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\begin{document}
\title{Octet and Decuplet Baryon Magnetic Moments in the Chiral Quark Model}
\author{Harleen Dahiya  and Manmohan Gupta}
\address {Department of Physics, Centre of Advanced Study in Physics,
Panjab University,Chandigarh-160 014, India.} 
\date{\today}
 \maketitle
\begin{abstract}
Octet and decuplet baryon magnetic moments have been formulated within
the chiral quark model, incorporating the  polarization of the sea quarks and
their orbital angular momentum through the Cheng-Li mechanism with
coupling breaking and mass breaking terms.
 In the case of octet magnetic moments, the results not only show improvement 
over the nonrelativistic quark model results but also give a non zero 
value for  the right hand side of Coleman-Glashow sum rule, usually
zero in most of the models. 
In the case of decuplet magnetic
moments, we obtain a good overlap with the data for 
$\Delta^{++}$ and $\Omega^-$.
In the case of octet, the predictions of the  
$\chi$QM with the Cheng-Li mechanism improve further
when effects of spin-spin forces and ``mass adjustments'' due to
confinement effects are included for both the
NMC as well as the E866 data for the $\bar u-\bar d$ asymmetry.
In the case of latest E866 data, we get
an excellent fit for the octet magnetic moments, specifically in the  
case of $p, \Sigma^+$, $\Xi^o$ and in 
the violation of Coleman Glashow sum rule 
we almost get a perfect fit, whereas in almost all the other cases the results 
are within 5\% of the data.
In the case of decuplet baryons, the measurement of the magnetic moments of
$\Delta^+$, $\Delta^-$, $\Sigma^+$ and $\Sigma^-$ would have important 
implications  for the Cheng-Li mechanism.
\end{abstract}

\section{Introduction}

The European Muon Collaboration (EMC)  measurements \cite{EMC} of
polarized structure functions of proton 
had shown that only about 30\% of the proton's spin is carried by the
valence quarks. This ``unexpected'' conclusion from the point of view of 
nonrelativistic quark model (NRQM), usually referred to as the
``proton spin crisis'',
becomes all the more intriguing
when it is realized that the NRQM is able to give a reasonably
good description of magnetic moments using the assumption
that magnetic moments of quarks are proportional to the spin carried
by them. Further, this issue regarding spin and magnetic moments becomes 
all the more difficult to
understand when it is realized that the magnetic moments of
baryons receive contributions not only from the magnetic moments carried
by the valence quarks but also from various complicated effects,
such as orbital excitations \cite{{orbitex}}, 
relativistic and exchange current effects \cite{{mgupta1},{excurr}}, pion cloud
contributions \cite{{pioncloudy}},  effect of the confinement
on quark masses \cite{{effm1},{effm2}}, effects of the spin-spin
forces \cite{{mgupta1},{effm2},{Isgur}}, sea quark polarization
\cite{{cheng},{chengsu3},{cheng1},{song},{johan}}, 
pion loop corrections \cite{loop}, etc.. 
Recently, it has been emphasized \cite{{johan},{cg1}} that the problem
regarding magnetic moments gets further complicated when one realizes
that the Coleman Glashow sum rule (CGSR) for octet baryons \cite{cg},
valid in a large variety of models, is convincingly violated
by the data \cite{PDG}.   

The authors, in a recent Rapid Communication \cite{hdorbit}, 
have shown that
within the chiral quark model ($\chi$QM)
\cite{cheng,wein,manohar,eichten}, 
when orbital angular momentum of the ``sea'' quarks along with 
their polarization is considered through 
the Cheng-Li mechanism \cite{cheng1}, one is able to get a
non zero value for the violation of CGSR ($\Delta$CG) \cite{CGSR}  
apart from
improving the NRQM predictions for magnetic moment of octet baryons. 
This fact, when viewed in the context of success of $\chi$QM for the
explanation of ``proton spin crisis'' {\cite{EMC}}, $\bar u-\bar d$
asymmetry {\cite{{GSR},{NMC},{E866}}}, existence of significant
strange quark content \cite{{st q},{ao}}, 
quark flavor and spin distribution functions \cite{{adams}}, 
hyperon decay parameters \cite{decays} etc., 
strongly suggests that constituent quarks,  
weakly interacting Goldstone bosons (GBs) and $q \bar q$ pairs
provide the appropriate degrees of freedom in the scale between chiral
symmetry breaking ($\chi_{SB}$) and the confinement scale.
This is further borne out by the fact that when the Cheng-Li
mechanism is coupled with the effects of configuration mixing, known to
be improving the predictions of NRQM
\cite{{mgupta1},{Isgur},{DGG},{yaouanc},{photo}}, as well as
compatible with $\chi$QM \cite{{riska},{chengspin},{prl}} and
``mass adjustments'' arising due to confinement of quarks  
\cite{{effm1},{effm2}}, leads to an
almost perfect fit for $\Delta$CG and an excellent fit for octet
magnetic moments \cite{hdorbit}, strongly suggesting that $\chi$QM not only
provides the appropriate degrees of freedom in this scale 
but also gives vital clues for the dominant dynamics of the constituents
in the nonperturbative regime. In view of this, it is desirable to
broaden the scope of Ref \cite{hdorbit} as well as to get into the
detailed implications of magnetic moments of octet and decuplet
baryons on coupling breaking terms, configuration mixing
generated by spin-spin forces
\cite{{mgupta1},{effm2},{Isgur},{DGG},{yaouanc},{photo},{riska},{chengspin},{prl}}, 
``mass adjustments'' \cite{{effm1},{effm2}} etc.. 

The purpose of the present paper is to formulate the octet and
decuplet magnetic moments in $\chi$QM (with and without spin-spin
forces) including the details of  the Cheng-Li mechanism with emphasis on
coupling breaking and mass breaking terms. Further, it is also the
purpose to investigate the implications of the different inputs
pertaining to $\bar u-\bar d$ asymmetry on $\chi$QM parameters and
magnetic moments. Furthermore, we also intend to study the
implications of variation of quark masses on magnetic moments.

The plan of the paper is as
follows. In Sec \ref{detail}, we detail the formulation of octet and
decuplet baryon magnetic moments with emphasis on the Cheng-Li
mechanism.
In Sec \ref{spin},  the modifications due to the spin-spin
forces on valence quarks, sea quark polarizations and the sea
orbital angular momentum have been discussed. 
Sec \ref{inputs} includes a discussion on the various inputs
used in the analysis particularly the parameters of the $\chi$QM
obtained after carrying out a fit of spin and quark distribution
functions by including the latest results of the E866 for $\bar u-\bar
d$ asymmetry. In Sec \ref{results}, we present the numerical results
and their discussion. Sec \ref{summary} comprises the
summary and the conclusions. 


\section{Magnetic moments in the $\chi$QM with the Cheng-Li mechanism}
 \label{detail}
In the $\chi$QM, the ``proton spin crisis'' finds its explanation in
the fact that the spin quenching of valence quarks takes place through
the existence of sea consisting of GBs and $q \bar q$ pairs, 
polarized opposite to the valence quark spin polarization.
Cheng and Li \cite{cheng1}, 
in order to explain the success of NRQM in explaining
the magnetic moments even when the valence quarks carry only about
30\% of the total spin, found that in $\chi$QM the polarization of the
sea quarks is ``balanced'' by the orbital motion of the sea quarks. 
In fact, in the Cheng-Li mechanism, the sea quarks are
negatively polarized with respect to the valence spin, whereas the
orbital angular momentum of the sea quarks contribute with the same
sign. In the case of nucleon magnetic moment, they have shown that the
contribution of these without coupling breaking and mass breaking
terms  mutually cancel, thus leaving
the predictions of the NRQM in broad agreement with the data. In the
following, we detail the contribution of the sea quark polarization and
their orbital angular momentum with coupling breaking and mass breaking
terms for all the octet and decuplet baryons. However, before 
doing that we detail some of the essentials of $\chi$QM
for the sake of notations and conventions as well as to 
facilitate discussion.

The basic process in the $\chi$QM is the
emission of a GB by a constituent quark which further splits into a $q
\bar q$ pair, for example,                          

\be
  q_{\pm} \rightarrow {\rm GB}^{0}
  + q^{'}_{\mp} \rightarrow  (q \bar q^{'})
  +q_{\mp}^{'},                              \label{basic}
\ee
where $q \bar q^{'}  +q^{'}$ constitute the ``quark sea''
  \cite{{chengsu3},{cheng1},{song},{johan}}. 
The effective Lagrangian describing interaction between quarks
and the octet GB and singlet $\eta^{'}$ is

\be
{\cal L} = g_8 \bar q \phi q,
\ee
where $g_8$ is the coupling constant,
\[ q =\left( \ba{c} u \\ d \\ s \ea \right)\]
and
\[ \phi = \left( \ba{ccc} \frac{\pi^o}{\sqrt 2}
+\beta\frac{\eta}{\sqrt 6}+\zeta\frac{\eta^{'}}{\sqrt 3} & \pi^+
  & \alpha K^+   \\
\pi^- & -\frac{\pi^o}{\sqrt 2} +\beta \frac{\eta}{\sqrt 6}
+\zeta\frac{\eta^{'}}{\sqrt 3}  &  \alpha K^o  \\
 \alpha K^-  &  \alpha \bar{K}^o  &  -\beta \frac{2\eta}{\sqrt 6}
 +\zeta\frac{\eta^{'}}{\sqrt 3} \ea \right). \]


SU(3) symmetry breaking is introduced by considering
different quark masses $M_s > M_{u,d}$ as well as by considering
the masses of GBs to be nondegenerate
 $(M_{K,\eta} > M_{\pi})$ {\cite{{chengsu3},{cheng1},{song},{johan}}}, whereas 
  the axial U(1) breaking is introduced by $M_{\eta^{'}} > M_{K,\eta}$
{\cite{{cheng},{chengsu3},{cheng1},{song},{johan}}}.
The parameter $a(=|g_8|^2$) denotes the transition probability
of chiral fluctuation
of the splittings  $u(d) \rightarrow d(u) + \pi^{+(-)}$, whereas 
$\alpha^2 a$, $\beta^2 a$ and $\zeta^2 a$ respectively 
denote the probabilities of transitions of
$u(d) \rightarrow s  + K^{-(o)}$, $u(d,s) \rightarrow u(d,s) + \eta$,
 and $u(d,s) \rightarrow u(d,s) + \eta^{'}$.

Following Cheng and Li \cite{cheng1}, the magnetic moment of a given baryon
which receives contributions from valence quarks, ``sea'' quarks and
the orbital angular momentum of the ``sea'' is expressed as
\be
\mu(B)_{{\rm total}} = \mu(B)_{{\rm val}} + \mu(B)_{{\rm sea}} +
\mu(B)_{{\rm orbit}}.     \label{totalmag}
\ee
The valence and the sea contributions, in terms of quark spin
polarizations, can be written as 
\be
\mu(B)_{{\rm val}}=\sum_{q=u,d,s} {\Delta q_{{\rm val}}\mu_q}, ~~~ 
\mu(B)_{{\rm sea}}=\sum_{q=u,d,s} {\Delta q_{{\rm sea}}\mu_q},    \label{mag}
\ee
where $\mu_q= \frac{e_q}{2 M_q}$ ($q=u,d,s$) is the quark magnetic moment.
$e_q$ and $M_q$ are the electric charge and the mass respectively for the
quark $q$. Similarly, the orbital angular momentum 
contribution of the sea, $\mu(B)_{{\rm orbit}}$, can be expressed in
terms of the valence   
quark polarizations and the orbital moments of the sea quarks, 
the details of which would be given in Section \ref{seaorbit}. 
Following references {\cite{{cheng},{cheng1},{johan}}}, 
the quark spin polarization can be defined
as 
\be
\Delta q= q_{+}- q_{-}+
\bar q_{+}- \bar q_{-},   \label{spin contr}
\ee
where $q_{\pm}$ and $\bar q_{\pm}$ can be 
calculated by using the number operator defined as
\be
N=n_{u_{+}}u_{+} + n_{u_{-}}u_{-} + n_{d_{+}}d_{+} + n_{d_{-}}d_{-} +
n_{s_{+}}s_{+} + n_{s_{-}}s_{-}, 
\ee  
with the coefficients of the $q_{\pm}$ giving the number of 
$q_{\pm}$ quarks.

To calculate $\mu(B)_{{\rm val}}$, we need to calculate the valence
spin polarizations $\Delta q_{{\rm val}}$. 
 These can be calculated from the spin structure 
of a given baryon and are presented in Tables \ref{ocval} and
\ref{deval} for the octet and decuplet baryons respectively. 
For ready reference some essential details of the calculations for
valence quark polarizations are presented in Appendix A. 

\subsection{ Contribution of the ``sea'' quark polarization to
the magnetic moments}
The ``quark sea'' contribution to the magnetic moment, 
$\mu(B)_{{\rm sea}}$ can be evaluated 
if one is able to find ${\Delta q_{{\rm sea}}}$ for a given baryon. In
order to make the paper self contained as well as to facilitate its
extension to the case where effects of spin-spin forces 
\cite{{mgupta1},{effm2},{Isgur}} are
included, following \cite{{chengsu3},{cheng1},{song},{johan}},
 we summarize the essentials of these calculations.
The spin structure for the process given in Eq. (\ref{basic}), 
after one interaction, can be obtained  by  substituting for every
valence quark, for example,  

\be
 q_{\pm} \rightarrow \sum P_q q_{\pm} +
 |\psi(q_{\pm})|^2, \label{q}
\ee
where $\sum P_q$ is the probability of emission of GB from a $q$ quark and the 
probabilities of transforming a $q_{\pm}$ quark are
$|\psi(q_{\pm})|^2$. The relevant details pertaining to the
 calculations of ${\Delta q_{{\rm sea}}}$ using Eq. (\ref{q}) are
 presented in the Appendix A. The expressions for ${\Delta
 q_{{\rm sea}}}$ in the case of proton are as follows
\bea
\Delta u_{sea}&=&-\frac{a}{3} (7+4 \alpha^2+
 \frac{4}{3}\beta^2 +\frac{8}{3} \zeta^2), \\
\Delta d_{sea}&=&-\frac{a}{3} (2-\alpha^2
-\frac{1}{3}\beta^2 -\frac{2}{3} \zeta^2), \\
 \Delta s_{sea}&=&-a \alpha^2.
\eea
The detailed expressions for all the other octets are presented in 
Table \ref{ocsea}. 


The sea quark spin polarizations for the decuplet baryons
can be calculated  in a similar manner as that of octet baryons. For
example, the general expressions for the spin structure of the decuplet 
baryons of the types $B^*(xxy)$, $B^*(xxx)$ and $B^*(xyz)$, using
Eq. (\ref{q}), are respectively given as
\be
B^*(xxy)= 2 \left(\sum P_x x_{+} +{|\psi(x_{+})|}^2\right)
+ \left(\sum P_y y_{+}+ {|\psi(y_{+})|}^2\right),
\ee
\be
B^*(xxx)= 3 \left(\sum P_x x_{+} + {|\psi(x_{+})|}^2\right),
\ee
\be
B^*(xyz)= \left(\sum P_x x_{+} + {|\psi(x_{+})|}^2\right)+
\left(\sum P_y y_{+} + {|\psi(y_{+})|}^2\right)
+ \left(\sum P_z z_{+} + {|\psi(z_{+})|}^2\right),
\ee
where $x$, $y$ and $z$ correspond to any of the $u$, $d$ and $s$ quarks.
The detailed expressions for the spin polarizations $\Delta q_{{\rm sea}}$, 
corresponding to different decuplet baryons, are again presented in 
Table \ref{desea}. 


\subsection{Contribution of the ``sea'' orbital angular momentum to
the magnetic moments} \label{seaorbit} 

The contribution of the angular momentum of the sea to the magnetic
moment of a given quark, following  Cheng and Li \cite{cheng1}, is
given as 
\be
\mu (q_{+} \rightarrow {q}_{-}^{'}) =\frac{e_{q^{'}}}{2M_q}
\langle l_q \rangle +
\frac{{e}_{q}-{e}_{q^{'}}}{2 {M}_{{\rm GB}}}\langle {l}_{{\rm GB}} \rangle,
\ee
where 
\be
\langle l_q \rangle=\frac{{M}_{{\rm GB}}}{M_q+{M}_{{\rm GB}}} ~{\rm and} 
~\langle l_{{\rm GB}} \rangle=\frac{M_q}{M_q+{M}_{{\rm GB}}},
\ee
$\langle l_q, l_{{\rm GB}} \rangle$ and ($M_q$, ${M}_{{\rm GB}}$) are the 
orbital angular momenta and masses of quark and GB respectively.
The orbital moment of each process is then multiplied by the probability
for such a process to take place to yield the magnetic moment due to
all the transitions starting with a given valence quark, for example,
\be [ \mu (u_{\pm}(d_{\pm}) \rightarrow )]=  \pm a
\left [\mu (u_{+}(d_{+}) \rightarrow d_- (u_-)) +
\alpha^2 \mu (u_+(d_+) \rightarrow s_-) 
+(\frac{1}{2} +\frac{1}{6} \beta^2+ \frac{1}{3} \zeta^2)
\mu (u_{+}(d_{+}) \rightarrow u_- (d_-))\right ],
\ee

\be
[\mu (s_{\pm} \rightarrow )]=  \pm a
\left [\alpha^2 \mu (s_{+} \rightarrow u_-) +
\alpha^2 \mu (s_+ \rightarrow d_-) +
(\frac{2}{3} \beta^2+ \frac{1}{3} \zeta^2)
\mu (s_{+} \rightarrow s_- ) \right ].
\ee
The above equations, derived by Cheng and Li, can easily be
generalized in $\chi$QM to include the coupling breaking and mass
breaking terms. For example, in terms of the parameters $\alpha$,
$\beta$ and $\zeta$, the  orbital moments of $u$, $d$ and $s$ quarks 
respectively are
\bea
\mu(u_+ \rightarrow) & =& 
a \left [\frac{-M^2_{\pi}+3 M^2_{u}}{2 {M}_{\pi}(M_u+{M}_{\pi})}
-\frac{\alpha^2(M^2_{K}-3 M^2_{u})}{2 {M}_{K}(M_u+{M}_{K})} 
+ \frac{(3+\beta^2+2 \zeta^2)M^2_{\eta}}{6 {M}_{\eta}(M_u+{M}_{\eta})}
 \right ]{\mu}_N, \\
\mu(d_+ \rightarrow) & =& 
a \frac{M_u}{M_d}\left [\frac{2 M^2_{\pi}-3 M^2_{d}}
{2 {M}_{\pi}(M_d+{M}_{\pi})}-
 \frac{\alpha^2 M^2_{K} }{2 {M}_{K}(M_d+{M}_{K})}   
- \frac{(3+\beta^2+2 \zeta^2)M^2_{\eta}}{12 {M}_{\eta}(M_d+{M}_{\eta})}
 \right ]{\mu}_N, \\
\mu(s_+ \rightarrow) & =& 
a \frac{M_u}{M_s}\left [ \frac{\alpha^2 (M^2_{K}-3 M^2_s) }
{2 {M}_{K}(M_s+{M}_{K})}  -
\frac{(2 \beta^2+\zeta^2)M^2_{\eta}}{6 {M}_{\eta}(M_s+{M}_{\eta})}
 \right ]{\mu}_N,
\eea
where $\mu_N$ is the Bohr magneton.
The orbital contribution of the octet baryon of the type $B(xxy)$ is
given as
\be
\mu(B)_{{\rm orbit}} =\Delta x_{{\rm val}} \left[\mu (x_+ \rightarrow) \right]+
\Delta y_{{\rm val}} \left[\mu (y_+ \rightarrow) \right],  \label{orbit}
\ee
wherein the valence spin polarizations can be read from 
Table \ref{ocval}. 

Similarly, the orbital contribution of the decuplet baryons 
$B^*(xxy)$,  $B^*(xxx)$ and $B^*(xyz)$ are respectively given as
\be
\mu(B^*)_{{\rm orbit}} =\Delta x_{{\rm val}} [\mu (x_+ \rightarrow)] + 
\Delta y_{{\rm val}}[\mu (y_+ \rightarrow)],
\label{de orbit}
\ee
\be
\mu(B^*)_{{\rm orbit}} = \Delta x_{{\rm val}} [\mu (x_+ \rightarrow)],
\ee
\be
\mu(B^*)_{{\rm orbit}} =  \Delta x_{{\rm val}} [\mu (x_+ \rightarrow)] + 
\Delta y_{{\rm val}}[\mu (y_+ \rightarrow)]
+\Delta z_{{\rm val}} [\mu (z_+ \rightarrow)],
\ee
wherein the valence spin polarizations can again be read from 
Table \ref{deval}. 


\section {Cheng-Li mechanism with spin-spin forces} \label{spin}

Spin-spin forces,  known to be compatible with the $\chi$QM
\cite{{riska},{chengspin},{prl}}, generate
configuration mixing \cite{{Isgur},{DGG},{yaouanc}} for the
octet baryons. 
The configuration mixing in $\chi$QM effectively leads to modification
of the valence quark and sea spin distribution functions \cite{hd}.
In the present case, configuration mixing leads to modifications 
in the spin distribution functions.
From Eqs. (\ref{mag}) and (\ref{orbit}), it is evident that
the effects of spin-spin forces on magnetic moments
can be included if one is able to
estimate the same on the valence and sea contributions.
Configuration mixing, generated by the spin-spin forces, can be
expressed for the octet baryons as \cite{{Isgur}}
\be
|B \rangle=\left(|56,0^+\rangle_{N=0} \cos \theta +|56,0^+ \rangle_{N=2}  
\sin \theta \right) \cos \phi 
+  \left(|70,0^+\rangle_{N=2} \cos \theta +|70,2^+\rangle_{N=2}  
\sin \theta \right) \sin \phi, \label{full mixing}
\ee
where $\theta$ and $\phi$ are the mixing angles with
\bea
 |56,0^+\rangle_{N=0,2} &=& \frac{1}{\sqrt 2}(\chi^{'} \phi^{'} +
\chi^{''} \phi^{''}) \psi^{s}(0^+), \label{56}   \\      
|70,0^+\rangle_{N=2} &=&  \frac{1}{2}[(\phi^{'} \chi^{''} +\phi^{''}\chi^{'})
\psi^{'}(0^+) + (\phi^{'} \chi^{'} -\phi^{''} \chi^{''})\psi^{''}(0^+)], 
\label{70}  \\ 
|70,2^+\rangle_{N=2} &=&  \frac{1}{\sqrt 2}[ \phi^{'} \chi^{s} \psi^{'}(2^+)
+ \phi^{''} \chi^{s} \psi^{''}(2^+)].                                    
\eea
The spin wave functions  are as follows

\[ \chi^{'} =  \frac{1}{\sqrt 2}(\uparrow \downarrow \uparrow
-\downarrow \uparrow \uparrow),~~~
\chi^{''} =  \frac{1}{\sqrt 6} (2\uparrow \uparrow \downarrow
-\uparrow \downarrow \uparrow -\downarrow \uparrow \uparrow). \] \\
In general, the isospin wave functions for the octet baryons of the 
type $B(xxy)$ are   
\[\phi^{'}_B = \frac{1}{\sqrt 2}(xyx-yxx),~~~
\phi^{''}_B = \frac{1}{\sqrt 6}(2xxy-xyx-yxx),\]
and for $\Lambda (uds)$ we have
\[ \phi^{'}_{\Lambda} = \frac{1}{2 \sqrt 3}(usd+sdu-sud-dsu-2uds-2dus),\]
\[ \phi^{''}_{\Lambda} = \frac{1}{2}(sud+usd-sdu-dsu).\]
 For the definition of the spatial  wave functions
 ($\psi^{s}, \psi^{'}, \psi^{''})$ as well as the
 definitions of the overlap integrals, we  refer the
 reader to reference {\cite{{yaoubook}}.

In view of the fact that in $\chi$QM it is the spin structure of the
valence quarks which is important, therefore, the above mixing can
effectively be reduced to non trivial mixing 
{\cite{{mgupta1},{effm2},{yaouanc}}} and the corresponding
``mixed'' octet of baryons is expressed as

\begin{equation}
|B\rangle 
\equiv \left|8,{\frac{1}{2}}^+ \right> 
= \cos \phi |56,0^+\rangle_{N=0}
+ \sin \phi|70,0^+\rangle_{N=2},  \label{mixed}
\end{equation} 
henceforth configuration mixing given in Eq. (\ref{full mixing})
and the ``mixed'' octet would be used interchangeably.
This leads to the changes in the valence quark spin
structure, for example, the spin polarizations for 
proton  are
\bea
\Delta u_{{\rm val}} &=&{\cos}^2 \phi \left[\frac{4}{3} \right] 
   + {\sin}^2 \phi \left[\frac{2}{3}  \right], \\
\Delta d_{{\rm val}} &=&{\cos}^2 \phi \left[-\frac{1}{3} \right]  +
  {\sin}^2 \phi \left[\frac{1}{3}  \right], \\
\Delta s_{{\rm val}} &=& 0.
\eea 
These expressions would replace $\Delta q_{{\rm val}}$
in Eq. (\ref{mag}) and (\ref{orbit}) for  calculating the effects of 
configuration mixing on the valence and the orbital part. 
The valence quark spin polarizations with configuration mixing for
all the octet baryons are presented in Table \ref{ocval}.


The sea quark polarization also gets modified with the
inclusion of configuration mixing and 
can easily be calculated \cite{hd}. The details of the calculations
are given in Appendix A. For the case of proton, these are expressed as
\bea
\Delta u_{sea}&=&-{\cos}^2 \phi \left[\frac{a}{3} (7+4 \alpha^2+
 \frac{4}{3}\beta^2 +\frac{8}{3} \zeta^2)\right]
-{\sin}^2 \phi \left[\frac{a}{3} (5+2 \alpha^2
+\frac{2}{3}\beta^2 +\frac{4}{3} \zeta^2)\right], \\
\Delta d_{sea}&=&-{\cos}^2 \phi \left[\frac{a}{3} (2-\alpha^2
-\frac{1}{3}\beta^2 -\frac{2}{3} \zeta^2)\right]
-{\sin}^2 \phi \left[\frac{a}{3} (4+\alpha^2
+\frac{1}{3}\beta^2 +\frac{2}{3} \zeta^2)\right], \\ 
 \Delta s_{sea}&=&-a \alpha^2.
\eea 
The sea quark spin polarizations for the other octet baryons can
similarly be calculated and are presented in \linebreak
Table \ref{ocsea}. 

There is no mixing in the case of decuplet baryons \cite{Isgur,yaouanc}, 
thus the decuplet baryon wave function is given as 
 \be
|B^*\rangle \equiv |56,0^+\rangle_{N=0} =\chi^{s}\phi^{s} \psi^{s}(0^+),
\label{56decuplet}
\ee
where the spin wave function for the  decuplet baryon is
\[\chi^{s} =  (\uparrow \uparrow \uparrow). \]
The isospin wave functions for the decuplet baryons of the types
$B^*(xxx)$, $B^*(xxy)$ and $B^*(xyz)$ respectively are
\[\phi^{s}_{B^*} = xxx, \]
\[\phi^{s}_{B^*} = \frac{1}{\sqrt 3}(xxy + xyx + yxx), \]
\[\phi^{s}_{B^*} = \frac{1}{\sqrt 6}(xyz + xzy + yxz + yzx + zxy + zyx), \]
where $x$, $y$ and $z$ correspond to any of the $u$, $d$ and $s$ quarks.
The valence and sea quark spin polarizations for the decuplet baryons
are presented  in Tables \ref{deval} and \ref{desea}.


\section {Inputs} \label{inputs}
To facilitate the understanding of different inputs based on 
Eq. (\ref{totalmag}), in Appendix A we have
presented the complete expression for two of the octet
baryon magnetic moments, $p$ and $\Lambda$, whereas in the
case of decuplet baryons we have considered the example of $\Delta^+$. 
The other octet and decuplet magnetic moments can
similarly be formulated. As is evident from the Appendix, to calculate
magnetic moments we need inputs related to $\chi$QM, 
mixing anle $\phi$  and quark masses.

The parameters characterizing the $\chi$QM are $a$, $\alpha^2$,
$\beta^2$ and $\zeta^2$  denoting 
the transition probability of chiral fluctuation
of the splittings  $u(d) \rightarrow d(u) + \pi^{+(-)}$, 
$u(d) \rightarrow s  + K^{-(o)}$, 
$u(d,s) \rightarrow u(d,s) + \eta$ and
$u(d,s) \rightarrow u(d,s) + \eta^{'}$ respectively.
These parameters are usually fixed by considering 
the violation of Gottfried sum
rule  \cite{GSR} measured through the  $\bar u-\bar d$ asymmetry
\cite{{NMC},{E866}} as well as $\Delta u$, $\Delta d$, $\Delta s$
\cite{adams}, characterizing spin polarizations as measured in  deep
inelastic scattering. 
Since we are considering $\chi$QM with and without configuration
mixing, therefore the parameters $\alpha$, $\beta$ and $\zeta$ have
been fixed for both the cases using the data pertaining to 
$\bar u-\bar d$ asymmetry by
NMC \cite{NMC} and the latest E866 \cite{E866} data as well as the
experimental input regarding $\Delta u$, $\Delta d$, $\Delta s$
\cite{adams}. Without getting into the details of the analysis, which would
be presented elsewhere, in Table \ref{ud} we have
presented the key elements of the analysis having implications for the
$\chi$QM parameters $\alpha$ and $\beta$. In the present analysis we
have taken the  pion fluctuation parameter $a$ to be 0.1, 
in accordance with most of the other
calculations \cite{{cheng1},{song},{johan}}.
As is evident from the table, one can find out that $\chi$QM
parameters with and without configuration mixing take different
values, for example, in the case of $\chi$QM without configuration mixing
we have $\alpha=0.6$, $\beta=0.9$, whereas in the case of 
$\chi$QM with configuration mixing ($\chi$QM$_{gcm}$) we have $\alpha=0.4$,
$\beta=0.7$. 
As the explanation of the violation of the
Gottfried sum rule \cite{GSR} implies different parameterization 
in the case of NMC \cite{NMC} and the latest E866 \cite{E866}  data, 
therefore, we have used
parameter   $\zeta=-0.7-\beta/2$ and $\zeta=-0.3-\beta/2$ respectively 
for  the NMC and the E866 data. 


The orbital angular moment contributions  are  characterized by
the parameters of $\chi$QM as well as the masses of the GBs. 
For evaluating the contribution  of pions, 
we have used its on mass shell value in accordance with
several other similar calculations \cite{{mpi1}}. Similarly, for the
other GBs we have considered their on mass shell values, however their
contributions are much smaller compared to the pionic contributions.

In view of the fact that $\chi$QM with configuration  mixing involves
baryon wave functions which are perturbed by the spin-spin forces,
therefore, in principle one should employ the fully perturbed wave 
functions of the
octet baryons as derived by Isgur {\it et al.} \cite{Isgur} 
given in Eq. (\ref{full mixing}). 
However, as discussed
earlier we find the essential effect of configuration mixing can
be reproduced by considering non trivial mixing  
expressed through the ``mixed'' nucleon (Eq. (\ref{mixed})), wherein 
the angle $\phi$ is fixed from the consideration of 
neutron charge radius \cite{{yaouanc},{neu charge}}.

In accordance with the basic assumptions of $\chi$QM, the constituent
quarks are supposed to have only Dirac magnetic moments governed by
the respective quark masses.
In the absence of any definite guidelines for the constituent quark
masses, for the $u$ and $d$ quarks we have used  their most widely 
accepted values in hadron spectroscopy 
\cite{{Isgur},{chengspin},{yaoubook},{close},{mu1}}, 
for example $M_u=M_d=330$ MeV. Apart from taking the above quark masses, 
one has to consider the strange quark
mass implied by the various sum rules derived from the spin-spin
interactions for different baryons {\cite{{mgupta1},{Isgur},{yaouanc}}}, 
for example,
$ \Lambda-N =M_s-M_u$, $(\Sigma^*-\Sigma)/(\Delta-N)=M_u/M_s$ and 
$(\Xi^*-\Xi)/(\Delta-N)=M_u/M_s$, respectively fix 
$M_s$ for $\Lambda$, $\Sigma$ and $\Xi$ baryons. 
These quark masses and corresponding magnetic moments
have to be further adjusted by the quark confinement effects
\cite{{effm1},{effm2}}. In conformity with  additivity assumption, 
the simplest way to incorporate this adjustment \cite{{effm1},{effm2}} 
is to first express $M_q$ in the magnetic moment
operator in terms of $M_B$, the mass of the baryon obtained
additively from the quark masses, which then
is replaced  by $M_B+\Delta M$,
$\Delta M$
being the mass difference between the experimental value and $M_B$.
This leads to the following adjustments in the quark
magnetic moments:
$\mu_d = -[1-(\Delta M/M_B)] {\mu}_N$,
$\mu_s = -M_u/M_s [1-(\Delta M/M_B)]{\mu}_N$ and $\mu_u=-2 \mu_d$.
The baryon magnetic moments calculated after incorporating this effect
would be referred to as ``mass adjusted''. 

\section{Results and Discussions} \label{results}
Using Eq. (\ref{totalmag}) and the inputs discussed above as well as 
the expressions given in Tables \ref{ocval}, \ref{deval},  \ref{ocsea} 
and \ref{desea} , in Table  \ref{E866} we have
presented the results of octet magnetic moments without taking any of
the magnetic moments as inputs.
A brief discussion of the contents of Table \ref{E866}  
corresponding to the latest E866 data \cite{E866}
has already been carried out in Ref \cite{hdorbit}. In the present
case, however, we intend to discuss in detail the role of Cheng-Li
mechanism, spin-spin forces and ``mass adjustments'' in getting the
fit for octet magnetic moments. To this end
one can immediately find that
$\chi$QM with Cheng-Li mechanism,  however without spin-spin forces
and ``mass adjustments'', consistently improves the predictions of NRQM
as well as is able to generate a non zero value of $\Delta$CG. 
On closer examination of the results, several interesting points pertaining
to Cheng-Li mechanism emerge out.
The total contribution
to the magnetic moment is coming from several sources with similar and
opposite signs, for example, the orbital is contributing with the same
sign as that of the valence, whereas sea is contributing with opposite
sign. The ``sea'' and orbital contributions are fairly 
significant as compared to the valence contributions 
and they cancel in the right direction, for example,
the valence contributions of $p$, $\Sigma^+$ and $\Xi^o$ are
 higher in magnitude than the experimental
value but the sea contribution being higher in magnitude than the orbital
contribution reduces the valence contribution leading to a 
better agreement with data. Similarly, in the case of $n$ and 
$\Sigma^-$ the valence contribution in magnitude is lower 
than the experimental value but in these cases the sea contribution 
is lower than the orbital part so it adds on to the valence contribution 
again improving agreement with data. 
Thus, in a very interesting manner, the orbital and sea contributions
together add on to the valence contributions leading to better agreement
with data as compared to NRQM. This not only endorses the earlier
conclusion of Cheng and Li \cite{cheng1} but also suggests that the Cheng-Li
mechanism could perhaps provide the dominant dynamics of the
constituents in the nonperturbative regime of QCD on
which further corrections could be evaluated.
To this end, in Table \ref{E866}, we have presented the
results wherein the effects of spin-spin forces and ``mass adjustments''
have been included.
As is evident from the table, we have
been able to get an excellent fit for
almost all the baryons, it is almost perfect for $p, \Sigma^+$ and $\Xi^o$,
whereas in the case of $n, \Sigma^-$ and $\Lambda$ the value is reproduced
within 5\% of experimental data. Interestingly, we have been able to
get an almost perfect fit for  $\Delta$CG. These excellent results
reinforce our earlier conclusion that Cheng-Li
mechanism seems to be providing the dominant dynamics of the
appropriate degrees of freedom  in the nonperturbative regime of QCD.


In order to study closely the role of configuration mixing on octet
magnetic moments, in Table  \ref{E866} we have presented the
results with and without mixing, however with the inclusion of 
``mass adjustments''. 
As is evident from the table, one finds that the individual 
magnetic moments show improvements after the inclusion of configuration 
mixing, particularly  in the case of $p$, $n$, $\Sigma^+$, $\Xi^o$ and
for $\Lambda$ one observes a significant improvement. 
It may be noted that configuration mixing reduces
valence, sea and orbital contributions to the magnetic moments
and the results which are generally on the higher side get
corrected in the right direction by the inclusion of the spin-spin
forces.
This is particularly manifest in the case of $\Xi$ particles, for
example, the magnitude of $\Xi^o$ magnetic moment without the
spin-spin forces is lowered so as to achieve an almost perfect fit,
whereas in case of $\Xi^-$, a difficult case for most of the models,
the spin-spin contributions increase the magnitude for better
agreement with data.  
In contrast to general improvement in the case of individual magnetic 
moments, $\Delta$CG hardly gets affected by configuration mixing.
One may wonder 
whether $\Delta$CG could also be reproduced with the variation of 
mixing angle $\phi$. Our calculations in this regard show that variation 
of $\phi$ does not lead to any improvement in the magnetic moments as well as
$\Delta$CG, the angle  $\phi$  fixed from the 
neutron charge radius  \cite{{yaouanc},{neu charge}}
seems to be providing the best fit.


It would also perhaps be interesting to find out the 
implications of spin-spin forces for $\chi$QM without ``mass adjustments''.
Broadly speaking the individual magnetic moments can again be fitted, 
however $\Delta$CG leaves much to be desired. This can be easily checked
from Table \ref{NMC}, wherein we have presented these calculations with 
the NMC data, the E866 based fit  follows the same 
pattern.

The value of $\Delta$CG registers a remarkable improvement
when effects due to ``mass adjustments'' alongwith the effects
of spin-spin forces are included. This is not 
surprising as the large value of $\Delta$CG could come only from the
valence quark corrections,  duly provided by the ``mass
adjustments''. It would be desirable to know what level of fit can be
achieved without spin-spin forces, however with the inclusion of 
``mass adjustments''.
A closer examination of the table immediately brings out that in this 
case the  individual magnetic moments leave much to be desired
whereas one is able to reproduce $\Delta$CG, in accordance with our 
earlier conclusions. It may also be noted that the ``mass
adjustments'' generally lower the various contributions
except for the nucleon.
Further, even a variation of quark masses and
the $\chi$QM parameters does not achieve an overall present level of
fit.  
In short, we may emphasize that the final fit obtained here
cannot be achieved if any of the ingredients, for example,
Cheng-Li mechanism, spin-spin forces and ``mass adjustments, is
absent.


For the sake of completeness, as mentioned earlier also, we have
presented in Table \ref{NMC} the octet magnetic moments when the
$\chi$QM parameters are 
fitted by incorporating NMC data. This table also includes our results
wherein magnetic moments have been calculated with spin-spin forces
however without``mass adjustments'', not included in Table
\ref{E866}. From the table, one can immediately find out that the
basic pattern of results remain the same, however in general the
results are lower as compared to the case of E866 data. This is not
difficult to understand when one realizes that the contribution of sea
polarization in case of E866 and NMC data are quite different.
This can
be  understood easily when one realizes that the 
sea quark polarization is proportional to the  parameter $\zeta$.
Because of $|\zeta_{E866}|<|\zeta_{NMC}|$, one can easily
understand the corresponding lowering of the magnetic moments
in the case of NMC data, however both the calculations are in
good agreement with each other.


In Table \ref{decuplet}, we have presented the results of the decuplet 
baryons for the NMC and the latest E866 data. 
The calculations of decuplet magnetic moments have been carried out
with the same $\chi$QM parameters and quark masses 
as that of the octet magnetic moments. From
the table, it is evident that we have been able to obtain a fairly
good agreement pertaining to the case of $\Delta^{++}$ and $\Omega^-$,
for which the experimental data is available. In order to compare the
present results with other recent calculations, in the table 
we have also included  the results of X. Song \cite{song} and 
Linde {\it et al.} \cite{johan}.
A closer examination of the decuplet magnetic moments reveals several
interesting points which would have bearing on the Cheng-Li
mechanism. 
For example, in the case of $\Delta^-$ and $\Sigma^-$,
because the orbital part dominates over the sea quark
polarization,  the magnetic
moments are higher as compared to the results of NRQM, 
Song \cite{song} and Linde {\it et al.} \cite{johan}.
On the other hand, in the case of $\Delta^+$ and $\Sigma^+$, the
sea quark polarization dominates over the orbital part as a consequence of
which the magnetic moment contribution is more or less the same as
that of the results of NRQM, Song and Linde {\it et al.}.
In general, one can find that whenever there is an excess of $d$ quarks
the orbital part dominates, whereas when we have an excess of $u$ quarks, 
the sea quark polarization dominates.
A measurement of these magnetic moments, therefore, would have
important implications for the $\chi$QM as well as the Cheng-Li mechanism.

 
While carrying out the fit, as mentioned earlier, the quark masses
which have been employed for the calculations correspond to the
generally accepted values used for hadron spectroscopic calculations. 
It may be of interest to study the
variation of these masses on the magnetic moments. To this end, in
Table \ref{masses}, we have investigated the effect of varying
valence quark masses. As is evident from the table we find that
results worsen in both the cases, for example, when they are reduced or
increased compared to the ones considered earlier.  The violation
of CGSR is also fitted best for the genarally accepted mass values 
employed in our
calculations. These results remain true for E866 as well as the NMC data.
This looks to be surprising as the hadron spectroscopic predictions are
known to be somewhat insensitive to the  valence quark masses.
In view of this, it needs to be emphasized that an excellent fit achieved
here within $\chi$QM with Cheng-Li mechanism not only suggests that 
constituent quarks and weakly interacting GBs provide the appropriate 
constituents of nonperturbative regime of QCD but may also indicate 
a deeper significance of the values of various $\chi$QM parameters, 
mixing angle and quark masses employed here.


It may be of interest to mention some of the  effects which could 
lead to further improvements in the predictions of magnetic moments,
particularly in the case of  $\Xi^-$, 
beyond what is already achieved here.
As mentioned earlier, that there are several
effects such as relativistic and exchange currents
\cite{{mgupta1},{excurr}}, pion loop corrections \cite{loop} etc. 
which also affect
the magnetic moments. We believe the inclusion of these effects would
perhaps improve the present fit further, however before doing that one 
has to avoid double counting of various effects
considered. In fact, a cursory look at
\cite{pioncloudy} suggests that pion loop corrections would compensate
$\Xi^-$ much more compared to other baryons hence providing an almost
perfect fit.


\section{Summary and conclusion} \label{summary}
To summarize, baryon magnetic moments have been 
formulated in the $\chi$QM with a ``mixed'' octet of baryons 
generated by spin-spin forces,
incorporating the quark sea contribution as well as its orbital 
angular momentum through the Cheng-Li mechanism with emphasis on
coupling and mass breaking terms. 
With the inputs pertaining to the $\chi$QM parameters fixed by the
latest E866 data and with the generally accepted values of the quark
masses $M_q$, we find that the Cheng-Li mechanism is not only able 
to improve the baryon magnetic moments as compared to NRQM but also gives 
a non zero value for $\Delta$CG. 
The predictions of the  $\chi$QM with the Cheng-Li mechanism improve further
when effects of spin-spin forces and ``mass adjustments'' due to
confinement effects are included, for example, in the case of E866 data we get
an excellent fit for the octet magnetic moments and an almost perfect
fit for $\Delta$CG. 
Interestingly, we find that Cheng-Li mechanism coupled with the effects
of spin-spin forces plays a crucial role in fitting the individual 
magnetic moments, whereas ``mass adjustments'' alongwith Cheng-Li mechanism
play an important role in fitting $\Delta$CG.
Further, the difference in the sea polarization due to the NMC and E866 
data results in the baryon magnetic moments being on the lower side for the
former, however  both the calculations are in good agreement
with each other. 
Interestingly, we find that the masses $M_u=M_d=330$ MeV, after
corrections due to spin-spin forces and ``mass adjustments'', 
provide the best fit for the magnetic moments.
In the case of decuplet baryon magnetic moments, we find a good
agreement of $\Delta^{++}$ and $\Omega^-$ with the experimental data. 
On comparison of our results with the corresponding results of 
Song and Linde {\it et al.}, we
find that the measurement of the $\Delta^+$, $\Delta^-$, $\Sigma^+$,
$\Sigma^-$ would have implications for the Cheng-Li mechanism.


In conclusion, we would like to state that an excellent agreement of
the octet magnetic moments with the data achieved within $\chi$QM,
using Cheng-Li mechanism, effects of spin-spin forces and 
``mass adjustments'' due to confinement effects,
strongly suggests that the constituent quarks and the 
weakly interacting Goldstone bosons provide the  
appropriate degrees of freedom with the  Cheng-Li
mechanism being the dominant dynamics of the
constituents in the nonperturbative regime of QCD.

                                                              
              
\vskip .2cm
  {\bf ACKNOWLEDGMENTS}\\
The authors would like to thank S.D. Sharma and M. Randhawa for a few useful 
discussions.
H.D. would like to thank CSIR, Govt. of India, for
 financial support and the chairman,
 Department of Physics, for providing facilities to work
 in the department.

\appendix
\renewcommand{\theequation}{A-\arabic{equation}}
  \setcounter{equation}{0} 

\begin{center}
{\bf APPENDIX A} 
\end{center}

Following Eq. (\ref{totalmag}), the magnetic moment of a given
baryon in $\chi$QM with Cheng-Li mechanism can be expressed as 
\be
\mu(B)_{{\rm total}} = \mu(B)_{{\rm val}} + \mu(B)_{{\rm sea}} +
\mu(B)_{{\rm orbit}}. \label{totalmagmom}
\ee
To calculate $\mu(B)_{{\rm val}}$, we first express it in terms of
valence quark polarizations ($\Delta q_{{\rm val}}$) 
and the quark magnetic moments ($\mu_q$) for example,
\be
 \mu(B)_{{\rm val}}=\Delta u_{{\rm val}}\mu_u +\Delta d_{{\rm val}}\mu_d +
\Delta s_{{\rm val}}\mu_s. 
\ee
To calculate the quark polarizations, we first calculate the spin
structure of a baryon defined as $\hat B \equiv \langle B|N|B \rangle$, 
where $|B\rangle$
is the baryon wave function  and $N$ is the number operator defined as
\be
N=n_{u_{+}}u_{+} + n_{u_{-}}u_{-} + n_{d_{+}}d_{+} + n_{d_{-}}d_{-} +
n_{s_{+}}s_{+} + n_{s_{-}}s_{-}, 
\ee
the coefficients of the $q_{\pm}$ giving the number of 
$q_{\pm}$ quarks.

The spin structure of the ``mixed'' octet baryon is given as 
\be
\hat B \equiv \langle B|N|B \rangle={\cos}^2 \phi
\langle 56,0^+|N|56,0^+\rangle +
{\sin}^2 \phi\langle 70,0^+|N|70,0^+ \rangle. \label{spin st}
\ee
For $\phi=0$, we obtain the spin structure for the baryon $B$ without
spin-spin forces. For the case of proton, using Eqs. (\ref{56}) and
(\ref{70}) of the text, we have 
\bea 
\langle 56,0^+|N|56,0^+ \rangle&=&\frac{5}{3} u_{+} +\frac{1}{3} u_{-}+
\frac{1}{3} d_{+} +\frac{2}{3} d_{-}, \label{56proton} \\ 
 \langle 70,0^+|N|70,0^+ \rangle&=&\frac{4}{3} u_{+} +\frac{2}{3} u_{-}+
\frac{2}{3} d_{+} +\frac{1}{3} d_{-}. \label{70proton}
\eea
Thus for proton, $\mu(p)_{{\rm val}}$ is given as
\be
 \mu(p)_{{\rm val}}=\left[{\cos}^2 \phi \left(\frac{4}{3} \right)
+{\sin}^2 \phi \left(\frac{2}{3} \right) \right ]\mu_u +
\left [{\cos}^2 \phi \left(-\frac{1}{3} \right)  +
  {\sin}^2 \phi \left(\frac{1}{3}  \right) \right]\mu_d +
[0]\mu_s. \label{val p} 
\ee
For the case of $\Lambda$, we have
\bea
 \langle 56,0^+|N|56,0^+ \rangle&=&\frac{1}{2} u_{+} +\frac{1}{2} u_{-}+
\frac{1}{2} d_{+} +\frac{1}{2} d_{-}+
1 s_{+} +0 s_{-}, \label{56lambda} \\
 \langle 70,0^+|N|70,0^+ \rangle&=&\frac{2}{3} u_{+} +\frac{1}{3} u_{-}+
\frac{2}{3} d_{+} +\frac{1}{3} d_{-}+
\frac{2}{3} s_{+} +\frac{1}{3} s_{-}, \label{70lambda}
\eea
and  $\mu(\Lambda)_{{\rm val}}$ is given as
\be
\mu(\Lambda)_{{\rm val}}=\left[{\sin}^2 \phi \left(\frac{1}{3}
\right) \right]\mu_u 
+ \left[{\sin}^2 \phi \left(\frac{1}{3}  \right)\right]\mu_d +
\left[{\cos}^2 \phi (1) +{\sin}^2 \phi \left(\frac{1}{3} \right)
\right]\mu_s.  \label{val lambda} 
\ee
The expressions for the valence spin polarizations, $\Delta q_{{\rm val}}$, 
for other octet baryons can be similarly calculated and are given in
Table \ref{ocval}. 


Similarly, we obtain the contribution of $\mu(B)_{{\rm sea}}$
  which is expressed in terms of the sea quark polarizations  
($\Delta q_{{\rm sea}}$)
\be
 \mu(B)_{{\rm sea}}=\Delta u_{{\rm sea}}\mu_u+\Delta d_{{\rm sea}}\mu_d+
\Delta s_{{\rm sea}}\mu_s.
\ee
$\Delta q_{{\rm sea}}$ can be calculated as follows. 
The spin structure for the proton ``sea'', as obtained by substituting 
for $q_\pm$ from Eq. (\ref{q}) of the text  in Eqs. (\ref{56proton}) and 
(\ref{70proton}),
is given as

\[ \hat p={\cos}^2 \phi \left [ \frac{5}{3}(\sum P_u u_{+} + 
|\psi(u_{+})|^2)+
\frac{1}{3}(\sum P_u u_{-} + |\psi(u_{-})|^2)+
\frac{1}{3}(\sum P_d d_{+} + |\psi(d_{+})|^2) + \frac{2}{3}(\sum P_d d_{-} + 
|\psi(d_{-})|^2) \right ] \]
\be
+{\sin}^2 \phi \left [ \frac{4}{3}(\sum P_u u_{+} + 
|\psi(u_{+})|^2)+
\frac{2}{3}(\sum P_u u_{-} + |\psi(u_{-})|^2)+ 
\frac{2}{3}(\sum P_d d_{+} + |\psi(d_{+})|^2)+
\frac{1}{3}(\sum P_d d_{-} + |\psi(d_{-})|^2) \right ],
\ee


where
\[ \sum P_u= a\left( \frac{9+\beta^2+2 \zeta^2}{6} +\alpha^2\right)~~~
{\rm and}~~~
|\psi(u_{\pm})|^2=\frac{a}{6}(3+\beta^2+2 \zeta^2)u_{\mp}+
a d_{\mp}+a \alpha^2 s_{\mp},      \]

\[ \sum P_d= a\left( \frac{9+\beta^2+2 \zeta^2}{6} +\alpha^2\right)~~~
{\rm and}~~~ 
|\psi(d_{\pm})|^2=a u_{\mp}+
\frac{a}{6}(3+\beta^2+2 \zeta^2)d_{\mp}+ a \alpha^2 s_{\mp}, 
                                               \]

\[ \sum P_s= a\left( \frac{2 \beta^2+\zeta^2}{3}+2 \alpha^2\right)~~~
{\rm and}~~~
|\psi(s_{\pm})|^2=   a \alpha^2 u_{\mp}+
a \alpha^2 d_{\mp}+\frac{a}{3}(2 \beta^2+\zeta^2)s_{\mp}.
 \]
For the definitions of $\alpha, \beta$ and $\zeta$ we refer the 
readers to the text.
For proton, $\mu(p)_{{\rm sea}}$ is given as
\bea
 \mu(p)_{{\rm sea}}&=&\left[-{{\cos}}^2 \phi \left(\frac{a}{3} 
(7+4 \alpha^2+
 \frac{4}{3}\beta^2 +\frac{8}{3} \zeta^2)\right)-{\sin}^2 \phi
\left(\frac{a}{3} (5+2 \alpha^2 
+\frac{2}{3}\beta^2 +\frac{4}{3} \zeta^2)\right)\right]\mu_u \nonumber \\
+& &\left[-{\cos}^2 \phi \left(\frac{a}{3} (2-\alpha^2
-\frac{1}{3}\beta^2 -\frac{2}{3} \zeta^2)\right)-{\sin}^2 \phi \left(\frac{a}{3} (4+\alpha^2
+\frac{1}{3}\beta^2 +\frac{2}{3} \zeta^2)\right)\right]\mu_d+
\left[-a \alpha^2\right]\mu_s. \label{sea p}
\eea
Similarly, the spin structure for $\Lambda$ obtained by substituting 
for $q_\pm$  in Eqs. (\ref{56lambda}) and 
(\ref{70lambda}), is given as

\bea
 \hat \Lambda&=&{\cos}^2 \phi \left [ \frac{1}{2}\left ( \sum P_u u_{+} + 
|\psi(u_{+})|^2 +
\sum P_u u_{-} + |\psi(u_{-})|^2+
\sum P_d d_{+} + |\psi(d_{+})|^2+\sum P_d d_{-} + |\psi(d_{-})|^2 \right )
 \right .  \nonumber \\
 &+& \left .\sum P_s s_{+} +|\psi(s_{+})|^2 \right ] 
 + {\sin}^2 \phi \left [ \frac{2}{3} \left (\sum P_u u_{+}
+ |\psi(u_{+})|^2+\sum P_d d_{+} + |\psi(d_{+})|^2+
\sum P_s s_{+} +|\psi(s_{+})|^2 \right )  \right .\nonumber \\
&+& \left . 
\frac{1}{3} \left (\sum P_u u_{-} + 
|\psi(u_{-})|^2 + \sum P_d d_{-} + |\psi(d_{-})|^2+
\sum P_s s_{-} +|\psi(s_{-})|^2 \right )  \right ],
\eea
giving $\mu(\Lambda)_{{\rm sea}}$ as
\bea
 \mu(\Lambda)_{{\rm sea}}&=&
\left[-{\cos}^2 \phi \left(a \alpha^2\right)-{\sin}^2 \phi 
\left(\frac{a}{9}(9+6 \alpha^2+\beta^2+ 2
\zeta^2)\right)\right]\mu_u +
\left[-{\cos}^2 \phi \left(a \alpha^2\right)-{\sin}^2 \phi 
\left(\frac{a}{9}(9+6 \alpha^2+\beta^2+ 2
\zeta^2)\right)\right]\mu_d \nonumber \\
&+&\left[-{\cos}^2 \phi \left(\frac{a}{3}(6 \alpha^2+
4 \beta^2 +2 \zeta^2)\right)-{\sin}^2 \phi \left(\frac{4}{9}a(3
\alpha^2+2 \beta^2+ \zeta^2)\right)\right]\mu_s. \label{sea lambda}
\eea
The sea quark spin polarizations for the other baryons 
are given in Table \ref{ocsea}. 

The contribution of the orbital part to the total magnetic moment 
of proton as given by Eq. (\ref{orbit}) is
\be
 \mu(p)_{{\rm orbit}} ={\cos}^2 \phi\left [ \frac{4}{3} 
[\mu (u_+ \rightarrow)] - \frac{1}{3} [\mu (d_+ \rightarrow)] \right]+
{\sin}^2 \phi\left[ \frac{2}{3} [\mu (u_+ \rightarrow)]
+\frac{1}{3} [\mu (d_+ \rightarrow)] \right ],  \label{orbit p}
\ee
and for $\Lambda$ it is
\be
 \mu(\Lambda)_{{\rm orbit}} ={\cos}^2 \phi [ \mu (s_+
\rightarrow)]
+{\sin}^2 \phi\left[\frac{1}{3} [\mu (u_+ \rightarrow)]
+\frac{1}{3}[\mu (d_+ \rightarrow)+
\frac{1}{3}[\mu (s_+ \rightarrow) ] \right ]. \label{orbit lambda}
\ee
Substituting Eqs. (\ref{val p}), (\ref{sea p}) and (\ref{orbit p}) in
Eq. (\ref{totalmagmom}) we get the total magnetic moment of proton
and from Eqs. (\ref{val lambda}), (\ref{sea lambda}) and (\ref{orbit
lambda}) we get the magnetic moment of $\Lambda$.
Similarly, we have calculated the valence, sea and orbital 
contributions to the magnetic moments for the other octet baryons. 

We also present the magnetic moment of $\Delta^{+}$, as an example of the 
decuplet baryon.
As there is no mixing in the case of decuplet baryons, therefore, the
spin structure for $\Delta^{+}$
using Eqs. (\ref{56decuplet}) of the text is given as
\be
 \langle56,0^+|N|56,0^+\rangle=2 u_{+}+ d_{+}. \label{56delta}
\ee
The valence contribution $\mu(\Delta^+)_{{\rm val}}$  to the total magnetic 
moment is 
\bea
\mu(\Delta^+)_{{\rm val}}&=&\Delta u_{{\rm val}}\mu_u +
\Delta d_{{\rm val}}\mu_d +\Delta
s_{{\rm val}}\mu_s \nonumber \\
& =& 2 \mu_u +1\mu_d+ 0 \mu_s.  \label{dec val}
\eea

The contribution of the ``sea''  to the total magnetic moment in terms of 
the sea quark polarizations and $\mu_q$ is
\be
 \mu(\Delta^+)_{{\rm sea}}=\Delta u_{{\rm sea}}\mu_u+
\Delta d_{{\rm sea}}\mu_d+\Delta s_{{\rm sea}}\mu_s. 
\ee
The spin structure for the sea quarks of $\Delta^+$ is obtained by substituting for $q_\pm$ in Eq. (\ref{56delta}), therefore
\be
  \hat {\Delta^{+}}=2 (\sum P_u u_{+} + 
|\psi(u_{+})|^2)+
(\sum P_d d_{+} + |\psi(d_{+})|^2),
\ee
and
\be
 \mu(\Delta^+)_{{\rm sea}}=\left[- a( 5+2 \alpha ^2 + \frac{2}{3} \beta^2+\frac{4}{3} \zeta^2) \right]\mu_u+
\left[- a( 4 + \alpha ^2 + \frac{1}{3} \beta^2+\frac{2}{3}\zeta^2)
\right]\mu_d+\left[-3a \alpha^2 \right]\mu_s.   \label{dec sea}
\ee
The contribution of the ``sea'' to the magnetic moment of other
decuplet baryons can
be written by using the expressions for the spin polarizations from
Table \ref{desea}.

The contribution of the orbital part to the total magnetic moment 
as given by Eq. (\ref{de orbit}) is
\be
 \mu(\Delta^+)_{orbit} =2 [\mu (u_+ \rightarrow)] +
[\mu (d_+ \rightarrow)]. \label{3dec orbit}
\ee
Substituting Eqs. (\ref{dec val}), (\ref{dec sea}) and 
(\ref{3dec orbit}) in Eq. (\ref{totalmagmom}) we get the total 
magnetic moment of $\Delta^+$.


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\pagebreak

%\renewcommand{\thetable}{\arabic{table}}
\begin{table}
\begin{center}
\begin{tabular}{cccc}      
Octet & & & \\
baryons & $\Delta u_{{\rm val}}$ &$\Delta d_{{\rm val}}$ &  
$\Delta s_{{\rm val}}$ \\ \hline 
$p(uud)$ & ${\cos}^2 \phi \left[\frac{4}{3} \right]
+{\sin}^2 \phi \left[\frac{2}{3} \right]$
& ${\cos}^2 \phi \left[-\frac{1}{3} \right]  +
  {\sin}^2 \phi \left[\frac{1}{3}  \right]$ & 0 \\  
$n(udd)$& ${\cos}^2 \phi \left[-\frac{1}{3} \right]  +
  {\sin}^2 \phi \left[\frac{1}{3}  \right]$ &
${\cos}^2 \phi \left[\frac{4}{3} \right]
+{\sin}^2 \phi \left[\frac{2}{3} \right]$
& 0 \\  
 $\Sigma^+(uus)$ &${\cos}^2 \phi \left[\frac{4}{3} \right]
+{\sin}^2 \phi \left[\frac{2}{3} \right]$
& 0 &
${\cos}^2 \phi \left[-\frac{1}{3} \right]  +
  {\sin}^2 \phi \left[\frac{1}{3}  \right]$ \\ 
$\Sigma^-(dds)$ & 0 & ${\cos}^2 \phi \left[\frac{4}{3} \right]
+{\sin}^2 \phi \left[\frac{2}{3} \right]$ &
${\cos}^2 \phi \left[-\frac{1}{3} \right]  +
  {\sin}^2 \phi \left[\frac{1}{3}  \right]$ \\ 
$\Xi^o(uss)$ &${\cos}^2 \phi \left[-\frac{1}{3} \right]  +
  {\sin}^2 \phi \left[\frac{1}{3}  \right]$ & 0 &
${\cos}^2 \phi \left[\frac{4}{3} \right]
+{\sin}^2 \phi \left[\frac{2}{3} \right]$ \\
$\Xi^-(dss)$ &0 &${\cos}^2 \phi \left[-\frac{1}{3} \right]  +
  {\sin}^2 \phi \left[\frac{1}{3}  \right]$ &
${\cos}^2 \phi \left[\frac{4}{3} \right]
+{\sin}^2 \phi \left[\frac{2}{3} \right]$ \\ 
$\Lambda(uds)$ & ${\cos}^2 \phi[0]+ {\sin}^2 \phi \left[\frac{1}{3}  \right]$
&${\cos}^2 \phi[0]+ {\sin}^2 \phi \left[\frac{1}{3}  \right]$ &
${\cos}^2 \phi \left[1 \right]
+{\sin}^2 \phi \left[\frac{1}{3} \right]$ \\ 

\end{tabular}
\end{center}
\caption{Valence quark spin polarizations for the ``mixed'' octet baryons.
The spin structure of the baryon $B$ without
spin-spin forces can be obtained by taking  $\phi=0$. }
\label{ocval}
\end{table}


\begin{table}
\begin{center}
\begin{tabular}{cccc}       
Decuplet & & & \\
baryons & $\Delta u_{{\rm val}}$ &$\Delta d_{{\rm val}}$ &  
$\Delta s_{{\rm val}}$ \\ \hline 

$\Delta^{++}(uuu)$ & 3  & 0  & 0 \\  
$\Delta^{+}(uud)$ & 2  & 1  & 0 \\  
$\Delta^{0}(udd)$ & 1  & 2  & 0 \\  
$\Delta^{-}(ddd)$ & 0  & 3  & 0 \\  
$\Sigma^{*^+}(uus)$ &2  &0 & 1  \\
$\Sigma^{*^o}(uds)$ & 1 & 1 & 1 \\
$\Sigma^{*^-}(dds)$ & 0 & 2 & 1 \\
$\Xi^{*^o}(uss)$  &  1  & 0 & 2 \\
$\Xi^{*^-}(dss)$  &  0  & 1 & 2 \\
$\Omega^{-}(sss)$ & 0 & 0 & 3 \\  

\end{tabular}
\end{center}
\caption{Valence quark spin polarizations for the decuplet  baryons. }
\label{deval}
\end{table}



\begin{table}
\begin{center}
\begin{tabular}{cccc}       
Octet & & & \\
baryons & $\Delta u_{{\rm sea}}$ &$\Delta d_{{\rm sea}}$ &  
$\Delta s_{{\rm sea}}$ \\ \hline 
$p(uud)$ &$-{\cos}^2 \phi \left[\frac{a}{3} (7+4 \alpha^2+
 \frac{4}{3}\beta^2 +\frac{8}{3} \zeta^2)\right]$
  &$-{\cos}^2 \phi \left[\frac{a}{3} (2-\alpha^2
-\frac{1}{3}\beta^2 -\frac{2}{3} \zeta^2)\right]$
 & $-a \alpha^2$ \\ 
 & $-{\sin}^2 \phi \left[\frac{a}{3} (5+2 \alpha^2
+\frac{2}{3}\beta^2 +\frac{4}{3} \zeta^2)\right]$
&$-{\sin}^2 \phi \left[\frac{a}{3} (4+\alpha^2
+\frac{1}{3}\beta^2 +\frac{2}{3} \zeta^2)\right]$
& \\  

$n(udd)$ &$-{\cos}^2 \phi \left[\frac{a}{3} (2-\alpha^2
-\frac{1}{3}\beta^2 -\frac{2}{3} \zeta^2)\right]$
&$-{\cos}^2 \phi \left[\frac{a}{3} (7+4 \alpha^2+
 \frac{4}{3}\beta^2 +\frac{8}{3} \zeta^2)\right]$
 & $-a \alpha^2$ \\ 
& $-{\sin}^2 \phi \left[\frac{a}{3} (4+\alpha^2
+\frac{1}{3}\beta^2 +\frac{2}{3} \zeta^2)\right]$
& $-{\sin}^2 \phi \left[\frac{a}{3} (5+2 \alpha^2
+\frac{2}{3}\beta^2 +\frac{4}{3} \zeta^2)\right]$ & \\  

 $\Sigma^+(uus)$ & $-{\cos}^2 \phi \left[\frac{a}{3}
   (8+3 \alpha^2+ \frac{4}{3}\beta^2+ \frac{8}{3} \zeta^2)\right]$ & 
$-{\cos}^2 \phi \left[\frac{a}{3}(4-\alpha^2) \right]$
 & $-{\cos}^2 \phi \left[\frac{a}{3} (2
\alpha^2-\frac{4}{3}\beta^2 -\frac{2}{3} \zeta^2)\right]$ \\
 &$-{\sin}^2 \phi \left[\frac{a}{3} (4+3 \alpha^2+
 \frac{2}{3}\beta^2 +\frac{4}{3} \zeta^2)\right]$ 
& $-{\sin}^2 \phi \left[\frac{a}{3}(2+\alpha^2) \right]$
& $-{\sin}^2 \phi  \left[\frac{a}{3} (4 \alpha^2+
 \frac{4}{3}\beta^2 +\frac{2}{3} \zeta^2)\right]$ \\ 

$\Sigma^-(dds)$ &$-{\cos}^2 \phi \left[\frac{a}{3}(4-\alpha^2) \right]$
 & $-{\cos}^2 \phi \left[\frac{a}{3}
   (8+3 \alpha^2+ \frac{4}{3}\beta^2+ \frac{8}{3} \zeta^2)\right]$ &
$-{\cos}^2 \phi \left[\frac{a}{3} (2
\alpha^2-\frac{4}{3}\beta^2 -\frac{2}{3} \zeta^2)\right]$ \\
& $-{\sin}^2 \phi \left[\frac{a}{3}(2+\alpha^2) \right]$
&$-{\sin}^2 \phi \left[\frac{a}{3} (4+3 \alpha^2+
 \frac{2}{3}\beta^2 +\frac{4}{3} \zeta^2)\right]$ 
& $-{\sin}^2 \phi  \left[\frac{a}{3} (4 \alpha^2+
 \frac{4}{3}\beta^2 +\frac{2}{3} \zeta^2)\right]$ \\ 


 $\Xi^o(uss)$ &$-{\cos}^2 \phi \left[\frac{a}{3}
   (3 \alpha^2-2- \frac{1}{3}\beta^2 -\frac{2}{3} \zeta^2)\right]$
&$-{\cos}^2 \phi \left[\frac{a}{3}(4\alpha^2-1) \right]$
&$-{\cos}^2 \phi \left[\frac{a}{3} (7 \alpha^2+
\frac{16}{3}\beta^2 +\frac{8}{3} \zeta^2)\right]$ \\
&  $-{\sin}^2 \phi \left[\frac{a}{3} (2+3 \alpha^2+
\frac{1}{3}\beta^2 +\frac{2}{3} \zeta^2)\right]$
&$ -{\sin}^2 \phi \left[\frac{a}{3}(1+2 \alpha^2) \right] $
& $-{\sin}^2 \phi \left[\frac{a}{3} (5 \alpha^2+
\frac{8}{3}\beta^2 +\frac{4}{3} \zeta^2)\right]$\\

$\Xi^-(dss)$&$-{\cos}^2 \phi \left[\frac{a}{3}(4\alpha^2-1) \right]$
& $-{\cos}^2 \phi \left[\frac{a}{3}
   (3 \alpha^2-2- \frac{1}{3}\beta^2 -\frac{2}{3} \zeta^2)\right]$
&$-{\cos}^2 \phi \left[\frac{a}{3} (7 \alpha^2+
\frac{16}{3}\beta^2 +\frac{8}{3} \zeta^2)\right]$ \\
&$ -{\sin}^2 \phi \left[\frac{a}{3}(1+2 \alpha^2) \right] $
&  $-{\sin}^2 \phi \left[\frac{a}{3} (2+3 \alpha^2+
\frac{1}{3}\beta^2 +\frac{2}{3} \zeta^2)\right]$
& $-{\sin}^2 \phi \left[\frac{a}{3} (5 \alpha^2+
\frac{8}{3}\beta^2 +\frac{4}{3} \zeta^2)\right]$\\ 

$\Lambda(uds)$&$-{\cos}^2 \phi \left[a \alpha^2\right]$ &
$-{\cos}^2 \phi \left[a \alpha^2\right]$ &
$-{\cos}^2 \phi \left[\frac{a}{3}(6 \alpha^2+
4 \beta^2 +2 \zeta^2)\right]$ \\
&$-{\sin}^2 \phi \left[\frac{a}{9}(9+6 \alpha^2+\beta^2+
2 \zeta^2)\right]$ &
$-{\sin}^2 \phi \left[\frac{a}{9}(9+6 \alpha^2+\beta^2+
2 \zeta^2)\right]$ &
$-{\sin}^2 \phi \left[\frac{4}{9}a(3 \alpha^2+2 \beta^2+
\zeta^2)\right]$ \\


\end{tabular}
\end{center}
\caption{Sea quark spin polarizations for the ``mixed'' 
octet baryons in terms of the coupling  breaking terms.
The spin structure of the baryon $B$ without
spin-spin forces can be obtained by taking $\phi=0$.} 
\label{ocsea}
\end{table}



\begin{table}
\begin{center}
\begin{tabular}{cccc}     
Decuplet & & & \\
 baryons & $\Delta u_{{\rm sea}}$ &$\Delta d_{{\rm sea}}$ &  
$\Delta s_{{\rm sea}}$ \\ \hline 

$\Delta^{++}(uuu)$ & $- a( 6 + 3 \alpha ^2 +\beta^2+2 \zeta^2)$  & $-3a$ &  $-3a \alpha^2$ \\  
$\Delta^{+}(uud)$ & $- a( 5+2 \alpha ^2 + \frac{2}{3} \beta^2+\frac{4}{3} \zeta^2)$  &
   $- a( 4 + \alpha ^2 + \frac{1}{3} \beta^2+\frac{2}{3}\zeta^2)$ & $-3a \alpha^2$ \\  
$\Delta^{o}(udd)$ & $- a( 4 + \alpha ^2 + \frac{1}{3} \beta^2+\frac{2}{3}\zeta^2)$   & 
$- a( 5 + 2 \alpha ^2 + \frac{2}{3} \beta^2+\frac{4}{3} \zeta^2)$  &  $-3a \alpha^2$ \\  
$\Delta^{-}(ddd)$ & $-3a$  & $- a( 6 + 3 \alpha ^2 +\beta^2+2 \zeta^2)$    &  $-3a \alpha^2$ \\  
$\Sigma^{*^+}(uus)$ & $- a( 4+3 \alpha^2 + \frac{2}{3} \beta^2+\frac{4}{3}\zeta^2)$  &
 $- a( \alpha ^2 + 2)$ & $- 2 a( 2 \alpha ^2 + \frac{2}{3} \beta^2+\frac{1}{3} \zeta^2)$  \\
$\Sigma^{*^o}(uds)$ & $- a( 3 + 2\alpha ^2 +\frac{1}{3} \beta^2+\frac{2}{3} \zeta^2)$ & 
$- a( 3 + 2\alpha ^2 +\frac{1}{3} \beta^2+\frac{2}{3} \zeta^2)$ & 
 $- 2a( 2\alpha ^2 +\frac{2}{3} \beta^2+\frac{1}{3} \zeta^2)$ \\
$\Sigma^{*^-}(dds)$ & $- a( \alpha ^2 + 2)$  &  
$- a( 4 + 3 \alpha ^2 + \frac{2}{3} \beta^2+\frac{4}{3} \zeta^2)$ & 
$- 2 a( 2 \alpha ^2 + \frac{2}{3} \beta^2+\frac{1}{3} \zeta^2)$ \\
$\Xi^{*^o}(uss)$  &  $-a (2+3 \alpha^2+\frac{1}{3}\beta^2+\frac{2}{3}\zeta^2)$  &  
$-a(2 \alpha^2 +1)$ & $- a(5 \alpha^2 + \frac{8}{3} \beta^2+\frac{4}{3} \zeta^2)$ \\
$\Xi^{*^-}(dss)$  &   $-a(2 \alpha^2 +1)$  & 
$-a (2+3 \alpha^2+\frac{1}{3}\beta^2+\frac{2}{3} \zeta^2)$   & $- a(5 \alpha^2 + \frac{8}{3} \beta^2+\frac{4}{3} \zeta^2)$ \\
$\Omega^{-}(sss)$ &  $-3a \alpha^2$ &  $-3a \alpha^2$ & 
$- 6a(\alpha^2 + \frac{2}{3} \beta^2+\frac{1}{3} \zeta^2)$ \\   

\end{tabular}
\end{center}
\caption{Sea quark spin polarizations for the decuplet  baryons in terms of 
the coupling  breaking terms. }
\label{desea}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{cccccc}     
              
 &  & \multicolumn{2}{c}{$\chi$QM} & 
\multicolumn{2}{c} {$\chi$QM$_{gcm}$} \\  
Parameter & Data & \multicolumn{2}{c}{$\alpha=0.6,\beta=0.9 $} & 
\multicolumn{2}{c} {$\alpha=0.4 ,\beta=0.7$} \\ \cline{3-6} 
&&NMC &E866 & NMC & E866 \\ \hline


 $\Delta$ u & 0.85 $\pm$ 0.05 {\cite{adams}} & 0.88 & 0.92  & 0.91 & 0.93 \\

 $\Delta$ d & -0.41  $\pm$ 0.05  {\cite{adams}} & -0.35  &-0.36 & -0.33 & -0.34 \\

$\Delta$ s &-0.07  $\pm$ 0.05  {\cite{adams}} & -0.05 & -0.05 & -0.02 & -0.02 \\
$G_A/G_V$ & 1.267  $\pm$ .0035  {\cite{PDG}} & 1.23 & 1.28  & 1.24 & 1.27 \\
$(=\Delta u-\Delta d)$ &&&&& \\
$\bar d-\bar u$ & 0.147 $\pm$ .024 {\cite{NMC}}  &
0.147 & 0.12 & 0.147 & 0.12  \\
                & 0.118 $\pm$ .015 \cite{E866} &&&& \\

$\bar d/\bar u$ & 1.96 $\pm$ 0.246 {\cite{baldit}} &  1.89 & 1.59  &
1.89 &  1.59 \\
                & 1.41 $\pm$ 0.146 \cite{E866} && && \\

$f_s$ &  0.10 $\pm$ 0.06 {\cite{ao}} & 0.15 & 0.13 & 0.07 & 0.05  \\


$f_3/f_8$ & 0.21 $\pm$ 0.05 {\cite{cheng}} & 0.25 & 0.25 &
0.21 &   0.21 \\     

\end{tabular}
\end{center}
\caption{$\chi$QM parameters obtained by fitting the spin and quark
distribution functions  with and
without spin-spin forces, $\chi$QM$_{gcm}$ corresponds to the case
where the ``mixed'' nucleon has been used.} \label {ud}
\end{table}


\begin{table}
{\footnotesize
\begin{center}
\begin{tabular}{ccccccccccccccc}   
 &  & & \multicolumn{4}{c} { } & \multicolumn{4}{c} {} & 
\multicolumn{4}{c} {$\chi$QM with mass }\\  

&  & & \multicolumn{4}{c} {$\chi$QM} & \multicolumn{4}{c}
{$\chi$QM with} &  \multicolumn{4}{c} {adjustments and}\\ 
&  & & \multicolumn{4}{c} {} & \multicolumn{4}{c}
{mass adjustments} &  \multicolumn{4}{c} {configuration
mixing}\\ 
\cline{4-7}  \cline{8-11} \cline {12-15}

Octet & Data & NRQM &  Valence  & Sea & Orbital &  Total &  
Valence  & Sea & Orbital &  Total &  Valence  & Sea & Orbital &  Total \\
 baryons & ~\cite{PDG} &  &   & &&&&&&&&&& \\  \hline

p          & 2.79  & 2.72 & 3.00 & -0.70 & 0.54 & 2.84 &  
3.17   & -0.59&    0.45 & 3.03 &  
2.94    & -0.55& 0.41 & 2.80   \\
n          & -1.91 & -1.81 & -2.00 & 0.34 & -0.41 & -2.07 & 
-2.11    & 0.24  & -0.37 & -2.24 & 
 -1.86  &  0.20 & -0.33  & -1.99   \\

$\Sigma^-$ & -1.16 &  -1.01 & -1.12 & 0.13 & -0.29 & -1.28 & 
-1.08 & 0.08& -0.26 &   -1.26 &
   -1.05    & 0.07 &  -0.22 & -1.20  \\
$\Sigma^+$ & 2.45  & 2.61 & 2.88 & -0.69 & 0.45 & 2.64 &  
2.80  &  -0.55 &  0.37 & 2.62 & 
 2.59 & -0.50 & 0.34 &  2.43  \\


$\Xi^o$    & -1.25 & -1.41 & -1.53 & 0.37 & -0.23 & -1.39 & 
-1.53 & 0.22 &  -0.16 &  -1.47 & 
 -1.32 & 0.21 &-0.13 & -1.24  \\
$\Xi^-$    & -0.65 & -0.50 & -0.53 & 0.09 & -0.06 & -0.50 & 
-0.59 & 0.06  & -0.01 & -0.54 &  
 -0.61 & 0.06&  -0.01 &   -0.56  \\

$\Lambda$ & -0.61  & -0.59 & -0.65 & 0.10 & -0.08 & -0.63 & 
-0.69  & 0.05  & -0.04  &-0.68  &
-0.59 & 0.04 &-0.04  & -0.59   \\
 \hline
$\Delta$CG  & 0.49 $\pm$ 0.05 & 0 &  & & & 0.10 & 
& &&  0.46 &  & &&     0.48 \\  

\end{tabular}
\end{center}}
\caption{ Octet baryon magnetic moments in units of $\mu_N$ for the latest E866 data.}  \label{E866}
\end{table}


\begin{table}
{\footnotesize
\begin{center}
\begin{tabular}{ccccccccccccccc}     
 &  & & \multicolumn{4}{c} { } & \multicolumn{4}{c} {} & 
\multicolumn{4}{c} {$\chi$QM with mass }\\  

&  & & \multicolumn{4}{c} {$\chi$QM} & \multicolumn{4}{c}
{$\chi$QM with} &  \multicolumn{4}{c} {adjustments and}\\ 
&  & & \multicolumn{4}{c} {} & \multicolumn{4}{c}
{configuration mixing} &  \multicolumn{4}{c} {configuration
mixing}\\ 
\cline{4-7}  \cline{8-11} \cline {12-15}

Octet & Data & NRQM &  Valence  & Sea & Orbital &  Total &  
Valence  & Sea & Orbital &  Total &  Valence  & Sea & Orbital &  Total \\
 baryons & ~\cite{PDG} &  &   & &&&&&&&&&& \\  \hline

p          & 2.79  & 2.72 & 3.00 & -0.79 & 0.53 & 2.74 &  
2.76   & -0.62&    0.48 & 2.62 &  
2.94    & -0.65& 0.41 & 2.70   \\
n          & -1.91 & -1.81 & -2.00 & 0.30 & -0.29 & -1.99 & 
-1.76    & 0.25  & -0.39 & -1.90 & 
 -1.86  &  0.27 & -0.34  & -1.93   \\

$\Sigma^-$ & -1.16 &  -1.01 & -1.12 & 0.16 & -0.30 & -1.26 & 
-1.09 & 0.10& -0.25 &   -1.24 &
   -1.05    & 0.14 &  -0.26 & -1.17  \\
$\Sigma^+$ & 2.45  & 2.61 & 2.88 & -0.77 & 0.43 & 2.54 &  
2.67  &  -0.65 &  0.40 & 2.42 & 
 2.59 & -0.59 & 0.36 &  2.36  \\


$\Xi^o$    & -1.25 & -1.41 & -1.53 & 0.45 & -0.21 & -1.29 & 
-1.32 & 0.26 &  -0.16 &  -1.22 & 
 -1.32 & 0.26 &-0.14 & -1.20  \\
$\Xi^-$    & -0.65 & -0.50 & -0.53 & 0.08 & -0.01 & -0.46 & 
-0.56 & 0.09  & -0.01 & -0.48 &  
 -0.61 & 0.09&  -0.02 &   -0.54  \\

$\Lambda$ & -0.61  & -0.59 & -0.65 & 0.12 & -0.07 & -0.60 & 
-0.56  & 0.07  & -0.05  &-0.54  &
-0.59 & 0.07 &-0.05  & -0.57   \\
 \hline
$\Delta$CG  & 0.49 $\pm$ 0.05 & 0 &  & & & 0.10 & 
& &&  0.12 &  & &&     0.44 \\ 

\end{tabular}
\end{center}}
\caption{ Octet baryon magnetic moments in units of $\mu_N$ for the NMC data.}
 \label{NMC}
\end{table}


\begin{table}
{\footnotesize
\begin{center}
\begin{tabular}{cccccccccccc}       
              
Decuplet & Data & NRQM & X. Song  & Linde {\it et al.}  &  Valence  &
\multicolumn{2}{c}{Sea} &  
\multicolumn{2}{c}{Orbital} & \multicolumn{2}{c}{Total} \\ \cline{7-12}

 baryons & ~\cite{PDG} & &~{\cite{song}} & ~{\cite{johan}} &  & NMC &E866 &NMC
 &E866 &NMC & E866 \\ \hline 

$\Delta^{++}$ & 3.7 $<$ $\mu_{\Delta^{++}}$ $<$ 7.5& 5.43 & 5.55 &
5.21 & 6.36 & -1.59 & -1.31 & 0.94 & 0.92  & 5.71 & 5.97   \\
$\Delta^{+}$  & - & 2.72 & 2.73 & 2.45& 3.18 & -0.94 & -0.79 & 0.38 & 0.37 
& 2.62 & 2.76   \\
$\Delta^{o}$  & - & 0 & -0.09 & -0.30 & 0 & -0.28 & -0.28 & -0.18 & -0.18 
& -0.46 & -0.46  \\
$\Delta^{-}$  & - & -2.72 & -2.91 & -3.06 & -3.18 & 0.37 &  0.23 &
-0.74  & -0.73  & -3.55 & -3.68  \\

$\Sigma^{*+}$  & - & 3.02 & 3.09 & 2.85 & 3.24 &  -0.88 & -0.73 &  0.58 & 0.56 
& 2.94  & 3.07  \\ 
$\Sigma^{*o}$  & - & 0.30 & 0.27 & 0.09 & 0.33 & -0.28 &-0.26 & 0.01 & 0.01
& 0.06 & 0.08  \\
$\Sigma^{*-}$  & - & -2.41 & -2.55 & -2.66 & -2.58 & 0.32  & 0.20 &
-0.54 & -0.54  & -2.80 & -2.92  \\


$\Xi^{*o}$     & - & 0.60 & 0.63 & 0.49 & 0.52 & -0.27 & -0.24 & 0.21 & 0.21 
& 0.46  & 0.49  \\
$\Xi^{*-}$     & - & -2.11 & -2.19 & -2.27 & -2.30 & 0.31  & 0.21 &
-0.35 & -0.34  & -2.33  & -2.43  \\
$\Omega^{-}$  & -2.02 $\pm$ 0.005 & -1.81 & -1.83 & -1.87 & -2.07 &
0.30 & 0.21 & -0.14 &  -0.15 & -1.91 & -2.01  \\          


\end{tabular}
\end{center}}
\caption{Decuplet magnetic moments for NMC and E866 data.} \label{decuplet}
\end{table}


\begin{table}
\begin{center}
\begin{tabular}{ccccccccc}      
              
Octet  & Data & NRQM &\multicolumn{2}{c} {$M_u,M_d=310$ MeV} & 
\multicolumn{2}{c} {$M_u,M_d=340$ MeV}& 
\multicolumn{2}{c} {$M_u,M_d=330$ MeV} \\ \cline{4-9}   
baryons& ~\cite{PDG} & & NMC &E866 &NMC & E866 & NMC & E866 \\ \hline
p          & 2.79  &  2.72 & 2.48 & 2.60 & 2.69 & 2.84 &   2.70 & 2.80   \\
n          & -1.91 &  -1.81 &  -1.79 & -1.88 & -1.96 & -2.06  &   
-1.93 &  -1.99    \\

$\Sigma^-$ & -1.16 &  -1.01  & -1.16& -1.20 & -1.28 & -1.32 &   
-1.17&  -1.20   \\
$\Sigma^+$ & 2.45  &  2.61&  2.20 & 2.31  &  2.42 &2.54  &  
2.36 & 2.43     \\


$\Xi^o$    & -1.25 &  -1.41 &  -1.10 & -1.16 &  -1.26 & -1.32 &   
-1.20&  -1.24    \\
$\Xi^-$    & -0.65 &  -0.50 &  -0.48 & -0.50&  -0.56 & -0.59&
 -0.54 &   -0.56  \\
$\Lambda$ & -0.61  &  -0.60 &  -0.54 &   -0.57 & -0.63 & -0.64 &  
-0.57 & -0.59 \\ \hline
$\Delta$CG  & 0.49 $\pm$ 0.05 & 0 & 0.29 & 0.31 &0.25  & 0.31 & 
0.44 &  0.48  \\ 

                    
\end{tabular}
\end{center}
\caption{Comparison of the results of $\chi$QM with Cheng-Li mechanism, 
spin-spin forces and ``mass adjustments'' for different sets of  
quark masses.} \label{masses}
\end{table}

\end{document}

