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\begin{frontmatter}

\title{The baryon magnetic moments of the octet and the decuplet using
  different limits of the  SU(3) flavor group}

\author[Merida]{J. G. Contreras},
\author[Merida]{R. Huerta} and
\author[Merida]{L. R. Quintero}

\address[Merida]{Departamento de F\'{\i}sica Aplicada,
CINVESTAV--IPN, Unidad M\'erida, A. P. 73 Cordemex, 97310 M\'erida, 
Yucat\'an, M\'exico}
 

\begin{abstract}
  Working within the non relativistic quark model a two parameter fit
  to the magnetic moments of the baryon octet is presented.  The model
  is based on taking different limits of the SU(3) flavor group to
  describe different magnetic moments. Using the values extracted from
  the fit the magnetic moments of the baryon decuplet have been
  predicted and an excellent agreement with the experimental
  measurements has been found.
\end{abstract}


\end{frontmatter}

PACS: 13.40.E, 13.30.E, 12.39.Jh

Keywords: Magnetic moments, Baryons, Flavor Symmetry

\section{Introduction}
There has recently been a renewed interest in the magnetic moments and
spin structure of baryons within a variety of models. For example, the
chiral quark model \cite{franklin0,linde}, quenched lattice gauge
theory \cite{leinweber} and the 1/N$_\mathrm{c}$ expansion
\cite{jenkins} to name a few.  These models are more ambitious than
the non-relativistic quark model (NQM).  Nonetheless it has been
argued that due to some subtle cancellations the NQM is a good
approximation to the magnetic moments \cite{cheng}, so a simple model
could be able to extract the physics of the problem more easily than a
complicated one.

It is well known since a long time that the magnetic moments of the
octet baryons can be described approximately via a SU(3) flavor group
\cite{coleman}.  Using this approach within the NQM, it is assumed
that the breaking of the flavor symmetry acts equally in all states of
the octet. Thus, for example, the agreement of the predicted ratio of
the magnetic moment of the proton to that of the neutron with the
experimental result is taken as a coincidence.

On the other hand it is evident that even knowing that the breaking of 
SU(3) flavor symmetry is about 30\% in the octet, it is naive to
expect such a discrepancy between the NQM and each of the magnetic
moments. Take for example  the case of the {\bf 20} in SU(4). There the
flavor symmetry breaking is quite bigger than in the case of
SU(3). Nonetheless there is no reason why this should greatly affect the
predictions of the magnetic moment of the proton or the neutron
within a model in SU(4) with respect to SU(3).

Using these thoughts as guidelines, the effect of
considering different 
limits of the SU(3) flavor group was
studied and applied to
calculate the magnetic moments of the octet. It was found that using a two
parameter fit to the experimental data a better agreement in
terms of $\chi^2$ was obtained,
 than other two parameter fits in the literature and a
comparable agreement to fits using four parameters (see for example
\cite{sehgal,pondrom}). Using the value obtained for the parameters 
 the magnetic moments for the baryons in the decuplet have been predicted. The
comparison to the existing experimental values is quite satisfactory.
%Introducing a third parameter via breaking of SU(2) symmetry we fine
%tuned our predictions.  The most important consequence of this last
%step is to predict a magnetic moment for $\Delta^0$ different from
%zero.


The current status on the experimental side is as follows. Seven of
the  magnetic moments are measured with around  1\% accuracy or
better \cite{pdg}. The transition magnetic moment for $\Sigma^0
\rightarrow \Lambda$ is known to a 5\% precision \cite{petersen}. From
the decuplet, the $\Omega^-$ was measured some time ago \cite{diehl}
and recently a new measurement has been presented
\cite{wallence}. Finally  the magnetic moment of the $\Delta^{++}$ has
also been measured \cite{bosshard}.

\section{The magnetic moments within different limits of the SU(3)
  flavor group}
\label{sec:model}

To date the way to explain the magnetic moments have been to look for
models to break the SU(3) symmetry. Here it was decided to, on the one
hand, to keep the flavor symmetry exact (in the following this case
will be labeled SU(3)$^e$), on the other hand to let only the mass of
the strange quark to go to $\infty$ and keep $m_u=m_d$ (in the following
this case will be labeled SU(2)$^\infty$). The driving idea behind
this approach is the {\em Ansatz} that baryons, and with them their
magnetic moments, prefer to stay near symmetric states. In this case
it means that for example the proton and the neutron not knowing
anything about the strange quark, may prefer to remain in a SU(2)$^\infty$
state more than staying in the exact SU(3) flavor state.

The magnetic moments for the octet of baryons within NQM are given in
table~\ref{tab:magmom}.  The case of exact SU(3) flavor symmetry
requires the masses of the quarks to be equal: $m_3\equiv m_\mathrm{u}
= m_\mathrm{d} = m_\mathrm{s}$.  This implies that the magnetic
moments $\mu_i$ obey the following equalities: $\mu_\mathrm{u}=
-2\mu_\mathrm{d}=-2\mu_\mathrm{s}$. Under these conditions the
magnetic moments for the octet are given as shown in the third column
of table \ref{tab:magmom} where the definition $\mu_3\equiv\mu_\mathrm{s}$ 
has been used.

The masses in the SU(2)$^\infty$ scenario fulfill $m_2\equiv
m_\mathrm{u} = m_\mathrm{d} \ll m_\mathrm{s}$. The magnetic moments of
the baryon octet were calculated in this case and then the limit
$m_\mathrm{s} \rightarrow \infty$ is taken. The expressions obtained
from this procedure are shown in the last column of table
\ref{tab:magmom} where the definition $\mu_2\equiv\mu_\mathrm{d}$ has
been used.

Now, one could write the formulas for both cases just one step before
taking the limit. For example the case of SU(2)$^\infty$ yields the
equations presented in the second column of table \ref{tab:corr}. Note
that in terms of $\mu_3/\mu_2=m_2/m_3$ the magnetic moments could be
group as those with small corrections {$p$, $n$, $\Sigma^+$,
$\Sigma^-$, $\Sigma^0\rightarrow\Lambda$} and those
with big corrections {$\Lambda$, $\Xi^0$ $\Xi^-$}. Repeating the
exercise for these three last magnetic moments in the case of
SU(3)$^e$ the equations of the last column of table \ref{tab:corr} are
found. Here the correction factors are again small.

From here one concludes that the magnetic moments of the baryons in
the first group like the SU(2)$^\infty$ limit better, whereas the rest
prefers to be close to the SU(3)$^e$ limit. To test this model a fit
to the experimental values --shown in table \ref{tab:meas}-- was
performed. In this fit the SU(2)$^\infty$ formulas were used for the
magnetic moments of the baryons $p$, $n$, $\Sigma^+$, $\Sigma^-$ and
the transition $\Sigma^0\rightarrow\Lambda$ and the SU(3)$^e$
equations for $\Lambda$, $\Xi^0$ y $\Xi^-$.

There is an important technical point, while performing the fit. The
magnetic moments of both the proton and the neutron have a very small
experimental error. This precision of more than one part per million
is huge when compared to the accuracy of the isospin symmetry of the
 ($p$,$n$) doublet. This turns meaningless a $\chi^2$ approach to the
fit. To avoid this problem, it was proposed in \cite{franklin} to add
in quadrature a common absolute error to all the moments. Following
this lead (see also \cite{pondrom}) an absolute error of
$\sigma=0.03\mu_N$ has been added in quadrature to the real
experimental error, and then the fit has been performed.

This two parameter fit can be viewed as two independent one parameter
fits. For the case of the SU(2)$^\infty$  limit a $\chi^2$ per degree of
freedom of 0.42 was found. The SU(3)$^e$ case yield $\chi^2/$ndf=1.9.
These produce a total $\chi^2/$ndf=1.4 when considering all eight
magnetic moments together. The fitted values of the parameters are
$\mu_2=-0.930\pm0.007$ and $\mu_3=-0.628\pm0.013$.
The values obtained for the magnetic moments using these parameters 
are shown in table~\ref{tab:meas}. The  errors shown are the maximum
spread in the values of the magnetic moments obtained by varying the
parameters within their errors.

\section{Discussion}


{\bf 1.} To be able to perform the fit an extra error of
$\sigma=0.03\mu_N$ has been added in quadrature to the experimental
error. This value makes sense as much as in the size of accuracy of
considering the proton and the neutron as a isospin doublet, as in
comparison to the errors of the other experimental errors. Nonetheless
to study the sensitivity of the results to this error, its value was
changed to 0.02 and 0.04 $\mu_N$. As expected, the main effect was in
the $\chi^2/$ndf which changed from 1.4 to 2.6 and 0.9 respectively.
The value of the parameters remained the same and their errors varied
between $\pm$0.93 to $\pm$0.17 for $\mu_2$ and $\pm$0.005 to
$\pm$0.009 for $\mu_3$. This shows that the fit is quite stable under
variation of this assumption. 
It must be noted that other analysis have used this extra error to
equalize the weights, within the fit, of the different magnetic
moments and to force a $\chi^2$/ndf of the order of one \cite{sehgal,karl}.
 
{\bf 2.} From this analysis it is clear that the effects of the flavor
symmetries and their breaking is different for different members of the
octet. This can be explained in a natural way from the wave function
of the baryons. Those grouped near the SU(3)$^e$ limit have either two
strange quarks (the $\Xi$s) or the influence of light quarks tends to
cancel each other ($\Lambda$). On the same way the rest of the baryons
do not have a dominance of the strange quark in their wave function
and thus cluster around the SU(2)$^\infty$  limit.

{\bf 3.} A similar analysis can be carried in the decuplet, either
studying their wave functions or writing their magnetic moments as a
function of the SU(2)$^\infty$, respectively SU(3)$^e$,
limit plus a correction
term. In this case as all the wave functions are symmetric, there is
no cancellation in the light quark sector, as there were in the case of
the $\Lambda$, and the grouping is quite natural. It is seen that the
magnetic moments of the $\Delta$s and the $\Sigma^*$s group near 
SU(2)$^\infty$ and the $\Xi^*$s and the $\Omega$ prefer the SU(3)$^e$
flavor limit. The predicted magnetic moments, using the parameters
found from the analysis of the octet are shown in table
\ref{tab:decu}. An excellent agreement with the experimental
measurements is found.


%{\bf 4.} As already mentioned there is no reason to expect all magnetic
%moments to respond in the same way to a breaking of the flavor
%symmetry. Thus the natural next step is to extend the analysis 
%to the breaking of the SU(2) flavor
%group. For the octet one expect that only the proton and the neutron
%are sensitive to such minor corrections because they have no strange
%quark whatsoever within the NQM.   .....?!

\section{Conclusions}

A two parameter fit to the magnetic moments of the baryon octet has
been shown. In terms of $\chi^2$ the model presented here has an
accuracy of the same order than other 4 parameter fits in the
literature. The value of the parameters has been used to predict the
magnetic moments of the decuplet and an excellent agreement with the
measured values has been found.


\begin{thebibliography}{9}

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  (1998), unpublished.

\bibitem{linde} 
J. Linde, O. Ohlsson and H. Snellman, Phys. Rev. {\bf D57}
 (1998) 452; Phys. Rev. {\bf D57} (1998) 5916.

\bibitem{leinweber} 
D. B. Leinweber, T. Draper and R. M. Woloshyn, Phys. Rev. {\bf D46}
(1992) 3067.

\bibitem{jenkins} 
E. Jenkins and A. V. Manohar, Phys. Lett. {\bf B335} (1994) 452.

\bibitem{cheng}
T.P. Cheng and Ling-Fong Li, Phys. Rev. Lett. {\bf 80} (1998) 2789.

\bibitem{coleman}
S. Coleman and S. L. Glashow, Phys. Rev. Lett. {\bf 6} (1961) 423.

\bibitem{sehgal} M. Casu and L. M. Sehgal, Phys. Rev. D 55 (1997)
  2644.

\bibitem{pondrom}
L. G. Pondrom, Phys. Rev. {\bf D53} (1996) 5322. 

\bibitem{pdg}
Particle Data Group, C. Caso {\em et al.}, Eur. Phys. J. {\bf C3}
(1998) 1.

\bibitem{petersen}
P. C. Petersen {\em et al.}, Phys. Rev. Lett. {\bf 57} (1986) 949.

\bibitem{diehl} 
H. T. Diehl {\em et al.}, Phys. Rev. Lett. {\bf 67}  (1991) 804.

\bibitem{wallence}
N. B. Wallence {\em et al.}, Phys. Rev. Lett. {\bf 74}  (1995) 3732.

\bibitem{bosshard}
 A. Bosshard {\em et al.}, Phys. Rev. {\bf D44}
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J. Franklin, Phys. Rev. {\bf D30} (1984) 1542.

\bibitem{karl} G. Karl, Phys. Rev. {\bf D45} (1992) 247.

\end{thebibliography}

\clearpage


\begin{table}
\centering
\caption{Expressions for the magnetic moments of the octet baryons for
  the NQM, and the  SU(3)$^e$ and SU(2)$^\infty$ flavor limits.}
\begin{tabular}{cccc}
\hline
\hline
Magnetic moment & NQM & SU(3)$^e$ & SU(2)$^\infty$  \\
\hline
$p$& $\frac{1}{3}(4\mu_u-\mu_d)$&$-3\mu_3$&$-3\mu_2$\\
$n$& $\frac{1}{3}(4\mu_d-\mu_u)$&$2\mu_3$&$2\mu_2$\\
$\Lambda$&$\mu_s$&$\mu_3$&0\\
$\Sigma^+$& $\frac{1}{3}(4\mu_u-\mu_s)$&$-3\mu_3$&$-8/3\mu_2$\\
$\Sigma^-$& $\frac{1}{3}(4\mu_d-\mu_s)$&$\mu_3$&$4/3\mu_2$\\
$\Xi^0$& $\frac{1}{3}(4\mu_s-\mu_u)$&$2\mu_3$&$2/3\mu_2$\\
$\Xi^-$& $\frac{1}{3}(4\mu_s-\mu_d)$&$\mu_3$&$-1/3\mu_2$\\
$\Sigma^0\rightarrow\Lambda$& $\frac{1}{\sqrt{3}}(\mu_d-\mu_u)$
&$\sqrt{3}\mu_3$&$\sqrt{3}\mu_2$\\
\hline
\hline
\end{tabular}
\label{tab:magmom}
\end{table}

\begin{table}
\centering
\caption{Expressions for the magnetic moments of the octet baryons for
   the SU(3)$^e$ and SU(2)$^\infty$ limits and a first correction to them.}
\begin{tabular}{ccc}
\hline
\hline
Magnetic moment & SU(2)$^\infty$ & SU(3)$^e$  \\
\hline
$p$&$-3\mu_2(1+0)$& \\
$n$& $2\mu_2(1+0)$& \\
$\Lambda$&0& $\mu_3(1+0)$\\
$\Sigma^+$&$-8/3\mu_2(1+\frac{\mu_3}{8\mu_2})$& \\
$\Sigma^-$&$4/3\mu_2(1-\frac{\mu_3}{4\mu_2})$& \\
$\Xi^0$& $2/3\mu_2(1+\frac{2\mu_3}{\mu_2})$&$2\mu_3[1-\frac{1}{3}
(1-\frac{\mu_2}{\mu_3})]$ \\
$\Xi^-$& $-1/3\mu_2(1-\frac{4\mu_3}{\mu_2})$&$\mu_3[1+\frac{1}{3}
(1-\frac{\mu_2}{\mu_3})]$ \\
$\Sigma^0\rightarrow\Lambda$&$\sqrt{3}\mu_2(1+0)$& \\
\hline
\hline
\end{tabular}
\label{tab:corr}
\end{table}

\begin{table}
\centering
\caption{Measured values for the magnetic moments of the baryon octet,
  along with the prediction of the model.}
\begin{tabular}{ccc}
\hline
\hline
Baryon & $\mu_\mathrm{exp}$ & $\mu_\mathrm{model} $  \\ 
\hline
$p$&2.79$\pm$6.3x10$^{-8}$&2.79$\pm$0.02\\
$n$&-1.91$\pm$4.5x10$^{-7}$&-1.86$\pm$0.01\\
$\Lambda$&-0.613$\pm$0.004&-0.63$\pm$0.01\\
$\Sigma^+$&2.46$\pm$0.01&2.48$\pm$0.02\\
$\Sigma^-$&-1.16$\pm$0.025&-1.24$\pm$0.01\\
$\Xi^0$&-1.25$\pm$0.014&-1.26$\pm$0.02\\
$\Xi^-$&-0.651$\pm$0.0025&-0.63$\pm$0.01\\
$\Sigma^0\rightarrow\Lambda$&-1.61$\pm$0.08&-1.61$\pm$0.01\\
\hline
\hline
\end{tabular}
\label{tab:meas}
\end{table}

\begin{table}
\centering
\caption{Prediction of the magnetic moments of the baryon decuplet and
  comparison with the measured values}
\begin{tabular}{cccc}
\hline
\hline
Baryon & Magnetic moment & $\mu_\mathrm{mod}$ & $\mu_\mathrm{exp}$  \\ 
\hline
$\Delta^{++}$&$-6\mu_2$&5.58$\pm$0.04&4.52$\pm$0.95\\
$\Delta^{+}$&$-3\mu_2$&2.79$\pm$0.02&--\\
$\Delta^{0}$&0&0&--\\
$\Delta^{-}$&$3\mu_2$&-2.79$\pm$0.02&--\\
$\Sigma^{*+}$&$-4\mu_2$&3.72$\pm$0.03&--\\
$\Sigma^{*0}$&$-\mu_2$&0.93$\pm$0.01&--\\
$\Sigma^{*-}$&$2\mu_2$&-1.86$\pm$0.01&--\\
$\Xi^{*0}$&0&0&--\\
$\Xi^{*-}$&$3\mu_3$&-1.88$\pm$0.04&--\\
$\Omega^-$&$3\mu_3$&-1.88$\pm$0.04&-2.02$\pm$0.06\\
\hline
\hline
\end{tabular}
\label{tab:decu}
\end{table}

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