%1st updating with effect from: 16 June 1992

%2ND UPDATING WITH EFFECT FROM: 14 JUNE 1993
%(changes include use of original commands of latex instead of
%applying amstex/plaintex commands in latex file)

%3RD UPDATING WITH EFFECT FROM: 28 JUNE 1993
%(for the purpose of making PlainTex file + Latex file identical)

%4TH UPDATING (BY RICHARD) WITH EFFECT FROM: 28 OCTOBER 1994
%Includes alphabetical (italic) footnotes in body text,
%symbol footnotes in preamble

%5TH UPDATING WITH EFFECT FROM: 4 JANUARY 1995
\documentstyle[12pt,world_sci]{article}

\def\slsh#1{\slash \!\!\! #1}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%             my macros
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\beqn}{\begin{eqnarray}}
\newcommand{\eeqn}{\end{eqnarray}}

\def\vev#1{{\langle#1\rangle}}
\def\calm{{\cal M}}
\def\sw{s_{\rm w}}
\def\cw{c_{\rm w}}
\def\sb{s_{\beta}}
\def\cb{c_{\beta}}
\def\abs#1{\left|#1\right|}
\def\tev{{\rm TeV }}
\def\gev{{\rm GeV }}
\def\Mev{{\rm MeV }}
\def\kev{{\rm keV }}
\def\ev{{\rm eV }}
\def\mev{{\rm meV }}



\def\spose#1{\hbox to 0pt{#1\hss}}
\def\lsim{\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$}}
     \raise 2.0pt\hbox{$\mathchar"13C$}}}
\def\gsim{\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$}}
     \raise 2.0pt\hbox{$\mathchar"13E$}}}
%\simpropto produces \propto with twiddle underneath
\def\simpropto{\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$}}
     \raise 2.0pt\hbox{$\propto$}}}

\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\def\etal{ {\it et al.} }
\def\ie{ {\it i.e.} }
\def\eg{ {\it e.g.} }
\def\ZPC#1#2#3{{\sl Z.~Phys.} {\bf C#1}, #2 (#3)}
\def\PTP#1#2#3{{\sl Prog. Theor. Phys.} {\bf #1}, #2 (#3)}
\def\PRL#1#2#3{{\sl Phys. Rev. Lett.} {\bf #1}, #2 (#3)}
\def\PRD#1#2#3{{\sl Phys. Rev.} {\bf D#1}, #2 (#3)}
\def\PLB#1#2#3{{\sl Phys. Lett.} {\bf B#1}, #2 (#3)}
\def\PREP#1#2#3{{\sl Phys. Rep.} {\bf #1}, #2 (#3)}
\def\NPB#1#2#3{{\sl Nucl. Phys.} {\bf B#1}, #2 (#3)}

\def\mgut{M_{\rm GUT}}
\def\mz{m_{\rm z}}
\def\mhl{m_{h^0}}
\def\mha{m_{A^0}}

%------------------NEW ADDITIONS TO EXISTING ARTICLE.STY------------------
\catcode`\@=11
\long\def\@makefntext#1{
\protect\noindent \hbox to 3.2pt {\hskip-.9pt  
$^{{\ninerm\@thefnmark}}$\hfil}#1\hfill}                %CAN BE USED 

\def\@makefnmark{\hbox to 0pt{$^{\@thefnmark}$\hss}}  %ORIGINAL 
        
\def\ps@myheadings{\let\@mkboth\@gobbletwo
\def\@oddhead{\hbox{}
\rightmark\hfil\ninerm\thepage}   
\def\@oddfoot{}\def\@evenhead{\ninerm\thepage\hfil
\leftmark\hbox{}}\def\@evenfoot{}
\def\sectionmark##1{}\def\subsectionmark##1{}}

%--------------------START OF PROCSLA.STY---------------------------------
% For symbolic footnotes indices in title/author preamble
\setcounter{footnote}{0}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}

%-------------------------------------------------------------------------
%NEWLY-DEFINED SECTION COMMANDS 
\newcounter{sectionc}\newcounter{subsectionc}\newcounter{subsubsectionc}
\renewcommand{\section}[1] {\vspace*{0.6cm}\addtocounter{sectionc}{1} 
\setcounter{subsectionc}{0}\setcounter{subsubsectionc}{0}\noindent 
        {\normalsize\bf\thesectionc. #1}\par\vspace*{0.4cm}}
\renewcommand{\subsection}[1] {\vspace*{0.6cm}\addtocounter{subsectionc}{1} 
        \setcounter{subsubsectionc}{0}\noindent 
        {\normalsize\it\thesectionc.\thesubsectionc. #1}\par\vspace*{0.4cm}}
\renewcommand{\subsubsection}[1]
{\vspace*{0.6cm}\addtocounter{subsubsectionc}{1}
        \noindent {\normalsize\rm\thesectionc.\thesubsectionc.\thesubsubsectionc. 
        #1}\par\vspace*{0.4cm}}
\newcommand{\nonumsection}[1] {\vspace*{0.6cm}\noindent{\normalsize\bf #1}
        \par\vspace*{0.4cm}}
                                                 
%NEW MACRO TO HANDLE APPENDICES
\newcounter{appendixc}
\newcounter{subappendixc}[appendixc]
\newcounter{subsubappendixc}[subappendixc]
\renewcommand{\thesubappendixc}{\Alph{appendixc}.\arabic{subappendixc}}
\renewcommand{\thesubsubappendixc}
        {\Alph{appendixc}.\arabic{subappendixc}.\arabic{subsubappendixc}}

\renewcommand{\appendix}[1] {\vspace*{0.6cm}
        \refstepcounter{appendixc}
        \setcounter{figure}{0}
        \setcounter{table}{0}
        \setcounter{equation}{0}
        \renewcommand{\thefigure}{\Alph{appendixc}.\arabic{figure}}
        \renewcommand{\thetable}{\Alph{appendixc}.\arabic{table}}
        \renewcommand{\theappendixc}{\Alph{appendixc}}
        \renewcommand{\theequation}{\Alph{appendixc}.\arabic{equation}}
%       \noindent{\bf Appendix \theappendixc. #1}\par\vspace*{0.4cm}}
        \noindent{\bf Appendix \theappendixc #1}\par\vspace*{0.4cm}}
\newcommand{\subappendix}[1] {\vspace*{0.6cm}
        \refstepcounter{subappendixc}
        \noindent{\bf Appendix \thesubappendixc. #1}\par\vspace*{0.4cm}}
\newcommand{\subsubappendix}[1] {\vspace*{0.6cm}
        \refstepcounter{subsubappendixc}
        \noindent{\it Appendix \thesubsubappendixc. #1}
        \par\vspace*{0.4cm}}

%---------------------------------------------------------------------------
%MACRO FOR ABSTRACT BLOCK
\def\abstracts#1{{
        \centering{\begin{minipage}{12.2truecm}\footnotesize\baselineskip=12pt\noindent
        \centerline{\footnotesize ABSTRACT}\vspace*{0.3cm}
        \parindent=0pt #1
        \end{minipage}}\par}} 

%---------------------------------------------------------------------------
%NEW MACRO FOR BIBLIOGRAPHY
\newcommand{\bibit}{\it}
\newcommand{\bibbf}{\bf}
\renewenvironment{thebibliography}[1]
        {\begin{list}{\arabic{enumi}.}
        {\usecounter{enumi}\setlength{\parsep}{0pt}
%1.25cm IS STRICTLY FOR PROCSLA.TEX ONLY
\setlength{\leftmargin 1.25cm}{\rightmargin 0pt}
%0.52cm IS FOR NEW DATA FILES
%\setlength{\leftmargin 0.52cm}{\rightmargin 0pt}
         \setlength{\itemsep}{0pt} \settowidth
        {\labelwidth}{#1.}\sloppy}}{\end{list}}

%---------------------------------------------------------------------------
%FOLLOWING THREE COMMANDS ARE FOR `LIST' COMMAND.
\topsep=0in\parsep=0in\itemsep=0in
\parindent=1.5pc

%---------------------------------------------------------------------------
%LIST ENVIRONMENTS
\newcounter{itemlistc}
\newcounter{romanlistc}
\newcounter{alphlistc}
\newcounter{arabiclistc}
\newenvironment{itemlist}
        {\setcounter{itemlistc}{0}
         \begin{list}{$\bullet$}
        {\usecounter{itemlistc}
         \setlength{\parsep}{0pt}
         \setlength{\itemsep}{0pt}}}{\end{list}}

\newenvironment{romanlist}
        {\setcounter{romanlistc}{0}
         \begin{list}{$($\roman{romanlistc}$)$}
        {\usecounter{romanlistc}
         \setlength{\parsep}{0pt}
         \setlength{\itemsep}{0pt}}}{\end{list}}

\newenvironment{alphlist}
        {\setcounter{alphlistc}{0}
         \begin{list}{$($\alph{alphlistc}$)$}
        {\usecounter{alphlistc}
         \setlength{\parsep}{0pt}
         \setlength{\itemsep}{0pt}}}{\end{list}}

\newenvironment{arabiclist}
        {\setcounter{arabiclistc}{0}
         \begin{list}{\arabic{arabiclistc}}
        {\usecounter{arabiclistc}
         \setlength{\parsep}{0pt}
         \setlength{\itemsep}{0pt}}}{\end{list}}

%---------------------------------------------------------------------------
%FIGURE CAPTION
\newcommand{\fcaption}[1]{
        \refstepcounter{figure}
        \setbox\@tempboxa = \hbox{\footnotesize Fig.~\thefigure. #1}
        \ifdim \wd\@tempboxa > 6in
           {\begin{center}
        \parbox{6in}{\footnotesize\baselineskip=12pt Fig.~\thefigure. #1}
            \end{center}}
        \else
             {\begin{center}
             {\footnotesize Fig.~\thefigure. #1}
              \end{center}}
        \fi}

%TABLE CAPTION
\newcommand{\tcaption}[1]{
        \refstepcounter{table}
        \setbox\@tempboxa = \hbox{\footnotesize Table~\thetable. #1}
        \ifdim \wd\@tempboxa > 6in
           {\begin{center}
        \parbox{6in}{\footnotesize\baselineskip=12pt Table~\thetable. #1}
            \end{center}}
        \else
             {\begin{center}
             {\footnotesize Table~\thetable. #1}
              \end{center}}
        \fi}

%--------------------------------------------------------------------------
%ACKNOWLEDGEMENT: this portion is from John Hershberger
\def\@citex[#1]#2{\if@filesw\immediate\write\@auxout
        {\string\citation{#2}}\fi
\def\@citea{}\@cite{\@for\@citeb:=#2\do
        {\@citea\def\@citea{,}\@ifundefined
        {b@\@citeb}{{\bf ?}\@warning
        {Citation `\@citeb' on page \thepage \space undefined}}
        {\csname b@\@citeb\endcsname}}}{#1}}

\newif\if@cghi
\def\cite{\@cghitrue\@ifnextchar [{\@tempswatrue
        \@citex}{\@tempswafalse\@citex[]}}
\def\citelow{\@cghifalse\@ifnextchar [{\@tempswatrue
        \@citex}{\@tempswafalse\@citex[]}}
\def\@cite#1#2{{$\null^{#1}$\if@tempswa\typeout
        {IJCGA warning: optional citation argument 
        ignored: `#2'} \fi}}
\newcommand{\citeup}{\cite}

\font\twelvebf=cmbx10 scaled\magstep 1
\font\twelverm=cmr10  scaled\magstep 1
\font\twelveit=cmti10 scaled\magstep 1
\font\elevenbfit=cmbxti10 scaled\magstephalf
\font\elevenbf=cmbx10     scaled\magstephalf
\font\elevenrm=cmr10      scaled\magstephalf
\font\elevenit=cmti10     scaled\magstephalf
\font\bfit=cmbxti10
\font\tenbf=cmbx10
\font\tenrm=cmr10
\font\tenit=cmti10
\font\ninebf=cmbx9
\font\ninerm=cmr9
\font\nineit=cmti9
\font\eightbf=cmbx8
\font\eightrm=cmr8
\font\eightit=cmti8
%-------------------------------------------------------------------------

%--------------------END OF PROCSLA.STY----------------------------------

%--------------------START OF DATA FILE----------------------------------
\textwidth 17cm
\textheight 25cm
\pagestyle{empty}
\topmargin -2.5cm
\oddsidemargin  -0.4cm
\evensidemargin -0.4cm
\parindent=1.5pc
\baselineskip=15pt
\begin{document}
\begin{flushright}
MPI/PhT/96--104\\
UCDPHY-96-27\\
September 1996
\end{flushright}

\centerline{}
\vskip3.2cm

%\centerline{\normalsize\bf WORLD SCIENTIFIC PUBLISHING COMPANY}
%\baselineskip=22pt
\centerline{\normalsize\bf NEUTRINO PROPERTIES AND SUSY WITHOUT $R$-PARITY
\footnote{
to be published in the Preceedings of the VIIIth Rencontres de Blois:
   "Neutrinos, Dark Matter and the Universe",
    Blois, France (June 8-12, 1996)}
}
%\centerline{\footnotesize\sf (For subsequent 20\% photoreduction
%to 17.8 $\times$ 11.9 cm text area)\footnote{The \LaTeX\ source
%file for this document may be used as a template for your
%article, and can be requested by e-mailing {\sf
%worldscp@singnet.com.sg}.}}


%\vfill
%\vspace*{0.6cm}

\centerline{}
\centerline{}
\centerline{\footnotesize Ralf Hempfling}
\baselineskip=13pt
\centerline{\footnotesize\it Max-Planck-Institut f\"ur Physik,
Werner-Heisenberg-Institut,}
\baselineskip=12pt
\centerline{\footnotesize\it F\"ohringer Ring 6, 80805 Munich, Germany}
\centerline{\footnotesize\it and}
\centerline{\footnotesize\it 
Univ. of California at Davis, Dept. of Physics, Davis, CA 95616}
\centerline{\footnotesize E-mail: hempf@bethe.ucdvis.edu}
\vspace*{0.3cm}
%\centerline{\footnotesize and}
%\vspace*{0.3cm}
%\centerline{\footnotesize SECOND AUTHOR'S NAME}
%\baselineskip=13pt
%\centerline{\footnotesize\it Group, Company, Address, City, State ZIP/Zone,
%Country}

%\vfill
\vspace*{8.2cm}
\abstracts{
In supersymmetric models without $R$-parity
neutrinos naturally become massive and mix with each other.
We explore the predictions of a very restricted 
model with only three free parameters
and find that this model naturally yields masses and mixing angles
compatible with experimental results from solar and atmospheric neutrino
experiments.
Furthermore, there is a tiny region
in parameter space where the solution to the solar neutrino puzzle
is compatible with either the LSND result or
the existence of significant hot dark matter neutrinos.
}
\clearpage 
\vspace*{0.6cm}
\normalsize\baselineskip=15pt
\setcounter{footnote}{0}
\renewcommand{\thefootnote}{\alph{footnote}}
%\section{Introduction}

\begin{figure}
\vspace*{13pt}
\vspace*{6.7truein}      %ORIGINAL SIZE=1.6TRUEIN x 100% - 0.2TRUEIN
\special{psfile=figa.ps
  voffset= 240 hoffset=   -40 hscale=50 vscale=50 angle = 0}
\special{psfile=figb.ps
  voffset= 240 hoffset=   220 hscale=50 vscale=50 angle = 0}
\special{psfile=figc.ps
  voffset= -40 hoffset=   90 hscale=50 vscale=50 angle = 0}
\caption{Histogram
of the number of models that yield a particular prediction
for $m_{\nu_{\mu}}^2- m_{\nu_{e}}^2$ 
assuming (a) small angle and (b) large angle solution to solar
neutrino problem. In (c) we 
solve the solar neutrino problem via small angle $e$--$\tau$ 
oscillations and check whether this is compatible with
the LSND result.
% for $e$--$\mu$ oscillation.
}
\label{fig}
\end{figure}

In the Standard Model of elementary particles (SM)
both lepton number ($L$) and baryon number ($B$) are
conserved due to an accidental symmetry,
{\sl i.e.} there is no renormalizable, gauge-invariant
term that would break the symmetry.
In the minimal supersymmetric extension of the SM (MSSM)
the situation is different. Due to a the variety
of scalar partners the MSSM allows for a host of new 
interactions many of which violate $B$ or $L$.


Since neither $B$ nor $L$ violation has been
observed in present collider experiments
these couplings are constrained from above.
%\cite{collider-c}.
More constraints arise from neutrino
physics
% [\citenum{neutrino-c}]
or cosmology.
%\cite{cosmology-c}.
Thus, all lepton and baryon number violating
interaction are often eliminated by imposing
a discrete, multiplicative symmetry called
$R$-parity,\cite{r-parity}
$R_p \equiv (-1)^{2S+3B+L}$, where $S$ is the spin.
One very attractive feature of 
$R_p$ conserving models is that
the lightest supersymmetric particle (LSP)
is  stable and a good cold dark matter candidate.\cite{cdm}

However, while the existence of a dark matter candidate
is a very desirable prediction, it does not prove
$R_p$ conservation and 
one should consider more general models.
Here, we will investigate
the scenario where $R_p$ is broken explicitly via
the terms\cite{suzuki} $W = \mu_i L_i H$,
where $H$   is the Higgs coupling to up-type fermions
and $L_i$ ($i = 1,2,3$) are the left-handed lepton doublets.
Clearly, these Higgs-lepton mixing terms violate
lepton-number. As a result, majorana masses will be generated for
one neutrino at tree-level and for the remaining
two neutrinos at the one-loop level.
These masses were calculated in the frame-work of minimal supergravity
in ref.~\citenum{npb} and the numerical results will be
briefly summarized here.

 
There are three $R_P$ violating parameters which can be used to fix 
1) the tree-level neutrino mass,
2) the $\mu$--$\tau$ mixing angle and
3) the $e$--$\mu$ mixing angle.
The question of whether e.g. the solar\cite{solarn}
and the atmospheric\cite{atmosphericn} neutrino puzzle
can be solved simultaneously depends on the prediction of
$m_{\nu_\mu}^2-m_{\nu_e}^2$.
In fig.~1 we have scanned the entire SUSY parameter space
consisting of the Higgsino (gaugino) mass parameter,
$\mu$ ($m_{1/2}$), the trilinear scalar interaction parameter $A_0$,
and the ratio of Higgs VEVs, $\tan\beta$. The universal 
scalar mass parameter $m_0$ is fixed by minimizing the potential.
Plotted is the number of models yielding a particular prediction for
$m_{\nu_\mu}^2-m_{\nu_e}^2$ for
(a) sin$^2 2 \theta_{e \nu_\mu} = 0.008$ and
(b) sin$^2 2 \theta_{e \nu_\mu} = 1$.
We fix $m_{\nu_\tau}=0.1$~eV and 
sin$^2 2 \theta_{\mu \nu_\tau} = 1$ in order to solve the 
atmospheric neutrino problem.
We see that both
long wave-length oscillation (LWO)\cite{lwo}
($m_{\nu_\mu}^2-m_{\nu_e}^2=10^{-10}$~eV$^2$)
and MSW effect\cite{msw-effect}
($m_{\nu_\mu}^2-m_{\nu_e}^2=10^{-5}$~eV$^2$)
can be accommodated.
In fig.~1(c) we solve the solar neutrino problem via $e$--$\tau$
oscillations and we fix sin$^2 2 \theta_{e \nu_\mu} = 0.004$
in order to accommodate the LSND result.\cite{lsnd}
We see that most models are already ruled out
by collider constraints and even more by dark matter (DM) constraints.
However, a very small (but non-zero) number of models
yields a prediction compatible with the LSND
result (the dotted line is lower limit of LSND).


\noindent{\bf Acknowledgements}
This work was supported in parts by the DOE under
Grants No. DE-FG03-91-ER40674 and by the
Davis Institute for High Energy Physics.

\vskip0.3cm
\noindent{\bf References}
\begin{thebibliography}{9}

\bibitem{r-parity}
N. Sakai and T. Yanagida, \NPB{197}{533}{1982}.

\bibitem{cdm} J. Ellis \etal,
%, J.S. Hagelin, D.V. Dimopoulos, K.A. Olive and M. Srednicki, 
\NPB{238}{453}{1984}.

\bibitem{suzuki} L.J. Hall and M. Suzuki, \NPB{231}{419}{1984}.

\bibitem{npb} R. Hempfling, MPI-PhT/95-59, 
 {\sl Nucl. Phys.} {\bf B}, to appear.

\bibitem{solarn}
%R. Davis \etal, \PRL{20}{1205}{1968};
%K.S. Hirata \etal, \PRL{65}{1297}{1990};
%S.A. Bludman, N. Hata, C.D. Kennedy and P.G. Langacker,
%\PRD{47}{2220}{1993};
P. Anselmann \etal, \PLB{327}{234}{1994}.

\bibitem{atmosphericn}
%K.S. Hirata \etal, \PLB{280}{146}{1992}.
%R. Becker-Szendy \etal, \PRL{69}{1010}{1992};\PRD{46}{1992}{3720}.
Y. Fukunda \etal, \PLB{335}{237}{1994}.

\bibitem{lwo}
V. Gribov and B. Pontecorvo, \PLB{28}{493}{1969};
%S.M. Bilenky and B. Pontecorvo, \PREP{41}{225}{1978};
V. Barger, R.J.N. Phillips and K. Whisnant, \PRD{24}{538}{1981};
\PRL{69}{3135}{1992}.

\bibitem{msw-effect} L. Wolfenstein, \PRD{17}{2369}{1978};
  {\bf 20}, 2634 (1979);
  S.P. Mickheyev and A. Yu Smirnov, {\sl Yad. Fiz.}
 {\bf 42}, 1441 (1985) [{\sl Sov. J. Nucl. Phys.} {\bf 42}, 913 (1986)].

\bibitem{lsnd} 
C. Athanassopoulos \etal, \PRD{75}{2650}{1995}; LA-UR-96-1326.

\end{thebibliography}
\end{document}


\bibitem{collider-c}
H. Dreiner and G.G. Ross, \NPB{365}{597}{1991};
H. Dreiner and R.J.N. Phillips, \NPB{367}{591}{1991};
J. Butterworth and H. Dreiner, \NPB{397}{3}{1993};
C.E. Carlson, P. Roy and M. Sher, \PLB{357}{99}{1995};
G. Bhattacharyya, D. Choudhury and K. Sridhar, \PLB{355}{193}{1995}.


\bibitem{neutrino-c}
K. Enqvist, A. Masiero and A. Riotto, \NPB{373}{95}{1992}.
J.C. Romao and J.W.F. Valle, \NPB{381}{87}{1992}
I. Umemura and K. Yamamoto, \NPB{423}{405}{1994}.
C.E. Carlson, P. Roy and M. Sher, \PLB{357}{99}{1995}. 

\bibitem{cosmology-c}
B.A. Campbell, S. Davidson, 
J. Ellis and K.A. Olive, \PLB{256}{457}{1991};
H. Dreiner and G.G. Ross, \NPB{410}{88}{1993}.

