%Paper: 
%From: schang@phya.snu.ac.kr (Sanghyun Chang)
%Date: Thu, 11 Mar 93 15:26:29 KST

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\begin{titlepage}
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\begin{center}
\today
\hfill SNUTP 92-87\\
\null\hfill \\
\vskip 1.5cm
{\large \bf   Constraints from
Nucleosynthesis and SN1987A }
\\
{\large \bf
on  Majoron Emitting  Double Beta Decay}
\vskip 1.7cm
{Sanghyeon Chang}
\vskip .05cm
{ Department of Physics and Center for Theoretical Physics}
\vskip .05cm
{Seoul National University, Seoul, 151-742 Korea}
\vskip 0.5cm
{Kiwoon Choi}
\vskip .05cm
{ Department of Physics, Chonbug National University}
\vskip 0.05cm
{Chonju, 560-756 Korea}

\vskip 1.5cm

{\bf Abstract}
\end{center}
\begin{quotation}
\baselineskip .73cm
We examine whether observable majoron emission in double beta decay
can be compatible with the big-bang  nucleosynthesis (NS)
and the observed neutrino flux from SN1987A.
It is found that
 the NS upper bound on $^4$He abundance implies  that
 the majoron-neutrino Yukawa coupling constant   $g\leq 9\times 10^{-6}$
 and its maximal value is
allowed only when the scalar quartic coupling constant $\lambda$
is extremely small, $\lambda\leq 100g^2$.
 It is also observed that, although quite less restrictive,
SN1987A also provides independent  constraints  on coupling constants.

\end{quotation}
\baselineskip .75cm
\end{titlepage}
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Majoron is the massless Goldstone boson
associated with    spontaneous lepton number violation $\cite{majoron}$.
If exists, it would lead to many interesting phenomenological
consequences.
Amongst them, one particularly interesting
phenomenon is  ``majoron emitting neutrinoless double beta decay"
($\beta\beta_J$) $\cite{beta}$.
Recently it was pointed out that the potential anomaly in the
double beta decay spectra of several elements may be explained by
$\beta\beta_J$ $\cite{cline}$. The desired
value of the Yukawa coupling constant
is roughly $g_{ee}\simeq 10^{-4}$.
Regardless of  this observation,
if $g_{ee}$ is not very small, \eg \, not less than $10^{-5}$,
one may be able to observe $\beta\beta_J$ in the near future.
As was shown in ref. $\cite{valle}$,
it is not difficult to construct majoron models  which
provides such a  value of $g_{ee}$  while
satisfying all known experimental constraints.
However in view of the strong cosmological and astrophysical
implications of majoron $\cite{kim}$, it is desirable to examine
whether observable $\beta\beta_J$ can be compatible
with the big-bang cosmology and also with astrophysical observations.
In this paper, we consider possible constraints from
the big-bang nucleosynthesis  and the supernova
 SN1987A on majoron models
in which observable $\beta\beta_J$ is possible.

If the
lepton number symmetry $L$ is spontaneously broken at an energy  scale $v$,
 majoron-neutrino Yukawa coupling
matrix $g_{\alpha\beta}$ ($\alpha$,$\beta$=$e,\mu,\tau$)
 is related to the neutrino mass matrix $m_{\alpha\beta}$
as $g_{\alpha\beta}=m_{\alpha\beta}/v$.
Current data on $\nu$-less $\beta\beta$ decay implies
$m_{ee}\leq 1$ eV and $g_{ee}\leq {\rm few}\times 10^{-4}$
with  uncertainties arising from   nuclear matrix elements
$\cite{beta}$.
For $\beta\beta_J$ to be observable in the near  future,
 we may need $g_{ee}\geq 10^{-5}$.
This implies that $L$-breaking scale is
very  small compared to the Fermi scale,
\beq
v\leq 100 \, \, {\rm keV},
\eeq
leading to  a  fine tuning problem in general $\cite{foot1}$.
Here we will  not concern this theoretical difficulty.
We rather concentrate on models with such a low $L$-breaking scale
 to see  whether observable $\beta\beta_J$ can be compatible with
the standard nucleosynthesis model $\cite{ktbook}$ and the observed neutrino
flux from SN1987A $\cite{1987a}$.


In  majoron models adopting low energy $L$-violation (1) to provide
observable $\beta\beta_J$,
low energy  majoron-neutrino interactions
can be described by  the effective lagrangian:
\bea
{\cal L}_{eff}&=&\bar{\nu}_{\alpha}i\gamma\!\cdot\!\partial\nu_{\alpha}
+|\partial_{\mu}\chi|^2-
\frac{1}{\sqrt{2}}(g_{\alpha\beta}\chi^{\ast}\bar{\nu}^c_{\alpha}\nu_{\beta}
+{\rm h.c.}) \nonumber \\
&&-\lambda (\chi\chi^{\ast}-\frac{v^2}{2})^2+...
\eea
where $\chi$ is a (mostly) gauge singlet Higgs field carrying
the lepton number two,
 and the ellipsis denotes generic nonrenormalizable interactions
suppressed by the powers of  cutoff  scale $\Lambda$ which is
usually taken to be the Fermi scale.
 The majoron field $J$  appears in $\chi$ as
\beq
\chi =\frac{1}{\sqrt{2}}(v+\rho+iJ),
\eeq
and the mass eigenstates neutrino $\nu_i=\sum_{\alpha}
U_{i \alpha}\nu_{\alpha}$
 ($i=1,2,3$) has the Yukawa interaction with $J$ and $\rho$
 whose coupling constant is given by $g_i=|\sum_{\alpha\beta}
 U^{\ast}_{i\alpha}U^{\ast}_
 {i\beta}g_{\alpha\beta}|$ for a unitary mixing matrix $U_{i\alpha}$.

Let us now consider possible constraints on
majoron-neutrino interactions  from the big-bang nucleosynthesis
(NS) $\cite{bertolini}$.   As is well known, in the standard model of NS,
 the energy density of exotic  particles
at the NS epoch ($T_{NS}\simeq 1$ MeV) is severely constrained
$\cite{ns}$.
In the case that $L$ is restored
and also there is no sizable lepton number excess,
  the observed $^4$He abundance implies
\beq
Y(T_{NS})\equiv \rho_{\chi}(T_{NS})/\rho_{\nu}(T_{NS})\leq 0.3,
\eeq
where $\rho_{\chi}$ and $\rho_{\nu}$ denote the energy
density of $\chi$ and a single species of left-handed
neutrino respectively $\cite{enqvist}$.
 Since $v< T_{NS}$, before  the NS
 the ratio of the Hubble expansion rate
$H$ to the $\nu$-$\chi$  interaction rate  behaves as
$\Gamma_{\rm int}/H\sim T^{-1}$.
Then $\chi$'s would be
rare at high temperature, but eventually enter into a
thermal equilibrium with neutrinos at some temperature
$T_{\rm eq}$ at which $\Gamma_{\rm int}\simeq H$.
For the NS constraint (4) to be satisfied,
we need  $T_{\rm eq}< T_{NS}$.  Here we will
directly  evaluate $\rho_{\chi}$  and  apply the constraint (4),
rather than using more naive condition $T_{\rm eq}< T_{NS}$.





To evaluate $\rho_{\chi}$, we first need to know
whether $L$ is restored around the NS epoch.
The effective mass of $\chi$ in the early universe
is  given by $\cite{finitetem}$
 \bea
 m^2_{\rm eff}&=
 &-\lambda v^2+\int \frac{d^3 k}{(2\pi)^3 E}(4\lambda f_{\chi}
 +2\, {\rm tr}(gg^{\dagger}) f_{\nu})   \nonumber \\
 &=&-\lambda v^2+
  4\lambda n_{\chi}\vev{1/E_{\chi}}+ {\rm tr}(gg^{\dagger}) T^2/12,
 \eea
where $f_X$ ($X=\chi,\nu$) denotes the phase space distribution
function of $X$ whose  number density is
defined as $n_X=\int d^3 k \, f_X/(2\pi)^3$, and
 neutrinos are assumed to be in thermal equilibrium.
To proceed, let us set
\beq
n_{\chi}\vev{1/E_{\chi}}=\xi Y^{\omega} n_{\nu}\vev{1/E_{\nu}}=\xi Y^{\omega}
T^2/24,
\eeq
where $Y=\rho_{\chi}/\rho_{\nu}$.
Clearly  the average energy of $\chi$
do {\it not} exceed that of $\nu$ and thus $n_{\chi}/n_{\nu}\geq
Y$,  implying  that $\xi Y^{\omega}\geq Y$.
Then for $v\leq 100$ keV and
 $Y$ saturating the NS bound (4), which is the most interesting
case for us,  $L$ is  restored
  around the NS epoch
 regardless of the value of ${\rm tr}(gg^{\dagger})$ $\cite{foot2}$.


 It may be necessary to further discuss on the parameters
 $\xi$ and $\omega$.  Self interactions among $\chi$'s do not
 change $\rho_{\chi}$, but can increase $n_{\chi}$ through
 the processes like $\chi\chi\ra\chi\chi\chi\chi^{\ast}$.
 The values of $\xi$ and $\omega$  depend on the strength
 of such ``number changing self interaction process" (NCP).
 In our case, NCP's can occur through
  the quartic coupling $\lambda\chi^2\chi^{\ast 2}$
  (loop effects) or through
   nonrenormalizable interactions
 like $\kappa\chi^3\chi^{\ast 3}/\Lambda^2$.
 If the NCP rate is weaker than the expansion rate,
  the average energy of $\chi$
 is roughly equal to that of $\nu$, and then $\xi\simeq\omega\simeq 1$.
In the opposite case,  $\chi$'s would achieve
 a thermal distribution, giving  $\xi\simeq 2$ and $\omega\simeq 1/2$.
 In the subsequent analysis,
  we will simply set $\xi\simeq \omega\simeq 1$ since   as we have
  argued  $Y\leq \xi Y^{\omega}$ and then this choice
   gives  more conservative result  for the $\chi$-production.

Clearly    ${\chi}$'s are mainly produced by the heaviest mass
eigenstate neutrino $\nu$  which has
 the largest Yukawa coupling  $g\equiv{\rm max} \, (g_i)$.
 For interaction terms  in  (2), the processes
which can dominantly produce $\chi$ are as follows:
(A) $\nu\nu\ra \chi$; (B) $\nu\bar{\nu}\ra\chi\chi^{\ast}$;
 (C)
$\nu\nu\ra\chi\chi\chi^{\ast}$ and $\bar{\nu}\bar{\nu}\ra
\chi\chi^{\ast}\chi^{\ast}$.
Before $\chi$'s enter into an equilibrium with
$\nu$'s, particularly when  the NS constraint (4) is satisfied,
 we can safely ignore
the inverse processes annihilating $\chi$'s.
Then the Boltzmann equation describing the evolution
of $\rho_{\chi}$ before the onset of the NS is given by $\cite{ktbook}$
\beq
\frac{d}{dt}\rho_{\chi}+4 H \rho_{\chi}= \sum_I
\int [d X] (2\pi)^4\delta^4(p_i-p_f) E_I |M_I|^2
 f,
\eeq
where $[dX]=\prod_a d^3 k_a/(2\pi^3) 2E_a$  ($a$ runs over particle species
participating in the process),
$E_I$ ($I=A, B, C$) is the total energy of $\chi$'s  in the final state
of the process $I$,
 $|M_I|^2$  is the amplitude squared including
 the symmetry factor for  identical particles,
and finally $f=\prod_i f_i$ for  the phase space distribution functions
$f_i$ of  particles in the initial state.
After a straightforward computation,
this Boltzmann equation  can be cast
into the following form:
\beq
-H\frac{d}{dT}Y=6.7\times 10^{-4}g^2\lambda[
(Z)_A
+(1.6 \frac{g^2}{\lambda}\ln(5/\lambda Z))_B+(10^{-2}\lambda)_C],
\eeq
where   $Z=Y+(g^2/2\lambda)-6(v/T)^2$.
Here the subscripts in the brackets
denote the contributing process. Note that
 the rate of (A) is proportional
to $m^2_{\rm eff}\propto Z$, while that of (B) includes the factor
$\ln ( m_{\rm eff}^2)$.  Of course the above equation
is valid only when $L$ is restored, viz $Z>0$.
At any rate, it indicates that  if $\lambda\geq g^2\ln g^{-2}$,
$\chi$'s are mainly produced
around the NS epoch by the inverse decay process (A) whose rate
 is dominated by the background matter induced piece which  is proportional
 to $Y$.
 However for $\lambda\leq g^2\ln g^{-2}$,
the process  (B) dominates over other processes.
If $\lambda \gg g$, the process (C) can be important at the very early stage
of $Y\ll 1$.


The Boltzmann equation (8)
 can be integrated to obtain
$Y(T_{NS})$. Applying the NS constraint (4), we then find
 \beq
 \lambda g^2\leq 7.2\times 10^{-19}\ln (1+\epsilon \lambda/g^2),
 \eeq
 where  $\epsilon=30/(\lambda^2/g^2
 -160\ln (g^2/5+0.1\lambda))$.
This gives
\beq
g\leq 9\times 10^{-6} R
\eeq
where $R\simeq 1$ for
 $\lambda\leq 100 g^2$, while $R\simeq  (r^{-1}\ln r)^{1/4}$
 for $r=\lambda/100 g^2\gg 1$.
Note that in the case of $\lambda> 100 g^2$,
 the $\chi$-production
is dominated by the process (A) whose rate is enhanced
by the  factor $\lambda/(g^2 \ln g^{-2})$ compared to the rate of
the process (B). This is the reason why we get
a stronger bound on $g$ in this case.



The implication of our NS result  for $\beta\beta_J$ is clear.
Since  $g$ is the majoron Yukawa coupling constant of the
heaviest mass eigenstate neutrino,  $ g_{ee} \leq g$
and thus any upper limit on $g$ applies also to $g_{ee}$.
Then taking into account possible uncertainties
of our analysis, we can conclude that
 $g_{ee}\simeq 10^{-5}$ for which
 observable $\beta\beta_J$
 is barely possible
  can be consistent with the standard NS  model,
   but only under very unlikely conditions
that the quartic coupling constant $\lambda$
is extremely small ($\leq 100 g^2$) and also
the majoron Yukawa couplings with $\nu_{\mu}$ and $\nu_{\tau}$
do {\it not} exceed that with $\nu_{e}$.
This conclusion is valid  as long as $v$ is  small enough,
\eg \,  less than few hundreds keV,
for  $L$  to be  restored around the NS epoch.


Although widely accepted and very natural,  the standard NS model
is {\it not} a unique  model
explaining observed cosmological data.
There may be other successful  models
in which  the constraint  (4) is not valid any more $\cite{ktbook}$,
implying that the NS limit  on $g$ does not have a strictly
firm foundation.
In this regard, it is still worthwhile  to
consider other implications of observable $\beta\beta_J$ with $g_{ee}
\geq 10^{-5}$, \eg \, for supernova dynamics.
It has already been studied how
majoron-like particles  can
 affect the explosion
 and the subsequent cooling of supernovae $\cite{choi,fuller}$,
but  the results of these studies do not show
 any meaningful implication  for $\beta\beta_J$.
Just after the observation
of SN1987A,
 Kolb and Turner $\cite{kolb}$
derived  a limit on the interactions between
supernova neutrinos and  ``cosmic background majorons" (CBM).
   Since the observed neutrino pulse from SN1987A
 is that of $\nu_1$,  the mass eigenstate neutrino  which is mostly $\nu_e$,
 the relevant Yukawa coupling constant here is $g_1$.
 Although in principle
   the parameter $g_{ee}$ describing $\beta\beta_J$ can be
 significantly different from $g_1$
 (note that $g_{ee}=\sum_{i} U^2_{i e} g_i$),
 it is somewhat natural to assume that neutrino mixing is small
enough for  $g_1$ to be close to $g_{ee}$. In this regard, the
  consideration  of CBM's may provide a useful information
 on  $\beta\beta_J$.

  If $\nu_1$'s  from the  supernova
are scattered off by
 CBM's,  there would  be
 a substantial decrease in the average neutrino energy,
 leading to an effective loss of  detectable flux.
  The  observed data
$\cite{1987a}$  indicates that the mean free path $l$ of
$\nu_1$ through the
CBM's should be  comparable to or greater than the distance
to the supernova, viz
\beq
l\geq 1.7\times 10^{23} \, {\rm cm}.
\eeq
 Using this,  it was found  in ref. $\cite{kolb}$ that
$g_1\leq 10^{-3}$.
However as we will see, one can in fact obtain a
significantly stronger  limit
 unless  $\lambda$ is  significantly less than $10^{-3}$.

Clearly for $g_{ee}$ in the range of
 observable $\beta\beta_J$, \ie \, $g_{ee}\geq 10^{-5}$,
  there was a period in the early
universe when $\rho$ and $J$ were at thermal equilibrium
with neutrinos. Later that, the relic majoron number density
can be increased by the decays of $\rho$ and $\nu$ and also
by the $\nu$ annihilations.
With these observations, we will set the majoron temperature
at present as $T_J\simeq 1.9$ K, which is a somewhat
conservative choice.
Then  the inverse mean free path of $\nu_1$  propagating through
CBM's is given by
\beq
l^{-1}\simeq \frac{\sqrt{2}}{4\pi^2}T_J^3\int_0^{\infty}
dw \, w^2 \frac{1}{e^w-1}\int_{-1}^{1} dz \, (1-z)^{1/2}
\sigma(s),
\eeq
where $\sigma (s)$ is the total
cross section for the reaction $\nu_1 J\ra \nu_1 J$
with the total energy-momentum squared  $s= 2ET_J w(1-z)+m_1^2$.
Here  $E$ ($\simeq 10$ MeV) and $m_1$
denote  the energy and the mass of the incident $\nu_1$.
There are three diagrams responsible for the reaction
$\nu_1J\ra \nu_1J$. Two of them are the Compton-type ones,
while the rest one involves the $\rho$-exchange.
The resulting cross section can be written as
\beq
\sigma (s)=\frac{g_1^4}{16\pi s}[ \, F(s)+y^{-1}G(s) \,]
\eeq
where
\bea
F(s)&\simeq& \frac{5}{2}+\frac{2}{x}-
\frac{2(1+2x)^2\ln (1+x)}{x^2(1+x)}+\frac{4+(1-x)\ln y}{1+x}
 \nonumber \\
G(s)&\simeq& \frac{\ln (1+x)}{x^2}-
\frac{1}{x(1+x)},
\nonumber
\eea
for $y=m_1^2/s$ and $x=s/m_{\rho}^2=s/2\lambda v^2$.
Here $F(s)$ represents the contribution
from the Compton-type  diagrams, $y^{-1}G(s)$
is from the $\rho$-exchange diagram.

 If $\lambda$ is so small,  \eg  \, $\lambda\ll
g_1\sqrt{s}/v$,
that the $\rho$-exchange diagram  can be ignored,
we have  $\sigma\simeq g_1^4 \ln (s/m_1^2)/16\pi s$. Then
applying the supernova constraint (11) yields  $g_1\leq 10^{-3}$
as was obtained in ref $\cite{kolb}$.
However in the other case of $\lambda\gg g_1\sqrt{s}/v$,
 the cross section is largely enhanced
by the $\rho$-exchange diagram, allowing
 us to obtain a more stringent limit on $g_1$.
 Our results summarized in fig. 1 shows that,
unless $\lambda\leq 10^{-3}$, some
 portion of the parameter region giving $g_1\leq 3\times  10^{-4}$
 is ruled out by the supernova constraint (11).
As was mentioned, if neutrino mixing is small, so that
 $g_{ee}\simeq g_1$,
 this supernova result can be
 directly applied to $\beta\beta_J$.

To conclude, we find that the standard nucleosynthesis model
strongly constrains the coupling constants in majoron models
in which  observable ``majoron emitting $\nu$-less double beta
decay" ($\beta\beta_J$) is possible
through a spontaneous lepton number violation
below few hundreds keV.
 Our results directly applies to the Yukawa coupling constant
$g_{ee}$ governing $\beta\beta_J$,
yielding  $g_{ee}\leq 9\times 10^{-6}$.
Here the maximal  value $9\times 10^{-6}$
is allowed  only when (i)
the scalar quartic coupling
is extremely weak, roughly  $\lambda\leq 10^{-8}$,
and (ii)  $g_{ee}$ is rather close to
the majoron Yukawa coupling constant of the heaviest
neutrino species.
We also find that the consideration
of the scatterings between  cosmic background majorons
and  supernova neutrinos  provides a
constraint on the majoron Yukawa coupling constant $g_1$
of the mass eigenstate neutrino which is mostly $\nu_e$.
Compared to the nucleosynthesis constraint,
this supernova constraint is much less restrictive,
but is independent of the validity of the standard nucleosynthesis model.
If $g_1\simeq g_{ee}$ and
$\lambda$ is {\it not} less than $10^{-3}$,
which is somewhat plausible, some part of the parameter region
giving observable $\beta\beta_J$ is excluded by the supernova data.


\vskip 1.5cm
\noindent
This work is supported in part by  KOSEF through
CTP at Seoul National University.



\vfill\eject

\begin{thebibliography}{99}

\bibitem{majoron}
Y. Chikashige, R. N. Mohapatra, and R. D. Peccei,
Phys. Lett. {\bf 98B}, 265 (1981); G. B. Gelimini and M. Roncadelli,
Phys. Lett. {\bf 99B}, 411 (1981);
H. Georgi, S. L. Glashow
and S. Nussinov, Nucl. Phys.
{\bf 193B}, 297 (1981).



\bibitem{beta}
H. V. Klapdor, J. Phys. G Suppl. {\bf 17}, 129 (1991);
M. Doi et al., Phys. Rev. {\bf D35}, 2575 (1988).


\bibitem{cline}
C. P. Burgess and J. M. Cline,  preprint
 McGill/92-22 (1992).


\bibitem{valle}
Z. G. Berezhiani, A. Yu. Smirnov and J. W. F. Valle,
preprint FTUV/92-20 (1992).

\bibitem{kim}
See for instance J. E. Kim, Phys. Rep. {\bf 150}, 1 (1987);
G. G. Raffelt, Phys. Rep. {\bf 198}, 1 (1990).


\bibitem{foot1}
In ref. $\cite{cline}$, a majoron model in which $L$
 is not spontaneously broken  was considered to  raise up
 $v$  up to few hundreds  MeV.


\bibitem{ktbook}
E. W. Kolb and M. S. Turner, {\it The Early Universe}
(Addison-Wesley, 1990).


\bibitem{1987a}
K. Hirata et al., Phys. Rev. Lett. {\bf 58}, 1490 (1987);
R. M. Bionta et al., Phys. Rev. Lett. {\bf 58}, 1494 (1987).


\bibitem{bertolini}
NS constraint on majoron models with
the 17 keV neutrino was fully discussed
by S. Bertolini and G. Steigmann, Nucl. Phys. {\bf B}387,
193 (1992).  For $m_{\nu}=17$ keV, they obtained   a lower limit on $v$
which is above 1 GeV.
Note that here our major concern is in $\beta\beta_J$,
and thus  we are interested in $v\leq 100$ keV for which
$L$ is restored around the NS epoch.


\bibitem{ns}
K. A. Olive {\it et al}., Phys. Lett. {\bf B}236, 454 (1990);
T. P. Walker {\it et al}., Ap. J. {\bf 376}, 51 (1991).


\bibitem{enqvist}
Note that once $L$ is restored,
the scheme considered by
K. Enqvist {\it et al}., Phys. Rev. Lett. {\bf 68}, 744 (1992)
to  relax the NS constraint (4) through
 $\nu_{\mu},\nu_{\tau}\ra\nu_e J$
can not be applied.


\bibitem{finitetem}
D. A. Kirzhnitz and A. D. Linde, Phys. Lett. {\bf 42B}, 471 (1972);
L. Dolan and R. Jackiw, Phys. Rev. {\bf D9}, 3320 (1974);
S. Weinberg, {\it ibid}, 3357 (1974).

\bibitem{foot2}
In fact $L$ is restored for all $T\geq T_{NS}$
for $Y$ satisfying  the Boltzmann
equation (8) with the b.c. $Y(T_{NS})=0.3$.



\bibitem{choi}
K. Choi and A. Santamaria, Phys. Rev. {\bf D42}, 293 (1990).

\bibitem{fuller}
E. W. Kolb, D. L. Tubbs, and D. A. Dicus,
Ap. J.  {\bf 225}, L57 (1982);
D. A. Dicus, E. W. Kolb and D. L. Tubbs, Nucl. Phys. {\bf B223}, 532 (1983);
G. M. Fuller, R. Mayle, and J. R. Wilson, Ap. J., {\bf 332}, 826 (1988).

\bibitem{kolb}
E. W. Kolb and M. S. Turner, Phys. Rev. {\bf D36}, 2895 (1987).



\end{thebibliography}



\newpage
\input prepictex
\input pictex
\input postpictex
\begin{figure}\hskip5mm
\beginpicture
\setcoordinatesystem units <35mm,15mm>
\setplotarea x from -3 to 0, y from -3 to 2
\axis bottom label {$m_1$/eV}
      ticks in withvalues $10^{-3}$ $10^{-2}$ $10^{-1}$ $10^{0}$ / quantity 4 /
\axis left label {$\displaystyle v\over{\rm keV}$}
      ticks in withvalues $10^{-3}$ $10^{-2}$ $10^{-1}$ $10^0$  $10^1$  $10^2$
/ quantity 6 /
\axis right ticks in quantity 6 /
\axis top ticks in quantity 4 /
\plot -3 -1.05 -2 -0.55 -1 -0.05 0 0.45 /
\plot -3    -1.5 -2.7  -1.19 -2.5  -1.0 -2.4  -0.92 -2.25 -0.8
-2    -0.63 -1.75 -0.48 -1.5  -0.33 -1.25 -0.19 -1    -0.06 -0.5   0.2 0
0.45 /
\plot -3     -2.6 -2.7   -2.05 -2.5   -1.7 -2.25  -1.32 -2     -1 -1.75  -0.73
-1.5   -0.5 -1.25  -0.3 -1     -0.13 -0.75   0.031 -0.5    0.17 -0.25   0.3 0
    0.44 /
\plot -3     -3.00 -2.5   -2.5 -2.25  -2.225 -2.125 -2.08 -2     -1.925 -1.9
-1.796
-1.75  -1.58 -1.5   -1.2 -1.35  -0.972 -1.25  -0.82 -1.125 -0.665 -1     -0.5
-0.75  -0.23
-0.6   -0.095 -0.5   -0.01 -0.35   0.115 -0.25   0.19 0       0.365 /
\setplotsymbol ({.}) \plotsymbolspacing=5pt
\setlinear \plot -3 -3 0 0 / \plot -3 -1 0 2 /
\endpicture
\vskip10mm
\caption{ Supernova constraint depicted on the plane of the $L$
breaking scale $v$ and the electron neutrino mass $m_1$.
The solid curves from bottom to top are obtained by applying
the condition (11) for $\lambda=10^{-3}, 10^{-2}, 10^{-1}$, and 1 respectively,
and the regions below them are excluded.
The two dotted lines  correspond to $g_1=10^{-3}$ and $10^{-5}$.  }
\end{figure}



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