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\begin{document}
\title{MSW WITHOUT MATTER}
\author{ T. GOLDMAN }
\address{Los Alamos National Laboratory\\Los Alamos, NM 87545, USA}
\author{ B. H. J. MCKELLAR }
\address{University of Melbourne\\Parkville, Victoria 3052, AUSTRALIA}
\author{G. J. STEPHENSON JR.}
\address{University of New Mexico\\ Albuquerque, NM 87131, USA}

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\maketitle\abstracts{ We examine the effects of a scalar field,
coupled only to neutrinos, on oscillations
among weak interaction current eigenstates.
The existence of a real scalar field is
manifested as effective masses for the
neutrino mass eigenstates, the same for
$\nbar$ as for $\n$.  Under some conditions,
this can lead to a vanishing of $\delta m^2$,
giving rise to MSW-like effects.  We present
an idealized example and show that it may be possible
to resolve the apparent discrepancy in spectra
required by r-process nucleosynthesis in the
mantles of supernovae and by Solar neutrino
solutions.}

We have recently examined~\cite{Clouds}
the possibility that, in addition to the 
Standard Model interactions, neutrinos interact 
with each
other through an extremely light 
scalar
field $\phi$.
The neutrinos with mass $m_i$ couple to
$\phi$ with constants $g_i$.  We showed 
that, consistent
with known phenomena,
neutrino clouds could form in the 
early
Universe, influence the
evolution of structures on stellar scales, and 
have observable consequences.  Here, we discuss
a consequence of this scalar interaction
that can occur whether clouds form or not.
 
Following the relativistic many-body theory
known as
Quantum Hadrodynamics (QHD)~\cite{QHD},
we define effective masses
\begin{equation}
m^*_j = m_j -g_j\phi
\end{equation}
With more than one mass eigenstate, it is possible 
for some $m_j^*$ to become negative.
The richness of the system can be demonstrated 
with a spherically symmetric model
in which the
various couplings are all equal to the same
constant $g$.  Consider, for simplicity, two 
mass eigenstates, let the vacuum mass of the 
heavier be
denoted by $m_h$ and that of the lighter by
$m_l$.
In this case, the
shift from the vacuum mass to the effective 
mass is the same for both neutrinos,
\Bea
\Delta m &=& g\phi \\
m^*_h &=& m_h - \Delta m\\
m^*_l &=& m_l - \Delta m
\Eea
For large enough shift
 this can lead to $m_l$
becoming very negative.  If
\Bea
m_l^* & = & -m_h^*,\quad\mbox{then}\\
{m_h^*}^2-{m_l^*}^2 & = & 0,
\Eea
and there is a degeneracy between the two
neutrinos arising from a very 
different mechanism
than that involved in the usual MSW
effect~\cite{msw}.  Since the
change in the effective mass is due
to a scalar interaction, it is the same
for both $\n$ and $\nbar$ and the
degeneracy will occur at same
density, hence the same
radius in a star, supernova or
other object, for both.

In the presence of  matter,
there is also a normal
MSW effect which, being an energy
shift due to a vector interaction,
has the opposite sign for $\n$ and
$\nbar$, hence degeneracies
will occur
at different radii.

To illustrate these points we have
generated the cartoons in Figure 1
%\vspace{7cm}
\begin{figure}[h]
\begin{minipage}{2.3in}
\psfig{file=toon1r.ps,width=2.2in,height=2.5in}
\end{minipage} \hfill
\begin{minipage}{2.3in}
\psfig{file=toon2r.ps,width=2.2in,height=2.5in}
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\caption{Cartoon of effective mass shifts
and energy shifts described in the text}
\end{figure}
by representing the results of solving
the nonlinear differential equation for
the selfconsistent effective
mass~\cite{Clouds}
with a simple analytic form
and assuming a linear effect
for the vector MSW (clearly, this is far
too simple for a real system, but the trends
are correctly represented).  In part A we
assume no scalar field and demonstrate 
that the $\nbar$ degeneracy, indicated by
the shorter vertical line, occurs at
a smaller radius than
the $\n$ degeneracy, indicated by the 
longer vertical line.   In B, we add the
scalar.  The middle vertical line indicates
the position of the ${m^*}^2$ degeneracy
ignoring the
vector MSW; the outer vertical line
indicates the position of the $\nbar$
degeneracy with both.  The position of
the $\n$ degeneracy is shown by the 
inner most vertical line.


This result has possible physical 
implications.  It has recently been
shown~\cite{Fuller} that r-process
nucleosynthesis in the exterior of
a supernova can give a credible 
account for abundances, provided
there is an excess of neutrons over
protons.  To achieve this, it is
desirable to have the $\nbar$ at a 
higher temperature than the $\n$
at the site of the r-process,
which can be achieved through 
enhanced flavor transitions if the
$\nbar$ transition occurs outside
the $\n$ transition~\cite{Fuller}.
These authors suggest that this
can be achieved by an inverted 
spectrum ($m_{\n_e}$ larger 
than some other mass); it could
also be achieved through a scalar
interaction.

The extension of these considerations
to three generations is straightforward
and will be presented 
elsewhere~\cite{mswwom}.


This work has been supported in part by the
United States National Science Foundation, 
the United States Department of Energy,
the Australian Research Council and the
Australian DIST.

\section*{References}
\begin{thebibliography}{99}
\bibitem{Clouds} G. J. Stephenson Jr., T. 
Goldman and
B. H. J. McKellar, /hep-ph/ 9603392; Los Alamos 
preprint LA-UR-96949; University of Melbourne
preprint UM-P-96/29.
\bibitem{QHD} Brian D. Serot and John Dirk 
Walecka, Adv.
in Nucl. Phys. Vol. \underline{16}, 1 (J. W. 
Negele and Eric
Vogt, eds. Plenum Press, NY 1986).
\bibitem{msw} L. Wolfenstein, Phys. Rev. 
\underline{D17},
2369 (1978); \underline{D20}, 2364 (1979);
S. P. Mikheyev and A. Yu. 
Smirnov,
Nuovo Cimento Soc. Ital. Fis. \underline{9C}, 17 
(1986)
\bibitem{Fuller} G. M. Fuller, J. R. Primack and 
Y. -Z. Qian, 
Phys. Rev. \underline{D52}, 1288 (1995); Y. -Z. 
Qian
and G. M. Fuller, Phys. Rev. \underline{D52}, 
656 (1995).
\bibitem{mswwom} G. J. Stephenson Jr., T. 
Goldman and B. H. J. McKellar, submitted
to Phys. Rev. Lett.
\end{thebibliography}
\end{document}
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