
G_{E,N}^{0,m_s^{1},{\rm wf}} ({\bm Q}^2) &=& 0, \\
G_{M,N}^{{0},m_{\rm s}^{0}} ({\bm Q}^2) &=&\frac{M_N}{|{\bm
    Q}|}\frac{{\cal X}_1 ({\bm Q}^2)}{I_1}\,S_3  \\
G_{M,N}^{{0},m_{\rm s}^{1},{\rm op}} ({\bm Q}^2) &=& 
\frac{M_N}{|{\bm Q}|}\frac{m_8\sqrt{3}}{15}
\left(6{\cal M}_1 ({\bm Q}^2) - 2
    \frac{K_1}{I_1} {\cal X}_1 ({\bm Q}^2) \right)\,S_3\\
G_{M,N}^{{0},m_s^{1},{\rm wf}} ({\bm Q}^2) &=& 0.
\label{Eq:singform}
\end{eqnarray}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results for G0, A4, and HAPPEX II experiments and Discussion}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section, we present the results obtained from the $\chi$QSM.
A detailed description on numerical methods is presented in
Ref.~\cite{Kim:1995mr,Christov:1996vm}.  The parameters of the model 
are the constituent quark mass $M$, the current quark mass $m_{\rm u}$, the
cut-off $\Lambda$ of the proper-time regularization, and the strange quark mass
$m_{\rm s}$. These parameters are not free but have to be adjusted to
independent observables in a judicious way: The $\Lambda$ and the $m_{\rm u}$
are  adjusted for a given $M$ in the mesonic sector.   
 The physical pion mass $m_\pi = 139$ MeV and the pion decay constant $f_\pi =
93$ MeV are reproduced by these parameters.  The strange quark mass is chosen 
to be $m_{\rm s} = 180$  MeV throughout the present work.
 The remaining parameter $M$ is varied from $400$ MeV to 
$450$ MeV.  The value of $420\mbox{ MeV}$, which for many years is known to
produce  the best fit to many observables~\cite{Christov:1996vm}, is selected
for our  final result in the baryonic sector.  We always assume isospin
symmetry.  Actually with this formalism we obtained
the results (within the admittedly large
experimental errors) in fairly good agreement with the data 
of SAMPLE and HAPPEX.

The formalism of the $\chi$QSM has been applied frequently to SU(3)
baryons.  In the present case, where explicitly a strange quantity is
considered, one meets a problem concerning the asymptotic behavior of
the kaon field.  While in SU(2) the soliton incorporates the
asymptotic pion behavior $\exp(-\mu r)/r$ with $\mu=m_\pi$ in a
natural way, the construction of the SU(3) hedgehog by Witten's
embedding causes all other Goldstone bosons to share the same
asymptotic behavior.  This, however, contradicts the common belief
that the asymptotic form form of the kaon field is given by 
$\exp(-\mu r)/r$ with $\mu=m_K$.  Therefore, in our procedure, there
is some sort of systematic error which we have to estimate.  We do
this, performing two separate calculations, first by choosing the
parameter $\bar{m}$ in Eq.(\ref{Eq:hamil}) such that the SU(3)
calculation yields a pionic tail for all the Goldstone boson fields
($\mu=m_\pi$), and second, by selecting $\bar{m}$ in order to get a
kaonic tail ($\mu=m_K$).  In both cases we compensate by modifying the
perturbative collective treatment of $m_{\rm s}$ (or $m_8$) by subtracting the
corresponding term.  Altogether we get for each strange observables
two values the differences of which gives a measure for the systematic
error of our solitonic calculation.  To get a feeling for the
dependence of our results on the other parameters of the model we also
present results for various constituent quark mass $M$ (always
yielding a correct $f_\pi = 93\,{\rm MeV}$ and $m_\pi = 139\,{\rm MeV}$, of
course).

The electric strange form factor $G_{\rm E}^{\rm s}(Q^2)$ for three
different values of $M$ is shown in Fig. 1 with the pionic asymptotics
and in Fig. 2 with kaonic asymptotics.  Although in the latter case the
variation of $M$ causes somewhat larger changes, altogether the effect
of this is not very important and smaller than the effect of having
different Yukawa tails.  This qualitative feature is true for all
other form factors, {\em i.e.} the magnetic strange one in Figs. 3 and 4,
the electric singlet one in Figs. 5 and 6, and the magnetic singlet
one in Figs. 7 and 8.  In fact, Figs. 1-8 contain all the information  
required to predict the outcome of G0, A4, and HAPPEX II, which we
summarized in Tables 1-6.  For this we consider only the results of
our model with $M=420$ MeV, since it has been shown in many
calculations~\cite{Christov:1996vm} that other baryonic observables
are reproduced best.  

In Table I we display the prediction of the $\chi$QSM for the G0
experiment.  Presented is the combination of the strange vector
form factors $G_{E}^{{\rm s}} + \beta G_{M}^{{\rm
s}}$.  Here, $\beta$ is defined as 
\begin{equation}
\beta (Q^2,\theta) \;=\; \frac{\tau G_{M}^{p\gamma}}{\epsilon
  G_{E}^{p\gamma}},
\end{equation}
where $\tau = Q^2/(4M_N^{2})$ and $\epsilon = 
[1+2(1+\tau)\tan^2(\theta/2)]^{-1}$ and the $G_M^{p\gamma}$ and
$G_{E}^{p\gamma}$ are taken from experiment.  For smaller $Q^2$ values, 
$G_{E}^{{\rm s}} + \beta G_{M}^{{\rm s}}$ is rather sensitive to 
which tail we use.  For example, the result at $Q^2=0.16\ {\rm GeV}^2$ shows 
$15$ to 50 $\%$ difference between kaon and pion tails, whereas at 
$Q^2=0.951\ {\rm GeV}^2$ we find $10$ to $15\ \%$ difference.
Thus, smaller $Q^2$ show more sensitivity to the tail.  The
difference between the results from the kaon and pion tails is 
comparatively smaller at backward angles. In any case this difference
indicates the size of the systematic error of our model.  

In Table II we list the predictions of the singlet form factor
$G_{E}^{0} + \beta G_{M}^0$ for the G0 experiment and in
in Table III we present the predictions of the strange form factor
$G_{E}^{{\rm s}} + \beta G_{M}^{{\rm s}}$ at $Q^2=0.227{\rm
GeV}^2$ for the A4 experiment.  Table IV is for the predictions of  
the singlet form factor for the A4 experiment.  In Table V we list 
the prediction of the strange form factor for the HAPPEX II
experiment, whereas table VI we predict the 
corresponding singlet form factor. Again the difference between the
numbers obtained using a pion- and also a kaon-tail indicates the
size of the systematic error of our model.

\section{Summary and Conclusion}
In the present work, we have investigated the strange vector form
factors $G_{E}^{{\rm s}}$ and $G_{M}^{{\rm s}}$and flavor singlet 
vector form factors within the framework of the SU(3) chiral quark-soliton
model, incorporating the symmetry-conserving quantization.  The
rotational $1/N_c$ and strange quark mass $m_{\rm s}$ corrections
were taken into account.  In order to get a feeling for the systematic
error of our approach in calculating such a sensitive quantity as a
strange form factor, we also have considered two different
asymptotic behaviors of the soliton in such a way that the tails of  
the soliton fall off according to the Yukawa mass of the pion and of
the kaon.  

We first have examined in detail the dependence of the strange form
factors on the constituent quark mass $M$ which is the only free
parameter we deal with.  The dependence on the $M$ turned out rather
mild in general and we chose $M=420 $ MeV for which many other
properties of the nucleon and the hyperons are reproduced. We also have
predicted the combination of the strange form factors, {\em i.e.} 
$G_{E}^{{\rm s}} + \beta G_{M}^{{\rm s}}$ and  $G_{E}^{0} + \beta G_{M}^0$
corresponding to kinematics of three different experiments, that is, the G0,
A4, and HAPPEX II experiments.  

For the presently used chiral quark-soliton model the derivation of
a strange contribution to the electromagnetic form factors of the
nucleon is a rather natural thing, since the theory can be
considered as a many body approach with a polarized Dirac sea.  In
fact, as one finds in other observables of the nucleon, about $5\%-10\%$ 
contribution comes from the strange $s\bar{s}$ excitation of the quark
sea~\cite{Christov:1996vm}.  If one compares the present approach with 
others in the literature, one finds a difference insofar that most of 
the theories yield a negative strange magnetic moment, whereas the 
present one produces a slightly positive one~\cite{Silva:2001st}.  The 
reason might lie in the fact that the present approach is the only one 
with quarks in a self-consistent static meson field, with a proper 
treatment of the symmetries in SU(3) including rotational 
corrections.  In particular, the meson field is closely related to the 
instanton liquid of the QCD vacuum.  So far the approach has been 
successful in SU(2) and here we have a sensitive test in SU(3).  It 
is planned also to calculate the asymmetries of parity-violating 
electron scattering directly.  For this we need axial-vector form 
factors calculated in SU(3) which is presently under way.  

\newpage
\section*{Acknowledgment}
The authors are grateful to Frank Maas (A4) for useful discussions.
AS acknowledges partial financial support from Praxis XXI/BD/15681/98.
The work of HCK is supported by the KOSEF (R01--2001--00014).  The
work has also been supported by the BMBF, the DFG, 
the COSY--Project (J\" ulich) and POCTI (MCT-Portugal).

\begin{appendix}
\section{Densities}
In this appendix, we provide the densities for the strange vector 
and flavor-singlet form factors given in Eqs.(\ref{Eq:semform},
\ref{Eq:singform}):
\begin{eqnarray} {\cal B}({\bm Q}^2) &=& \int d^3x j_0 ( Q r)
\left[\Psi_{\rm val}^\dagger
({\bm x}) \Psi_{\rm val}({\bm x}) - \frac12 \sum_n {\rm sgn}(E_n)
\Psi_n^{\dagger} ({\bm x}) \Psi_n ({\bm x})\right],\nonumber \\  
{\cal C} ({\bm Q}^2) &=& N_c \int d^3x j_0 ( Q r)
\int d^3 y \left[
\sum_{n\neq {\rm val}} \frac{\Psi_{n}^\dagger
({\bm x}) \Psi_{\rm val}({\bm x}) \Psi_n^{\dagger}({\bm y})\beta 
\Psi_{\rm val} ({\bm y})}{E_n - E_{\rm val}} \right. \nonumber \\
& & \hspace{3cm}\;-\;\left. \frac12 \sum_{n,m} 
\Psi_n^{\dagger} ({\bm x}) \Psi_m ({\bm x})
\Psi_m^{\dagger}({\bm y})\beta \Psi_n ({\bm y})
{\cal R}_{\cal M} (E_n, E_m)\right], \nonumber \\
{\cal I}_1({\bm Q}^2) & = & \frac{N_c}{6} 
\int d^3 x j_0 ( Q r) \;\int d^3 y
\left [\sum_{n\neq {\rm val}}\frac{\Psi^{\dagger}_{n} ({\bm x}) {\bm \tau}
\Psi_{\rm val} ({\bm x}) \cdot
\Psi^{\dagger}_{\rm val} ({\bm y}) {\bm \tau} \Psi_{n} ({\bm y})}
{E_n - E_{\rm val}} \right .
\nonumber \\  & & \hspace{3cm} \;+\; \left .
\frac{1}{2}\sum_{n,m}
\Psi^{\dagger}_{n} ({\bm x}) {\bm \tau} \Psi_{m} ({\bm x}) \cdot
\Psi^{\dagger}_{m} ({\bm y}) {\bm \tau} \Psi_{n} ({\bm y})
{\cal R}_{\cal I} (E_n, E_m) \right ],
\nonumber \\
{\cal I}_2({\bm Q}^2) & = &\frac{N_c}{4} 
\int d^3 x j_0 ( Q r) \;\int d^3 y
\left [\sum_{m^{0}}\frac{\Psi^{\dagger}_{m^{0}} ({\bm x}) \Psi_{\rm val} 
({\bm x}) \Psi^{\dagger}_{\rm val} ({\bm y}) \Psi_{m{^0}} ({\bm y})}
{E_{m^{0}} - E_{\rm val}} \right .
\nonumber \\  & & \hspace{3cm} \;+\;\left . \frac{1}{2}\sum_{n,m_0}
\Psi^{\dagger}_{n} ({\bm x}) \Psi_{m^{0}} ({\bm x})
\Psi^{\dagger}_{m^{0}} ({\bm y}) \Psi_{n} ({\bm y})
{\cal R}_{\cal I} (E_n, E_{m^{0}}) \right ],
\nonumber \\
{\cal K}_1({\bm Q}^2) & = & \frac{N_c}{6} 
\int d^3 x j_0 ( Q r) \; \int d^3 y
\left [\sum_{n}\frac{\Psi^{\dagger}_{n} ({\bm x})
{\bm \tau} \Psi_{\rm val} ({\bm x}) \cdot
\Psi^{\dagger}_{\rm val} ({\bm y}) \beta {\bm \tau} \Psi_{n} ({\bm y})}
{E_n - E_{\rm val}} \right .
\nonumber \\  & & \hspace{3cm} \;+\; \left . \frac{1}{2}\sum_{n,m}
\Psi^{\dagger}_{n} ({\bm x}) {\bm\tau} \Psi_{m} ({\bm x}) \cdot
\Psi^{\dagger}_{m} ({\bm y}) \beta {\bm \tau} \Psi_{n} ({\bm y})
{\cal R}_{\cal M} (E_n, E_m)
\right ],
\nonumber \\
{\cal K}_2({\bm Q}^2) & = & \frac{N_c}{4} 
\int d^3 x j_0 ( Q r)\; \int d^3 y
\left [\sum_{m^{0}}\frac{\Psi^{\dagger}_{m^{0}} ({\bm x}) \Psi_{\rm val} 
({\bm x}) \Psi^{\dagger}_{\rm val} ({\bm y}) \beta \Psi_{m{^0}} ({\bm y})}
{E_{m^{0}} - E_{\rm val}} \right .
\nonumber \\  & & \hspace{3cm} \;+\;\left . \frac{1}{2}\sum_{n,m_0}
\Psi^{\dagger}_{n} ({\bm x}) \Psi_{m^{0}} ({\bm x})
\Psi^{\dagger}_{m^{0}} ({\bm y}) \beta \Psi_{n} ({\bm y})
{\cal R}_{\cal M} (E_n, E_{m^{0}}) \right ],\\
{\cal Q}_0({\bm Q}^2)  & = &
\frac{N_c}{3}\int d^3 x \frac{j_1 ( Q r)}{r}\,
\left[ \Psi ^{\dagger}_{\rm val}({\bm x}) \gamma_{5}
\{{\bm r} \times {\bm\sigma} \} \cdot {\bm \tau}
\Psi_{\rm val} ({\bm x}) \right. \nonumber \\ & &
\left . \hspace{1cm} \;-\;
\frac{1}{2}  \sum_n {\rm sgn} (E_n)
\Psi ^{\dagger}_{n}({\bm x}) \gamma_{5}
\{{\bm r} \times {\bm\sigma} \} \cdot {\bm\tau}
\Psi_{n}({\bm x}) {\cal R}(E_n)\right ],
\nonumber \\
{\cal Q}_1({\bm Q}^2) & = &  \frac{iN_c}{2}
\int d^3 x \frac{j_1 ( Q r)}{r}\,
\int d^3 y \nonumber \\  & \times &
\left[\sum_{n}{\rm sgn} (E_n)
\frac{\Psi^{\dagger}_{n} ({\bm x}) \gamma_{5}
\{{\bm r} \times {\bm \sigma} \} \times {\bm\tau}
\Psi_{\rm val} ({\bm x}) \cdot
\Psi^{\dagger}_{\rm val} ({\bm y}) {\bm \tau} \Psi_{n} ({\bm y})}
{E_n - E_{\rm val}} \right .
\nonumber \\  & & %\hspace{1cm}
\;+\; \left . \frac{1}{2} \sum_{n,m}
\Psi^{\dagger}_{n} ({\bm x})\gamma_{5} \{{\bm r} \times {\bm\sigma} \}
\times {\bm\tau}  \Psi_{m} ({\bm x}) \cdot
\Psi^{\dagger}_{m} ({\bm y}) {\bm\tau} \Psi_{n} ({\bm y})
{\cal R}_{\cal Q} (E_n, E_m) \right ],
\nonumber \\
{\cal X}_1({\bm Q}^2)   & = & N_c
 \int d^3 x \frac{j_1 ( Q r)}{r} \,
\int d^3 y \left[\sum_{n}
\frac{\Psi^{\dagger}_{n} ({\bm x})\gamma_{5}
\{{\bm r} \times {\bm\sigma} \}
\Psi_{\rm val} ({\bm x}) \cdot
\Psi^{\dagger}_{\rm val} ({\bm y}) {\bm\tau} \Psi_{n} ({\bm y})}
{E_n - E_{\rm val}} \right .
\nonumber \\  &+ & \left. %\hspace{1cm} \;+\; \left .
\frac{1}{2} \sum_{n,m}
\Psi^{\dagger}_{n} ({\bm x})\gamma_{5} \{{\bm r} \times {\bm\sigma} \}
\Psi_{m} ({\bm x}) \cdot
\Psi^{\dagger}_{m} ({\bm y}) {\bm\tau} \Psi_{n} ({\bm y})
{\cal R}_{\cal M} (E_n, E_m) \right ],
\nonumber \\
{\cal X}_2({\bm Q}^2)   & = & N_c
 \int d^3 x \frac{j_1 ( Q r)}{r}\,
\int d^3 y \left[\sum_{m^0}
\frac{\Psi^{\dagger}_{m^0} ({\bm x})\gamma_{5}
\{{\bm r} \times {\bm\sigma} \} \cdot {\bm\tau}
\Psi_{\rm val} ({\bm x})
\Psi^{\dagger}_{\rm val} ({\bm y}) \Psi_{m^0} ({\bm y})}
{E_{m^0} - E_{\rm val}} \right .
\nonumber \\  & & \hspace{1cm}
\;+\; \left . \sum_{n,m^0}
\Psi^{\dagger}_{n} ({\bm x})\gamma_{5} \{{\bm r} \times {\bm\sigma} \}
\cdot {\bm\tau} \Psi_{m^0} ({\bm x})
\Psi^{\dagger}_{m^0} ({\bm y}) \Psi_{n} ({\bm y})
{\cal R}_{\cal M} (E_n, E_{m^0}) \right ],
\nonumber \\
{\cal M}_0 ({\bm Q}^2) & = &  N_c
\int d^3 x \frac{j_1 ( Q r)}{r}\,
\int d^3 y \left[ \sum_{n}
\frac{\Psi^{\dagger}_{n} ({\bm x}) \gamma_{5}
\{{\bm r} \times {\bm\sigma} \} \cdot {\bm\tau}
\Psi_{\rm val} ({\bm x})
\Psi^{\dagger}_{\rm val} ({\bm y}) \beta \Psi_{n} ({\bm y})}
{E_{n} - E_{\rm val}} \right .
\nonumber \\  & & \hspace{1cm}
\;+\; \left . \frac{1}{2} \sum_{n,m}
\Psi^{\dagger}_{n} ({\bm x})\gamma_{5} \{{\bm r} \times {\bm\sigma} \}
\cdot{\bm\tau}  \Psi_{m} ({\bm x})
\Psi^{\dagger}_{m} ({\bm y})\beta \Psi_{n} ({\bm y})
{\cal R}_{\beta} (E_n, E_m) \right ],
\nonumber \\
{\cal M}_1({\bm Q}^2)  & = & \frac{N_c}{3}
\int d^3 x \frac{j_1 ( Q r)}{r}\,
\int d^3 y \nonumber \\  & \times &
\left[\sum_{n} 
\frac{\Psi^{\dagger}_{n} ({\bm x})\gamma_{5}
\{{\bm r} \times {\bm\sigma} \}
\Psi_{\rm val} ({\bm x}) \cdot
\Psi^{\dagger}_{\rm val} ({\bm y}) \beta {\bm\tau} \Psi_{n} ({\bm y})}
{E_n - E_{\rm val}} \right .
\nonumber \\  &+ & %\hspace{1cm}
\left . \frac{1}{2} \sum_{n,m}
\Psi^{\dagger}_{n} ({\bm x})\gamma_{5} \{{\bm r} \times {\bm\sigma} \}
\Psi_{m} ({\bm x}) \cdot
\Psi^{\dagger}_{m} ({\bm y}) \beta {\bm\tau} \Psi_{n} ({\bm y})
{\cal R}_{\beta} (E_n, E_m) \right ],
\nonumber \\
{\cal M}_2({\bm Q}^2) & = &  \frac{N_c}{3} 
\int d^3 x \frac{j_1 ( Q r)}{r}\,
\int d^3 y \nonumber \\  & \times &
\left[\sum_{m^0}
\frac{\Psi^{\dagger}_{m^0} ({\bm x}) \gamma_{5}
\{{\bm r} \times {\bm\sigma} \} \cdot {\bm\tau}
\Psi_{\rm val} ({\bm x})
\Psi^{\dagger}_{\rm val} ({\bm y})\beta \Psi_{m^0} ({\bm y})}
{E_{m^0} - E_{\rm val}} \right .
\nonumber \\  &+ & %\hspace{1cm}
\left . \sum_{n,m^0}
\Psi^{\dagger}_{n} ({\bm x})\gamma_{5} \{{\bm r} \times {\bm\sigma} \}
\cdot {\bm\tau}  \Psi_{m^0} ({\bm x})
\Psi^{\dagger}_{m^0} ({\bm y}) \beta \Psi_{n} ({\bm y})
{\cal R}_{\beta} (E_n, E_{m^0}) \right ]  .
\label{Eq:mdens}
\end{eqnarray}
The regularization functions in Eq.(\ref{Eq:mdens}) are as follows:
\begin{eqnarray}
{\cal R}_{I} (E_n, E_m) & = & - \frac{1}{2\sqrt{\pi}}
\int^{\infty}_{0} \frac{du}{\sqrt{u}} \phi (u;\Lambda_i)
\left [ \frac{E_n e^{-u E^{2}_{n}} +  E_m e^{-u E^{2}_{m}}}
{E_n + E_m} \;+\; \frac{e^{-u E^{2}_{n}} - e^{-u E^{2}_{m}}}
{u(E^{2}_{n} - E^{2}_{m})} \right ],
\nonumber \\
{\cal R}_{\cal M} (E_n, E_m) & = &
\frac{1}{2}  \frac{ {\rm sgn} (E_n)
- {\rm sgn} (E_m)}{E_n - E_m},\nonumber \\
{\cal R} (E_n) & = & \int \frac{du}{\sqrt{\pi u}}
\phi (u;\Lambda_i) |E_n| e^{-uE^{2}_{n}},
\nonumber \\
{\cal R}_{\cal N} (E_n, E_m) & = &
\frac{1}{2}  \frac{ {\rm sgn} (E_n)
- {\rm sgn} (E_m)}{|E_n| + |E_m|},\nonumber \\
{\cal R}_{\cal Q} (E_n, E_m) & = & \frac{1}{2\pi} 
\int^{1}_{0} d\alpha \frac{\alpha (E_n + E_m) - E_m}
{\sqrt{\alpha ( 1 - \alpha)}}
\frac{\exp{\left (-[\alpha E^{2}_n + (1-\alpha)E^{2}_m]/
\Lambda^{2}_i  \right)}}{\alpha E^{2}_n + (1-\alpha)E^{2}_m},
\nonumber \\
{\cal R}_{\beta}  (E_n, E_m) & = &
\frac{1}{2\sqrt{\pi}} \int^{\infty}_{0}
\frac{du}{\sqrt{u}} \phi (u;\Lambda_i)
\left[ \frac{E_n e^{-uE^{2}_{n}} - E_m e^{-uE^{2}_{m}}}
{E_n - E_m}\right],
\label{Eq:regulm}
\end{eqnarray}
where the cutoff function $\phi(u;\Lambda_i)=\sum_i c_i \theta
\left(u - \frac{1}{\Lambda^{2}_{i}} \right)$ is
fixed by reproducing the pion decay constant and other mesonic properties
\cite{Christov:1996vm}.
\end{appendix}

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\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline{\large \bf FIGURES}
\vspace{1.2cm}

\includegraphics[height=12cm]{St_E_Mdeppi.eps}
\vspace{0.8cm}

\noindent {\bf FIG.1}: The dependence of the strange electric form factor as a
  function of $Q^2$ on the 
constituent quark mass with the pion asymptotic tail ($\mu=140$ MeV).
The solid curve is for $M=420$ MeV, the dashed one for $400$ MeV,
 and the dotted one for $450$ MeV.  The strange quark mass is 
$m_{\rm s}=180$ MeV.
\newpage

\includegraphics[height=12cm]{St_E_Mdepka.eps}
\vspace{0.8cm}

\noindent {\bf FIG.2}: The dependence of the strange electric form factor as a
function of $Q^2$ on the constituent quark mass with the kaon
asymptotic tail ($\mu=490$ MeV).
The solid curve is for $M=420$ MeV, the dashed one for $400$ MeV,
 and the dotted one for $450$ MeV.  The strange quark mass is 
$m_{\rm s}=180$ MeV.
\newpage 

\includegraphics[height=12cm]{St_M_Mdeppi.eps}
\vspace{0.8cm}

\noindent {\bf FIG.3}: The dependence of the strange magnetic form factor as a
function of $Q^2$ on the constituent quark mass with the pion
asymptotic tail ($\mu=140$ MeV).
The solid curve is for $M=420$ MeV, the dashed one for $400$ MeV,
 and the dotted one for $450$ MeV.  The strange quark mass is 
$m_{\rm s}=180$ MeV.  The experimental data are taken from 
SAMPLE~\cite{SAMPLE00s}.
\newpage

\includegraphics[height=12cm]{St_M_Mdepka.eps}
\vspace{0.8cm}

\noindent {\bf FIG.4}: The dependence of the strange magnetic form factor as a
function of $Q^2$ on the constituent quark mass with the kaon
asymptotic tail ($\mu=490$ MeV).
The solid curve is for $M=420$ MeV, the dashed one for $400$ MeV,
and the dotted one for $450$ MeV.  The strange quark mass is 
$m_{\rm s}=180$ MeV.  The experimental data are taken from 
SAMPLE~\cite{SAMPLE00s}.
\newpage 

\includegraphics[height=12cm]{G0E_Mdeph14.eps}
\vspace{0.8cm}

\noindent {\bf FIG. 5}: The dependence of the singlet electric form factor as a
  function of $Q^2$ on the 
constituent quark mass with the pion asymptotic tail ($\mu=140$ MeV).
The solid curve is for $M=420$ MeV, the dashed one for $400$ MeV,
 and the dotted one for $450$ MeV.  The strange quark mass is 
$m_{\rm s}=180$ MeV.
\newpage 

\includegraphics[height=12cm]{G0E_Mdeph49.eps}
\vspace{0.8cm}

\noindent {\bf FIG. 6}: The dependence of the singlet form factor as a
function of $Q^2$ on the constituent quark mass with the kaon
asymptotic tail ($\mu=490$ MeV).
The solid curve is for $M=420$ MeV, the dashed one for $400$ MeV,
 and the dotted one for $450$ MeV.  The strange quark mass is 
$m_{\rm s}=180$ MeV.
\newpage

\includegraphics[height=12cm]{G0M_Mdeph14.eps}
\vspace{0.8cm}

\noindent {\bf FIG. 7}: The dependence of the singlet magnetic 
form factor as a function of $Q^2$ on the 
constituent quark mass with the pion asymptotic tail ($\mu=140$ MeV).
The solid curve is for $M=420$ MeV, the dashed one for $400$ MeV,
 and the dotted one for $450$ MeV.  The strange quark mass is 
$m_{\rm s}=180$ MeV.
\newpage

\includegraphics[height=12cm]{G0M_Mdeph49.eps}
\vspace{0.8cm}

\noindent {\bf FIG. 8}: The dependence of the singlet magnetic form 
factor as a function of $Q^2$ on the constituent quark mass with the kaon
asymptotic tail ($\mu=490$ MeV).
The solid curve is for $M=420$ MeV, the dashed one for $400$ MeV,
 and the dotted one for $450$ MeV.  The strange quark mass is 
$m_{\rm s}=180$ MeV.
\newpage

\newpage
\centerline{\large \bf TABLES}
\vspace{1.2cm}

\begin{table}[ht]
\caption{Strange form factors: The prediction for the G0 experiment.  
The constituent quark mass $M$ is chosen to be $420$ MeV.  The range 
represents two different results with the pion and kaon tails, 
respectively, indicating the systematic error of the model.}
\begin{tabular}{|c|c|c|c|c|}
\hline
 & \multicolumn{2}{c|}{$\theta=10^\circ$ }
& \multicolumn{2}{c|}{$\theta=108^\circ$}
\\ \cline{1-5}
$Q^2\;[{\rm GeV}^2]$ & $\beta$ & $G_E^{{\rm s}} + \beta G_M^{{\rm
    s}}$ $(\mu = m_{\pi}\sim m_{\rm K})$ & $\beta$ 
&  $G_E^{{\rm s}} + \beta G_M^{{\rm s}}$ 
$(\mu = m_{\pi}\sim m_{\rm K})$
\\ \hline
$0.16$  & $0.13$ & $0.09\sim 0.05$ & $0.63$ & $0.11\sim 0.09$ \\
$0.24$  & $0.20$ & $0.10\sim 0.06$ & $0.99$ & $0.14\sim 0.11$ \\
$0.325$ & $0.26$ & $0.11\sim 0.07$ & $1.31$ & $0.14\sim 0.13$ \\
$0.435$ & $0.35$ & $0.11\sim 0.07$ & $1.81$ & $0.15\sim 0.14$ \\
$0.576$ & $0.47$ & $0.10\sim 0.07$ & $2.49$ & $0.14\sim 0.14$ \\
$0.751$ & $0.61$ & $0.08\sim 0.06$ & $3.35$ & $0.12\sim 0.13$ \\
$0.951$ & $0.81$ & $0.07\sim 0.06$ & $4.62$ & $0.11\sim 0.12$ \\
\hline
\end{tabular}
\end{table}

\begin{table}[ht]
\caption{Singlet form factors: The prediction for the G0 experiment.  
The constituent quark mass $M$ is chosen to be $420$ MeV.  The range 
represents two different results with the pion and kaon tails, 
respectively, indicating the systematic error of the model.}
\begin{tabular}{|c|c|c|c|c|}
\hline
 & \multicolumn{2}{c|}{$\theta=10^\circ$ }
& \multicolumn{2}{c|}{$\theta=108^\circ$}
\\ \cline{1-5}
$Q^2\;[{\rm GeV}^2]$ & $\beta$ & $G_E^{0} + \beta G_M^{0}$ 
$(\mu = m_{\pi}\sim m_{\rm K})$& 
$\beta$ &  $G_E^{0} + \beta G_M^{0}$ 
$(\mu = m_{\pi}\sim m_{\rm K})$
\\ \hline
$0.16$  & $0.13$ & $2.38\sim 2.53$ & $0.63$ & $3.05\sim 3.20$ \\
$0.24$  & $0.20$ & $2.13\sim 2.33$ & $0.99$ & $3.04\sim 3.27$ \\
$0.325$ & $0.26$ & $1.90\sim 2.14$ & $1.31$ & $2.91\sim 3.22$ \\
$0.435$ & $0.35$ & $1.65\sim 1.92$ & $1.81$ & $2.80\sim 3.21$ \\
$0.576$ & $0.47$ & $1.39\sim 1.69$ & $2.49$ & $2.63\sim 3.15$ \\
$0.751$ & $0.61$ & $1.13\sim 1.45$ & $3.35$ & $2.39\sim 3.03$ \\
$0.951$ & $0.81$ & $0.92\sim 1.24$ & $4.62$ & $2.21\sim 2.96$ \\
\hline
\end{tabular}
\end{table}

\begin{table}[ht]
\caption{Strange form factors: The prediction for the A4 experiment.  
The constituent quark mass $M$ is chosen to be $420$ MeV.  The range 
represents two different results with the pion and kaon tails, 
respectively, indicating the systematic error of the model.}
\begin{tabular}{|c|c|c|c|c|}
\hline
 & \multicolumn{2}{c|}{$\theta=35^\circ$ }
& \multicolumn{2}{c|}{$\theta=145^\circ$}
\\ \cline{1-5}
$Q^2\;[{\rm GeV}^2]$ & $\beta$ & $G_E^{{\rm s}} + \beta G_M^{{\rm
    s}}$ $(\mu = m_{\pi}\sim m_{\rm K})$& $\beta$ 
&  $G_E^{{\rm s}} + \beta G_M^{{\rm s}}$
$(\mu = m_{\pi}\sim m_{\rm K})$
\\ \hline
$0.10$  & $0.099$ & $0.07 \sim 0.04$ & $-$ & $-$ \\
$0.227$ & $0.22$ & $0.10\sim 0.06$ & $4.07$ & $0.28\sim 0.32$ \\
$0.47$ & $-$ & $-$ & $8.963$ & $0.33\sim 0.42$ \\
\hline
\end{tabular}
\end{table}

\begin{table}[ht]
\caption{Singlet form factors: The prediction for the A4 experiment.  
 The constituent quark mass $M$ is chosen to be $420$ MeV.  The range 
represents two different results with the pion and kaon tails, 
respectively, indicating the systematic error of the model.}
\begin{tabular}{|c|c|c|c|c|}
\hline
 & \multicolumn{2}{c|}{$\theta=35^\circ$ }
& \multicolumn{2}{c|}{$\theta=145^\circ$}
\\ \cline{1-5}
$Q^2\;[{\rm GeV}^2]$ & $\beta$ & $G_E^{0} + \beta G_M^{0}$ 
$(\mu = m_{\pi}\sim m_{\rm K})$ 
& $\beta$ &  $G_E^{0} + \beta G_M^{0}$
$(\mu = m_{\pi}\sim m_{\rm K})$
\\ \hline
$0.10$  & $0.099$ & $2.61\sim 2.72$ & $-$ & $-$ \\
$0.227$  & $0.22$ & $2.21\sim 2.40$ & $4.07$ & $6.72\sim 7.05$ \\
$0.47$ & $-$ & $-$ & $8.963$ & $7.91 \sim 9.04$ \\
\hline
\end{tabular}
\end{table}

\begin{table}[ht]
\caption{Strange form factors: The prediction for the HAPPEX II experiment.  
The constituent quark mass $M$ is chosen to be $420$ MeV.  The range 
represents two different results with the pion and kaon tails, 
respectively, indicating the systematic error of the model.}
\begin{tabular}{|c|c|c|}
\hline
 & \multicolumn{2}{c|}{$\theta=6^\circ$} 
\\ \cline{1-3}
$Q^2\;[{\rm GeV}^2]$ & $\beta$ & $G_E^{{\rm s}} + 
\beta G_M^{{\rm s}}$ $(\mu = m_{\pi}\sim m_{\rm K})$
\\ \hline
$0.11$  & $0.09$ & $0.07\sim 0.04$ \\
\hline
\end{tabular}
\end{table}
\begin{table}[ht]
\caption{Singlet form factors: The prediction for the HAPPEX II experiment.  
The constituent quark mass $M$ is chosen to be $420$ MeV.  The range 
represents two different results with the pion and kaon tails, 
respectively, indicating the systematic error of the model.}
\begin{tabular}{|c|c|c|}
\hline
 &  \multicolumn{2}{c|}{$\theta=0.09^\circ$} 
\\ \cline{1-3}
$Q^2\;[{\rm GeV}^2]$ & $\beta$ & $G_E^{0} + 
\beta G_M^{0}$ $(\mu = m_{\pi}\sim m_{\rm K})$
\\ \hline
$0.11$  & $0.09$ & $2.55\sim 2.62$ \\
\hline
\end{tabular}
\end{table}

\end{document}
\bye(

