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\begin{document}
\draft
\preprint{\vbox{
\hbox{IFT-P.060/93}
\hbox{IFUSP/P-1074}
\hbox{hep-ph/9310yyy}
\hbox{September 1993}
}}
\title{
Some consequences in weak processes of three generations \\mixing in
the leptonic sector
}
\author{O.L.G. Peres$^{a}$, V. Pleitez$^{a}$
\\and \\R. Zukanovich Funchal$^{b}$ }
\address{
$^{a}$Instituto de F\'\i sica Te\'orica \\
Universidade Estadual Paulista \\
Rua Pamplona, 145 \\
01405-900--S\~ao Paulo, SP \\
Brazil \\
$^{b}$ Instituto de F\'\i sica da Universidade de S\~ao Paulo\\
01498-970 C.P. 20516--S\~ao Paulo, SP\\
Brazil\\
}
\maketitle
\begin{abstract}
We investigate the sensitivity of some weak processes to a
Cabibbo--Kobayashi--Maskawa mixing in the leptonic sector. Values for
mixing angles and masses compatible with several experimental accelerator data
were found. We discuss in this context neutrino oscillations, cosmological
and astrophysical consequences as well.
\end{abstract}
\pacs{PACS numbers:14.60-z, 12.15.Ff, 13.35+s}
\newpage

\narrowtext

\section{Introduction} %sec.1
\label{sec:int}
The physics of the $\tau$ lepton will provide in the near future
evidence concerning the question if this lepton, with its
neutrino partner, is a sequential lepton or not.
Although the existing data indicate a positive answer to that
question, the structure of the weak  interaction of the third lepton
family deserves more  detailed experimental and theoretical studies.
As was stressed in Ref.~\cite{eppz}, all experiments are
internally consistent with the standard model. Notwithstanding, it is
well known that the accuracy of the $\tau$ data is still poor and it
should be possible that new physics will come up when the proposed
$\tau$-Charm factories give new and more accurate data about $\tau$
decays and properties~\cite{p}.

One important issue concerns lepton universality. The $e-\mu$
universality is well established in $\pi$-decays given a ratio~\cite{b2}
\begin{equation}
G_e/G_\mu=0.9985\pm0.0015.
\label{emuniv}
\end{equation}
On the other hand, the current data on $\tau-\mu$ universality have not
the same precision and one still can speculate on a possible departure from
$\tau-\mu$ universality.

A quantity which parametrizes the $\tau-\mu$
universality is
\begin{equation}
\left(\frac{G_\tau}{G_\mu}\right)^2=\left(\frac{\tau_\mu}
{\tau_\tau}\right)
\left(\frac{m_\mu}{m_\tau}\right)^5\frac{B^{\tau e}}{B^{\mu e}},
\label{2}
\end{equation}
where $G_\tau$ and $G_\mu$ are, respectively, the coupling constants
of the $\tau$
and $\mu$ to the charged weak current respectively. We will use the
notation $B^{ab}=B.R.(a\to b+X)$ where $X$ are appropriate particles.
A similar notation is used for the partial width $\Gamma^{ab}$.

The new values for
the $\tau$ mass $m_\tau=1776.9^{+0.4}_{-0.5}\pm 0.2$ MeV~\cite{bes} or
$m_\tau=1771.3\pm2.4\pm1.4$ MeV~\cite{argus} imply that a possible
deviation from
$\tau-\mu$  universality is reduced from 2.5$\sigma$ to 1.7$\sigma$.
Explicitly with the BES data one has
\begin{equation}
\left( \frac{G_\tau}{G_\mu}\right)^2=0.960\pm0.024.
\label{tmu}
\end{equation}
\newpage
Two other experimental quantities which we will use later on
are~\cite{pdg}
\begin{equation}
\frac{B^{\tau\mu}}{B^{\tau e}}=0.98\pm0.02.
\label{tmu2}
\end{equation}
and
\begin{equation}
\frac{B^{\pi e}}{B^{\pi\mu}}=(1.218\pm0.014) \times
10^{-4},
\label{pi}
\end{equation}
A more recent data for the ratio is given in (\ref{pi})
is~\cite{dib1}
\begin{equation}
1.2265\pm0.0034\,(\mbox{stat})\pm0.0044\,(\mbox{sys})\times10^{-4}
\label{recent}
\end{equation}
 however we will use, for consistence, the world average of
Ref.~\cite{pdg} for all quantities.


In this work, we will consider that if neutrinos are massive in the
standard electroweak model~\cite{ws},
a mixing similar to the Cabibbo-Kobayashi-Maskawa (CKM) one
in the quark sector~\cite{km}, may occur with three lepton
generations.
We show in this paper that such a mixing it is not ruled out by current
data and can be used to explain a possible deviation from $\tau-\mu$
universality.

We must nevertheless stress that the analysis we will put forward
in this work is valid even if $\tau-\mu$ universality is confirmed.
Only the numerical values that we give will change.

The introduction of a mixing among three generations makes room for a
third massive neutrino with a mass, as we will see, beyond the current
limit of what
is consider, without mixing, the ``tau neutrino''~\cite{argus,argus2}.

The existence of neutral heavy leptons has been discussed in the
literature since about twenty years ago~\cite{teo}. Usually, the
heavy neutrino is assumed to mix with the electron and muon families
with either $V-A$ or $V+A$ charged and neutral currents. The heavy
neutrino could be a higher generation neutrino, associated with a yet
unobserved heavy charged lepton, a right-handed neutrino,
etc.~\cite{mg1}. Another possibility usually considered, is that in
addition to the three left-handed lepton doublets one includes $k$
neutrinos singlets $\chi_j$, $j=1,...,k$ \cite{mg2,lsl}. This led to
neutrino mixing without affecting the $Z^0$ width.

Here we will suppose that only 3 right-handed singlets have been
added and that there is no Majorana mass terms among the
right-handed singlets. This is the simplest extension of the standard
model which includes massive Dirac neutrinos.
 The only supposition we will make is that
\begin{equation}
0\lower.7ex\hbox{$\;\stackrel{\textstyle<}{\sim}\;$}m_{\nu_1}\approx
m_{\nu_2}\ll m_{\nu_3}.
\label{aprox}
\end{equation}
This will be assumed not only for the sake of simplicity but because
it is an interesting possibility. The two light neutrinos could be
detected in oscillation experiments~\cite{cr}. We will not make any
further assumptions concerning the mixing matrix elements nor the
$m_{\nu_3}$ value but leave them to be constrained by several
experimental data. This means that we will fit in our analysis
, at the same time, two
matrix elements and a mass.

We stress that it is important to specify the model one is
considering when comparing with experimental data. A purely
model independent approach cannot be definitive.

The outline of this work is as follows. In Sec.~\ref{sec:lepmix} we
consider the effect of mixing in the partial
width for the muon decay, leptonic tau decays and pion decays. These
are the quantities which are calculated theoretically. In
Sec.~\ref{sec:ma} we compare our
theoretical results with experimental data (within $1\sigma$)
for $\tau$ and pion decays
deriving from this comparison regions of permitted mixing angles and
mass.  In Sec.~\ref{subsec:z0} we take into account the
constraints coming from the $Z^0$ invisible partial width.
In Sec. \ref{subsec:bound} we
discuss the current limit of ``$m_{\nu_\tau}$'' on the light of
the mixing angles obtained in Sec~\ref{sec:ma}, showing that it is
possible to go beyond the current limit of 31 MeV without
contradicting experimental results. Neutrino oscillations are consider in
Sec.\ref{subsec:osc},
cosmological and astrophysical
constraints are discussed in Sec. \ref{subsec:cosmo} and Sec.
\ref{subsec:astro}
respectively. Finally, the last section is devoted to our conclusions.

\section{Three generations mixing in the leptonic sector}%sec.2
\label{sec:lepmix}
In this paper we will consider only Dirac neutrinos.
The general effects of the Cabibbo-Kobayashi-Maskawa mixing in the
leptonic sector were considered in Ref.~\cite{kg,ro,ng}. In particular
the  effects of such a mixing for the case of leptonic decays of the
$\mu$ were explicitly considered in Ref.~\cite{pk1}, and in
Ref.~\cite{ssi} for the $\tau$ lepton case.

In fact, it has been shown by Shrock~\cite{ro} that if neutrinos
have nonzero masses the semi-leptonic decays $h'\to h+l+\nu_l$
consist of an incoherent sum of the separate modes $h'\to h+l+\nu_i$,
where $\nu_l=\nu_e,\nu_\mu,\nu_\tau$
are weak eigenstates and $\nu_i,\,i=1,2,3$ are the mass eigenstates
allowed by phase space. These states are related by
$\nu_l=\sum_iV_{li}\nu_i$, $V$ being the unitary leptonic
CKM matrix. Explicitly, this mixing
matrix is defined for the three generations case as
\begin{equation}
\left(\begin{array}{c}
\nu_e \\ \nu_\mu \\ \nu_\tau
\end{array}\right)=\left(
\begin{array}{ccc}
V_{e1}    & V_{e2}    & V_{e3} \\
V_{\mu1} & V_{\mu2}  & V_{\mu3} \\
V_{\tau1}& V_{\tau2} &V_{\tau3}
\end{array}\right)
\left(
\begin{array}{c}\nu_1\\ \nu_2 \\ \nu_3\end{array}\right).
\label{4}
\end{equation}

It is well known that there are several parametrizations of the
CKM matrix. Here we will use the Maiani one
{}~\cite{pdg,maiani}. This form has the feature that
when two of such angles vanish the mixing matrix reduces
to the usual Cabibbo-like mixing matrix of two generations. Using the
usual notation $c_{ij}=\cos\theta_{ij},s_{ij}=\sin\theta_{ij}$, with
$i$ and $j$ being generation labels $i,j=1,2,3$, setting the phase
$\delta_{13}=0$ for the sake of simplicity and making the
substitution $c_{12}\to c_\theta,c_{13}\to c_\beta$ and $c_{23}\to
c_\gamma$ we have
\begin{equation}
\left(
\begin{array}{ccc}
c_\theta c_\beta \; \; \; & s_\theta c_\beta  \; \; & s_\beta \\
-s_\theta c_\gamma-c_\theta s_\gamma s_\beta \; \; \; & \; \; \;
c_\theta c_\gamma
-s_\theta s_\gamma s_\beta\; \; \;  & \; \; \;
s_\gamma c_\beta  \\
s_\theta s_\gamma -c_\theta c_\gamma s_\beta \; \; \; & \; \; \;
-c_\theta s_\gamma
-s_\theta c_\gamma s_\beta\; \; \;  & \; \; \;
c_\gamma c_\beta
\end{array}
\right).
\label{ckm}
\end{equation}

Assuming that only one of the neutrinos,
say $\nu_3$, is sufficiently massive we can write for the decay
probability of a charged lepton $l'$ into the charged lepton $l$ and
neutrinos $\nu_i\bar\nu_j$, the following expression~\cite{ssi},
\begin{eqnarray}
\Gamma(l'\to l\bar\nu_l\nu_{l'})&=&\frac{G^2m^5_{l'}}{192\pi^3}\left\{
\left(\vert V_{l'1}\vert^2+\vert V_{l'2}\vert^2\right)
\left(\vert V_{l1}\vert^2+\vert
V_{l2}\vert^2\right)\,\Gamma^{l'l}_{00}\right.\nonumber \\
&+&
\left.\left(\vert V_{l3}\vert^2[\vert V_{l'1}\vert^2+\vert V_{l'2}\vert^2]
+\vert V_{l'3}\vert^2[\vert V_{l1}\vert^2+\vert V_{l2}\vert^2]
\right)\,\Gamma^{l'l}_{03}\right.\nonumber \\
&+&\left.\vert V_{l'3}\vert^2\vert
V_{l3}\vert^2\,\Gamma^{l'l}_{33}\right\},
\label{decay}
\end{eqnarray}
with $l'=\mu,\tau$ and $l=e,\mu$ for the $\tau$ decay
and $l=e$ for the muon decay. Notice that $G^2$, although it is still
defined as $G^2/\sqrt2=g^2/8m^2_W$ in Eq. (\ref{decay}) is not equal to
the muon decay constant, $G_\mu$. We shall return to this point later.
In (\ref{decay}) we have defined the integrals
\begin{equation}
\Gamma^{l'l}_{00}=2\int^{x_M}_{x_m}(x^2-B)^{\frac{1}{2}}[x(3k-2x)-B]dx
\label{r0}
\end{equation}
\begin{eqnarray}
\Gamma^{l'l}_{03} & = &
2\int^{x_M}_{x_m}(x^2-B)^{\frac{1}{2}}\frac{(k-\delta_{3 l'}^2-x)}{(k-x)^3}
\left[(k-\delta_{3 l'}^2-x)^2x(k-x) \right. \nonumber \\
 & + &\left. [(k-x)^2+\delta_{3 l'}^2(k-x)-2\delta_{3 l'}^4] (2kx-x^2-B)\right]
   dx
\label{r3}
\end{eqnarray}
\begin{eqnarray}
\Gamma^{l'l}_{33} & = &
2\int^{x_M}_{x_m}\frac{(x^2-B)^{\frac{1}{2}}}{(k-x)^{\frac{3}{2}}}
(k-x-4\delta_{3 l'}^2)^{\frac{1}{2}} \left[(k-x-4\delta_{3 l'}^2)x(k-x) \right.
\nonumber \\
 & + & \left. (k-x+2\delta_{3 l'}^2)(2kx-x^2-B) \right]dx \label{r33}
\end{eqnarray}
with

\begin{equation}
k=1+\delta^2_{ll'},\quad B=4(k-1),\quad
\delta_{3l'}=\frac{m_{\nu_3}}{m_{l'}},\quad
\delta_{ll'}=\frac{m_l}{m_{l'}},
\label{def1}
\end{equation}

\begin{equation}
x_m=2\delta_{ll'},\quad
x_M=k-\frac{(m_{\nu_i}+m_{\nu_j})^2}{m^2_{l'}}.
\label{def2}
\end{equation}
$\Gamma^{l'l}_{00}$, $\Gamma^{l'l}_{03}$ and $\Gamma^{l'l}_{33}$
are the contributions for the $l'\to l\bar\nu_i\nu_j$ decays
from two massless, one massive and two massive neutrinos,
respectively.

Using explicitly the parameterization in (\ref{ckm}) we obtain
\begin{eqnarray}
\Gamma(\mu\to e \nu_\mu\bar\nu_e)&=&\frac{G^2m^5_{\mu}}{192\pi^3}\left[
(s_\beta^2s_\gamma^2+c_\gamma^2)c_\beta^2\,\Gamma^{\mu e}_{00}
+(s_\beta^4s_\gamma^2+
s_\beta^2c_\gamma^2+c_\beta^4s_\gamma^2)\Gamma^{\mu e}_{03}\right.\nonumber \\
&+&
\left.s_\gamma^2c_\beta^2s_\beta^2\,\Gamma^{\mu e}_{33}
\right]
\label{mue}
\end{eqnarray}

\begin{eqnarray}
\Gamma(\tau\to e
\nu_\tau\bar\nu_e)&=&\frac{G^2m^5_{\tau}}{192\pi^3}\left[(
s_\beta^2c_\gamma^2+s_\gamma^2)c_\beta^2\,\Gamma^{\tau e}_{00}
+(s_\beta^4c_\gamma^2
+s_\beta^2s_\gamma^2+
c_\beta^4c_\gamma^2)\,\Gamma^{\tau e}_{03}\right.\nonumber \\ &+&\left.
c_\gamma^2c_\beta^2s_\beta^2\,\Gamma^{\tau e}_{33}
\right],
\label{taue}
\end{eqnarray}

\begin{eqnarray}
\Gamma(\tau\to \mu
\nu_\tau\bar\nu_\mu)&=&\frac{G^2m^5_{\tau}}{192\pi^3}\left[
(s_\beta^2c_\gamma^2+s_\gamma^2)(s_\beta^2s_\gamma^2+c_\gamma^2)\,
\Gamma^{\tau\mu}_{00}
\right.\nonumber
\\ &+&\left. \left(s_\gamma^2(s_\beta^2c_\gamma^2+
s_\gamma^2)+c_\gamma^2(s_\beta^2s_\gamma^2+c_\gamma^2) \right)c_\beta^2\,
\Gamma^{\tau\mu}_{03}
+c_\gamma^2s_\gamma^2 c_\beta^4\,\Gamma^{\tau\mu}_{33}\right]
\label{taumu}
\end{eqnarray}
Notice that Eqs.~(\ref{mue})-(\ref{taumu}) depend only on the angles
$\beta$ and $\gamma$. We see from Eq.~(\ref{mue}) that in fact
\begin{equation}
G^2_\mu=G^2\times factor
\label{gmu}
\end{equation}
where the $factor$ will depend on the kinematical region allowed for
a particular decay, so
that the constant $G^2$ appearing in other decays is that given by
Eq.~(\ref{gmu}). For instance, if $\nu_3$ is kinematically forbidden
in the muon decay, from (\ref{mue}) we obtain
\begin{eqnarray}
G_\mu^2&=&G^2(1-\vert V_{\mu3}\vert^2)(1-\vert V_{e3}\vert^2)
\nonumber \\ &= &G^2(s_\beta^2s_\gamma^2+c_\gamma^2)c_\beta^2.
\label{gmu2}
\end{eqnarray}
Hence, we have defined $G_\mu$ in such a way that it
coincides with the usual value.

We will write Eqs.~(\ref{mue}), (\ref{taue}) and (\ref{taumu}),
 respectively, in the following form
\begin{equation}
\Gamma(\mu\to e\nu_\mu\bar\nu_e)=\frac{G^2m^5_\mu}{192\pi^3}f^{\mu
e}(\beta,\gamma,\delta_{e\mu},\delta_{3\mu}),
\label{def3}
\end{equation}
for the partial rate of the muon decay into electron and
\begin{equation}
\Gamma(\tau\to e\nu_\tau\bar\nu_e)=\frac{G^2m^5_\tau}{192\pi^3}f^{\tau
e}(\beta,\gamma,\delta_{e\tau},\delta_{3\tau}),
\label{def4}
\end{equation}

\begin{equation}
\Gamma(\tau\to \mu\nu_\tau\bar\nu_\mu)=\frac{G^2m^5_\tau}{192\pi^3}f^{\tau
\mu}(\beta,\gamma,\delta_{\mu\tau},\delta_{3\tau}),
\label{def41}
\end{equation}
for the decay of $\tau$ into electron and muon, respectively.

Using Eqs.~(\ref{def3})-- (\ref{def41}) we obtain
\begin{equation}
\left(\frac{G_\tau}{G_\mu}\right)^2 = \frac{f^{\tau
e}(\beta,\gamma,\delta_{e\tau},\delta_{3\tau})}{f^{\mu
e}(\beta,\gamma,\delta_{e\mu},\delta_{3\mu})}.
\label{tu}
\end{equation}
and

\begin{equation}
\frac{B^\tau_\mu}{B^\tau_e}= \frac{f^{\tau
\mu}(\beta,\gamma,\delta_{\mu\tau},\delta_{3\tau})}{f^{\tau e}
(\beta,\gamma,\delta_{e\tau},\delta_{3\tau})}=\frac{\Gamma^\tau_\mu}
{\Gamma^\tau_e}.
\label{taufra}
\end{equation}
The left hand sides of (\ref{tu}) and (\ref{taufra}) have the value showed
in (\ref{tmu})  and (\ref{tmu2}) respectively.

We will also consider $\pi\to l+\nu_l,\;l=e,\mu$ decays. The partial
width with a massive neutrino is~\cite{ro}
\begin{equation}
\Gamma^{\pi l}_3=\frac{G^2f^2_\pi V^2_{KM} m^3_\pi}{8\pi}
\left[\delta^2_{l\pi}+\delta^2_{3l}-(\delta^2_{l\pi}-\delta^2_{3l})^2\right]
\lambda^{\frac{1}{2}}(1,\delta^2_{l\pi},\delta^2_{3l})
\label{piw}
\end{equation}
where $\lambda$ is the triangular function,
$\delta_{l\pi}=m_l/m_\pi$ and $V^2_{KM}$ is the appropriate
Cabibbo-Kobayashi-Maskawa matrix elements of the quark sector.
In particular we have
\begin{eqnarray}
\Gamma(\pi\to \mu\nu_\mu)&=&\left(\vert
V_{\mu1}\vert^2+ \vert V_{\mu2}\vert^2 \right)\Gamma^{\pi\mu}_0+
\vert V_{\mu3}\vert^2\Gamma^{\pi\mu}_3\nonumber \\&=&
\left(s_\gamma^2s_\beta^2+c_\gamma^2 \right)\Gamma^{\pi\mu}_0+
s_\gamma^2c_\beta^2
\Gamma^{\pi\mu}_3
\label{pimu}
\end{eqnarray}
\begin{eqnarray}
\Gamma(\pi\to e\nu_e)&=&\left( \vert
V_{e1}\vert^2 +\vert V_{e2}\vert^2\right)\Gamma^{\pi e}_0+
\vert V_{e3}\vert^2 \Gamma^{\pi e}_3\nonumber \\ &=&
c_\beta^2\Gamma^{\pi e}_0+ s_\beta^2 \Gamma^{\pi e}_3
\label{pie}
\end{eqnarray}
{}From (\ref{pimu}) and (\ref{pie}) we obtain
\begin{equation}
\frac{\Gamma^{\pi e}}{\Gamma^{\pi\mu}}=\frac{c_\beta^2\Gamma^{\pi e}_0+
s_{\beta}^2 \Gamma^{\pi e}_{3}}
{(s_\gamma^2s_\beta^2+c_\gamma^2)\Gamma^{\pi\mu}_0+s_{\gamma}^2
c_{\beta}^2 \Gamma^{\pi \mu}_3}=
\frac{B^{\pi e}}{B^{\pi\mu}},
\label{piemu}
\end{equation}

In the next section we will compare Eqs.~(\ref{tu}), (\ref{taufra})
and (\ref{piemu}) with (\ref{tmu})-(\ref{pi}) and other experimental
data.
\section{Experimental constraints on mixing angles and neutrino mass}%sec.3
\label{sec:ma}

The effects of a leptonic--mixing
in the decays of pseudoscalars mesons was studied by Shrock~\cite{ro}
and Ng~\cite{ng}. Since then, several experiments were performed
searching the effect of extra peaks due to massive neutrinos in the
spectrum of the positron in $\pi^+\to e^+\nu(\gamma)$~\cite{b1}. The
measurement of this branching ratio
confirm the hypothesis of the $e-\mu$ universality at the $0.2\%$ level
{}~\cite{b2}. The theoretical uncertainty arises from
structure--dependent loop effects, they could be important in the
$\pi\to e\nu_e\gamma$ decay, but are less than $0.1\,\%$~\cite{b2}.

Here we will point out
some considerations about these experiments. In most of them
when fitting experimental data with theoretical calculations
it have been assumed that the heavy
neutrino couples mainly with the
electron or muon. Hence, according to this what they are comparing
with experimental data is~\cite{dib2}
\begin{equation}
\frac{\Gamma(\pi\to l\nu_3)}{\Gamma(\pi\to l\nu_l)}\,
\propto \vert V_{l3}\vert^2
\label{expe}
\end{equation}
where $\nu_l$ is the conventional massless neutrino, $l=e,\mu$ and the
factor of proportionality is the kinematical factor including phase
space.

It has also been consider by Bryman et al. ~\cite{b1}, the  ratio
\begin{equation}
\frac{\Gamma(\pi\to e\nu_3)}{\Gamma(\pi\to \mu\nu_l)}
\label{}
\end{equation}
which depends on both $V_{e3},V_{\mu3}$. However, the value of $V_{\mu3}$
was taken from the work by Abela et al.~\cite{b1}. In fact, most of the
experimental data were fitted assuming a heavy
subdominant coupled (HSC) neutrino, i.e., a heavy neutrino with
$V_{li}\ll1$~\cite{nota1}.

In our analysis we will not make any assumption with
respect to the matrix elements of the mixing matrix and when we have
considered a kinematically allowed heavy neutrino in pion decays
it is necessary to use (\ref{pimu}) and (\ref{pie}) in which
$V_{e3,\mu3}$ appear (after using the unitarity condition on the
matrix $V$).


We have done several plots of Eqs.~(\ref{tu}), (\ref{taufra})
and (\ref{piemu}) as a
function of the angles $\beta$ and $\gamma$ for several values of
$m_{\nu_3}$ using the experimental value (within $1\sigma$) given by
(\ref{tmu}), (\ref{tmu2}) and (\ref{pi}). The values of the other
parameters are those of the Ref.~\cite{pdg}.

Let us consider the following intervals :

\noindent {\em i)} $G_\tau/G_\mu>1$ independently of the mixing angles.
For the experimental data we have used, this region corresponds to $m_{\nu_3}
<50$ MeV.

\noindent {\em ii)} $G_\tau/G_\mu\leq1$, but there is no
intersection with (\ref{tmu}). This corresponds to the
mass interval $50\,\mbox{MeV}<m_{\nu_3}<84\,\mbox{MeV}$.

\noindent {\em iii)} $G_\tau/G_\mu\leq1$, compatible with (\ref{tmu})
and (\ref{tmu2}) but not compatible with (\ref{pi}). In this case we have
$84\,\mbox{MeV}\leq m_{\nu_3}<155\,\mbox{MeV}$ and $m_{\nu_3}>800$ MeV.

\noindent {\em iv)} $G_\tau/G_\mu\leq1$, compatible with all
experimental data in the $m_{\nu_3}$ interval: $155\,\mbox{MeV}\leq
m_{\nu_3}\leq800\,\mbox{MeV}$.


In Fig. 1 we show contour plot  of (\ref{tu}),
(\ref{taufra}) and
(\ref{piemu}). The allowed values for the mixing angles we obtain,
using $m_{\nu_3}=165\,\mbox{MeV}$, are
\begin{equation}
11.54^\circ<\beta<12.82^\circ,\quad \gamma<4.05^\circ.
\label{angles}
\end{equation}
We have seen that, to be in agreement with the processes represented by the
quantities in Eqs.~(\ref{tu}), (\ref{taufra})
and (\ref{piemu}) the heavier neutrino could have a mass in the
interval:
\begin{equation}
155 \,\mbox{MeV}\lower.7ex\hbox{$\;\stackrel{\textstyle<}{\sim}\;$}
m_{\nu_3} \lower.7ex\hbox{$\;\stackrel{\textstyle<}{\sim}\;$}800\,
\mbox{ MeV}.
\label{m3}
\end{equation}

\section{Constraints coming from $Z$- invisible width and end point of
tau decay into five pions}
\label{zotau}
We have seen in the last section that the mass region allowed by
the processes (\ref{tmu}), (\ref{tmu2}) and (\ref{pi}) is given in
(\ref{m3}). However, it is  mandatory to consider the $Z^0$ invisible
width and also the end point of the spectrum of the $\tau\to
5\pi\nu_\tau$ decay which provide the current upper limit on the
``tau neutrino'' mass. Hence, we will continue
our analysis in order to constrain even
more the allowed values for the neutrino mass and mixing angles.
\subsection{$Z^0$ invisible width}
\label{subsec:z0}
Let us consider the constraints coming from the $Z^0$ invisible width
measured by the LEP experiments: $\Gamma^{inv}=502\pm9$ MeV~\cite{pdg}.
For two massless and one massive neutrinos we have~\cite{hollik}
\begin{equation}
\frac{\Gamma^{inv}}{\Gamma^0} =
\left[2+\sqrt{1-4\mu_F}\,(1-\mu_F)\right]
\frac{1}{[(s_\beta^2s_\gamma^2+c_\gamma^2)c_\beta^2]^{\frac{1}{2}}},
\label{width}
\end{equation}
where
$$\Gamma^0= \frac{M_Z^3 G_\mu}{12\sqrt{2}\pi},\quad
\mu_F=m^2_{\nu_3}/M^2_Z,$$
$$
M_Z = 91.173\pm 0.020 \, \mbox{GeV} \, \mbox{
and} \, G_\mu =
1.16639\times 10^{-5}\,\mbox{GeV}^{-2}.$$
and we have used (\ref{gmu2}).

The contour plot of Eq.(\ref{width}) for $m_{\nu_3}=165 $ MeV appears also
in Fig. 1.
For masses above 225 MeV there is no compatibility region among $Z^0$
data and the other experimental ratios we have considered
in Sec.~\ref{sec:ma}. Hence, instead of (\ref{m3}) we have now
\begin{equation}
155\,\mbox{MeV}\lower.7ex\hbox{$\;\stackrel{\textstyle<}{\sim}\;$} m_{\nu_3}
\lower.7ex\hbox{$\;\stackrel{\textstyle<}{\sim}\;$}225\, \mbox{ MeV}.
\label{m3b}
\end{equation}
For now on in our analysis we will use a typical value of $m_{\nu_3}
=165$ MeV unless otherwise stated.

\subsection{End point of $\tau\to 5\pi\nu_\tau$ in the mixing scenario}
\label{subsec:bound}
The current upper limits on neutrino masses are:
\begin{equation}
\begin{array}{c}
``m_{\nu_e}''<7.3\,\mbox{eV}\,\quad \cite{pdg},\\
``m_{\nu_\mu}''<0.27\,\mbox{MeV}\, \quad \cite{pdg},\\
``m_{\nu_\tau}''<31\,\mbox{MeV}\, \quad \cite{argus,argus2}.
\end{array}
\label{numass}
\end{equation}
However these values were obtained in experiments which
tried to observe the effects of neutrino masses without
mixing in the leptonic sector.

The current experimental upper bound on ``$m_{\nu_\tau}$'' given by
ARGUS~\cite{argus,argus2} come from the study of the end point of the
hadronic invariant mass
distribution in the decay $\tau\to 5\pi\nu_\tau$.
Their analysis is only valid when there is no mixing in the leptonic sector,
otherwise the massive neutrino will not manifest itself by a shift on the
end point but by a shoulder in the distribution. The end point in both
situation
   s
will be the same.

For the sake of simplicity we shall consider here only the $\tau$
neutrino case but a similar procedure could be applied to the other $e$
and $\mu$ neutrinos. Assuming that $\nu_\tau$ in the decay $\tau\to
n\pi\nu_\tau$ is a particle with definite mass it was
obtained~\cite{concha}
\begin{equation}
\frac{d\Gamma}{dq^2}=\frac{G^2V^2_{KM}}{8\pi m_\tau^3}
w(q^2,m_\tau^2,m_{\nu_\tau}^2)\lambda^{\frac{1}{2}}
(m_\tau^2,q^2,m_{\nu_\tau}^2)h(q^2),
\label{gamma1}
\end{equation}
where
\begin{equation}
w(q^2,m_\tau^2,m_{\nu_\tau}^2)=(m_\tau^2-q^2)(m_\tau^2+2q^2)-
m_{\nu_\tau}^2(2m_\tau^2-q^2-m_{\nu_\tau}^2),
\label{w}
\end{equation}
and
\begin{eqnarray}
\lambda^{\frac{1}{2}}(m_\tau^2,q^2,m_{\nu_\tau}^2)&=&\left[ \right.
\{m_\tau^2-[(q^2)^{\frac{1}{2}}+m_{\nu_\tau}]^2\}\nonumber \\ &
&\mbox{} \times\{m_\tau^2-[(q^2)^{\frac{1}{2}}-m_{\nu_\tau}]^2\}
\left.\right]^{\frac{1}{2}}.
\label{l}
\end{eqnarray}
In Eq.~(\ref{gamma1}) the
function $h(q^2)$ contains the hadronic structure and $V^2_{KM}$
denotes the quark mixing angles, in this case $V^2_{KM}=V^2_{ud}$.
Here we shall not write down
explicitly both factors~\cite{concha}.

At the end point of the
spectrum
\begin{equation}
m_{\nu_\tau}=m_\tau-m_{had},
\label{e}
\end{equation}
where $m_{had}=(q^2)^{\frac{1}{2}}$.
Assuming that the $\nu_3$ is heavier than $\nu_{1,2}$ we get, instead
of Eq.(\ref{gamma1})
\begin{equation}
\frac{d\Gamma}{dq^2}=\frac{G^2V^2_{KM}}{8\pi m_\tau^3}\left[ (\vert
V_{\tau1}\vert^2+\vert
V_{\tau2}\vert^2) F_0 +\vert
V_{\tau3}\vert^2 F_3
\right]h(q^2)
\label{gamma2}
\end{equation}
where
\begin{equation}
F_0=  w(q^2,m_\tau^2,0)\lambda^{\frac{1}{2}}
(m_\tau^2,q^2,0)
\label{0}
\end{equation}
and
\begin{equation}
F_3=w(q^2,m_\tau^2,m_{\nu_3}^2)
\lambda^{\frac{1}{2}}(m_\tau^2,q^2,m_{\nu_3}^2).
\label{3}
\end{equation}
Each part of the phase space functions $F_0,F_3$ have a different end point.

The second end point related to the massive contribution $m_{\nu_3}$
disappears when both massive and massless contributions are summed up,
remaining only a shoulder in the distribution and the end point is related
with the (almost) massless neutrinos. This is the end point observed
experimentally.

Using $m_{\nu_3}=165$, $225$ MeV and typical
values for the respective mixing angles in Eq.(\ref{gamma2}) we
obtain Fig. 2. We see that the $m_{\nu_3}$ values given in (\ref{m3b})
are allowed by the experimental data of the $\tau\to5\pi\nu$ decay.
We also show in Fig. 2 a blow-up of the region near the end point to emphasize
the fact that all the three curves for $m_{\nu_3}=0,165,225$ MeV would be
compatible with the three events found by ARGUS in this end point
area~\cite{argus,argus2}.



We have only studied the phase space contributions and have not taken
into account the hadronic effects contained in $h(q^2)$. This
function hides our ignorance on hadronic structure and it will, in
general, modify the shape of the distribution but not affect the
end point. However close to the end point the shape of the distribution
does not depend very much on the resonant structure. On the other
hand, the population of the spectrum in this region will be strongly
modified~\cite{concha}. For this reason, to be able to detect an
effect of mixing it is necessary to study the population of the
spectrum close to the end point.

\section{Other constraints}



%%%%%

\subsection{neutrino oscillations}
\label{subsec:osc}
The case of three light neutrino oscillations was consider in
Ref.~\cite{vbarger}. Here we will study the case of only two light
neutrinos and a heavy one as in Ref.~\cite{ng}. Let us start by
summarizing at this stage our results for the mixing angles:
\begin{equation}
\left(
\begin{array}{ccc}
0.977\,c_\theta & 0.977\, s_\theta  &  0.211 \\
-0.998\,s_\theta-0.016\,c_\theta  & 0.998\,c_\theta-0.016\,s_\theta  & 0.069
\\
0.071\,s_\theta-0.2\,c_\theta  & -0.071\,c_\theta-0.211\,s_\theta & 0.975
\end{array}\right).
\label{ckmn}
\end{equation}
The matrix above is orthogonal independently of the $\theta$.

As we said before, neutrinos produced in weak processes are linear combinations
of the mass eigenstates $\nu_l=\sum_iV_{li}\nu_i$ with $V_{li}$ given in
(\ref{ckm}) and in particular in (\ref{ckmn}). Here we will write
down the probability of finding a neutrino $\nu_e,\nu_\mu$ after a
path length $R$ if at the origin it was a $\nu_e$, that is
\begin{eqnarray}
P(\nu_e\to\nu_e)=
\vert\langle\nu_e(L)\vert\nu_e(0)\rangle\vert^2&=&
1-2c_\beta^2s_\beta^2\left(1-\cos\frac{2\pi R}{L}\right)\nonumber \\
&-&2c_\theta^2s_\theta^2c_\beta^4\left(1-\cos\frac{2\pi R}{L_{12}}\right),
\label{ee}
\end{eqnarray}
and
\begin{eqnarray}
P(\nu_e\to\nu_\mu)&=&
\vert\langle\nu_\mu(L)\vert\nu_e(0)\rangle\vert^2 \nonumber \\&=&
h_1(\beta,\gamma,\theta)-
h_2(\beta,\gamma,\theta)\cos\frac{2\pi
R}{L_{12}}-h_3(\beta,\gamma,\theta)\cos\frac{2\pi R}{L}
\label{emu}
\end{eqnarray}
where we have defined
\[
h_1(\beta,\gamma,\theta)=2c_\beta^2[c_\theta^2s_\theta^2c_\gamma^2+
(1-c_\theta^2s_\theta^2)s_\gamma^2s_\beta^2+
c_\gamma^2s_\gamma^2s_\beta^2c_\theta^2s_\theta^2(c_\theta^2-s_\theta^2)],
 \]
\[
h_2(\beta,\gamma,\theta)=2[c_\theta s_\theta(c_\gamma^2-s_\gamma^2s_\beta^2)
+c_\gamma s_\gamma s_\beta(c_\theta^2-s_\theta^2)]c_\theta s_\theta c_\beta^2,
\]
\[
h_3(\beta,\gamma,\theta)=2s_\gamma^2s_\beta^2c_\beta^2.
\]

As in Ref.~\cite{ng}, for the mass range we are considering here,
neutrino oscillations occur with essentially two wavelengths. We have
defined in (\ref{ee}) and (\ref{emu})
\[ L_{ij}=\frac{2\pi}{E_i-E_j}\]
that is,
\[L_{12}=\frac{4\pi p}{\delta m^2}=2.5\frac{p/\mbox{MeV}}{\delta
m^2/(\mbox{eV})^2}\,\mbox{meter}, \]
with
\[\delta m^2=m^2_{\nu_1}-m^2_{\nu_2},\]
and
\[
L_{13}=L_{23}=L=\frac{2\pi}{m_{\nu_3}}=\frac{1.24}{m_{\nu_3}/\mbox{MeV}}
\times10^{-12}\,\mbox{meter}.\]
The masses
$m_{\nu_1}, m_{\nu_2}$, and of course $\delta m^2$, are still
undetermined. Let us take,for instance, $\delta m^2$ of the order of a few
$\mbox{eV}^2$. Of course for smaller $\delta m^2$ values the cosine involving
$L_{12}$  in Eq.(~\ref{ee}) must be taken into account.
The short
component has an oscillation length of the order of $10^{-15}$ m.
  A more detailed study
concerning neutrino oscillation data will be published elsewhere.

Let us consider the solar neutrino data. In this case
$R\sim10^{11}$ m and $L_{12}\approx 0.25$ m if $\delta m^2\approx\,
1(\mbox{eV})^2$. With this condition the oscillation has averaged
out and one obtain,
\begin{equation}
P(\nu_e\to\nu_e)=1-2c_\beta^2s_\beta^2-2c_\theta^2s_\theta^2c_\beta^4.
\label{solarav}
\end{equation}
Using the values in (\ref{ckmn}) and the experimental data given in Table 1
for $P(\nu_e\to\nu_e)$~\cite{osc} we
obtain the allowed regions showed in Fig. 3.
In Fig. 4 we show values for the angles compatible with those in
Fig. 1.
Note that with
these angles the result of Davis, a suppression of about $27\%$ of the
solar standard model prediction,
is not fitted at the $1\sigma$ level.
The lowest value of Eq.(\ref{solarav}) is about $ 30\%$.

In
fact, neutrino oscillations are only sensitive to small mass
differences and cannot be used to detect mixing of heavy neutrinos
directly. However, we see from (\ref{solarav}) that even in the
averaged situation the effect of the mixing with the heavy neutrino
survives via the angle $\beta$.

The range for the $\theta$ angle compatible with the allowed area
showed in Fig.1, coming from Kamiokande II, Gallex and SAGE data is
\begin{eqnarray}
 & 0.25271<s_{\theta}^2< 0.29862, \nonumber \\
 & 0.70142<s_{\theta}^2<0.74729.
\label{kamioka}
\end{eqnarray}
%and, from the Gallex data
%\begin{equation}
%0.48<s_{\theta}^2<0.78,
%\label{gallex}
%\end{equation}
%SAGE data do not impose any limit on $\theta$ in this region.

Finally, the mixing matrix in the leptonic
sector which is consistent with all experimental data considered
by us, for $m_{\nu_3}=165$ MeV, is:
\begin{equation}
\left(
\begin{array}{ccc}
0.817-0.848\, &\, 0.490-0.536 \, & \, 0.200-0.222 \\
-(0.502-0.560)\,  &\, 0.828-0.865\,  & \,<0.069 \\
-(0.192-0.129)\,  &\, -(0.101-0.183)\, & \,0.973-0.980
\end{array}\right).
\label{kmf}
\end{equation}
\subsection{Cosmological constraints}
\label{subsec:cosmo}
{}From cosmological constraints, stable neutrinos must have masses less
than 40 eV or greater than 2 GeV~\cite{lw}. In our scenario, of course,
by construction neutrinos are not stable.

Earlier works about the
upper bounds on the lifetime of a massive neutrino assumed V-A
interactions in annihilation processes of the massive neutrinos and
that the principal decay mode is $\nu_3\to
\nu_e\gamma$~\cite{dicus,sarkar}. However, for a neutrino mass
$m_{\nu_3}\gg m_{e,\mu}$ this decay (and the not GIM suppressed $\nu_3\to
\nu_e\gamma\gamma$) are not dominant~\cite{sarkar}.

Note that as there are no changing flavor neutral currents at tree level,
constraints as those in Ref.~\cite{aan} do not apply to our case.

Here we will assume the mixing angles given in (\ref{kmf}) and
evaluate the lifetime for a heavy neutrino.
In the mass range we are considering for the heaviest neutrino,
$155-225$ MeV, the main decays are $\nu_3\to l^+l^-\nu_{1,2}$,
$l=e,\mu$. The lifetime of heavy neutrinos in our context
was studied by Kolb and Goldman~\cite{kg}. Here we
will treat again these decays. The partial widths are
\begin{equation}
\Gamma(\nu_3\to ll\nu)=\frac{G_\mu^2 m^5_{\nu_3}}{192\pi^3
(1-\vert V_{\mu3}\vert^2)(1-\vert V_{e3}\vert^2)}[\vert
V_{l3}\vert^2 (1-\vert V_{l3}\vert^2)F(m_l,m_{\nu_3})],
\label{wnu3e}
\end{equation}
And a similar expression for $\nu_3\to\mu e\nu$.

Notice that we have taken into account the appropriate modification of
the $G_\mu$ constant given in Eq.~(\ref{gmu2}). In fact, we leave in
(\ref{wnu3e}) all factors in such a way that it is clear that
some cancellation of matrix elements occurs, otherwise we would obtained
only the
factor $\vert
V_{e3}\vert^2 (1-\vert V_{e3}\vert^2)$ for the case of electron decay
 and $\vert
V_{\mu3}\vert^2(1-\vert V_{\mu3}\vert^2)$ for $\mu$ decay as in
ref.~\cite{kg}.

The function $F(m_l,m_{\nu_3})$
is defined as
\begin{eqnarray}
F(m_l,m_{\nu_3})& = &
2\int^{y_M}_{y_m}\frac{(y^2-B')}{(k'-y)^{3}}
(k'-y-\omega^2_l)^2 \left[(k'-y-\omega_l^2)y(k'-y) \right.
\nonumber \\
 & + & \left.\left[(k'-y)^2+\omega_l^2(k'-y)-2\omega_l^4\right](2k'y-y^2-B')
\right] dy \label{l33}
\end{eqnarray}
where
\[k'=1+\omega^2_l,\quad B'=4(k'-1),\quad
\omega_l=\frac{m_l}{m_{\nu_3}},\quad y_m=2\omega_l,\quad y_M=1.
\]
Here we will consider only the $\nu_3 \to ee \nu$ decay.
The respective lifetime scaled from muon decay is
\begin{eqnarray}
\tau(\nu_3\to ee\nu)&=&\tau_\mu\left(\frac{m_\mu}{m_{\nu_3}}\right)^5
\frac{1-\vert V_{\mu3}\vert^2}{\vert V_{e3}\vert^2}
\frac{\Gamma^{\mu e}_{0 0}}{F(m_e,m_{\nu_3})}\nonumber \\ &=&
\tau_\mu\left(\frac{m_\mu}{m_{\nu_3}}\right)^5
\frac{1-s_\gamma^2c_\beta^2}{s_\beta^2}
\frac{\Gamma^{\mu e}_{0 0}}{F(m_e,m_{\nu_3})},
\label{vidae}
\end{eqnarray}
where $\tau_\mu$ is the muon lifetime. From (\ref{vidae})
 and the angles in (\ref{kmf}) we see that the lifetime for the decay of
$\nu_3$ into electrons is of the order of $10^{-6}-10^{-5}$ s.

For the range (4.2) the decays
$\nu_3\to\pi^+e^-$ and $\nu_3\to\mu^+
e^-\nu$ are possible, they will not be considered here. Our purpose is only to
indicate that there is no strong constraints coming
from processes with cosmological consequences.

There are experiments which tried to observe neutrino decays~\cite{fb,bd},
however those experiments are sensitive to lifetimes of the order of
$10^{-2}-10^{2}$ s. Hence, their data are not directly
applicable to our case.

For instance,
in \cite{fb} it was assumed that the $\nu_\tau$ couples mainly to a
single mass eigenstate, then an upper limit of the square of the
matrix element $\vert V_{ei}\vert$ was obtained. In our case it is
necessary to consider at least two of such matrix elements at once.

On
the other hand, we must recall the following. In \cite{fb} the
expected neutrino flux
depends on the number of $D_s$ ($N_{D_s}$) and
of $D$ mesons ($N_D$)
produced by protons in the dump. In computing this number it was
assumed the following values for the branching ratio for the semileptonic
decays $B.R.(D\to\mu\nu_3)=0.1$ and $B.R.(D_s\to \mu\nu_3)=0.03$.
Notwithstanding, at present these branching ratios are
$B.R.(D\to\mu\nu_3)\leq7.2\times10^{-4}$~\cite{pdg} and
$B.R.(D_s\to\mu\nu_3)=(4.0^{+1.8+.8}_{-1.4-.6}\pm1.8)\times10^{-3}$~
\cite{refx}. It means that the neutrino flux is reduced by
a factor of a hundred with respect to the values used by the CHARM
experiment~\cite{fb}, and that in fact higher values for the upper bounds
on the mixing angles are allowed.
On the other hand, in Ref.~\cite{bd} the neutrino flux
was calculated using Monte Carlo
programs and we do not know if in this case the same branching ratios
for the decays of the $D$ mesons were used.

Anyway, we stress that in our case the lifetime of the heavy
neutrino is such that it would decay before the detector.

\subsection{astrophysical constraints}
\label{subsec:astro}
Here we will discuss, briefly, the possible effects of our scenario in
the solar neutrino flux and in the supernova 1987A data. In fact for neutrinos
with mass $1.1\,\mbox{MeV}<m<14\,\mbox{MeV}$ the experimental
detection of solar neutrinos have been considered in Ref.~\cite{dtfw}.
Neutrinos coming from the sun, generated through the decay $^8B\to
^8\!B_ee^+\nu$ must have a mass up to near 14 MeV which is the threshold
energy for this reaction. Those coming from $pp$ reaction have
$m<1.44$ MeV. Of
course, a heavy neutrino would decay in flight producing
$\nu_{e,\mu}$ and charged leptons.

The observation of $\bar\nu_e$'s coming from the supernova 1987A
implies a constraint on their lifetime
\begin{equation}
\tau_{\nu_e}/m_{\nu_e}\lower.7ex\hbox{$\;\stackrel{\textstyle>}{\sim}\;$}
6\times10^5s/eV.
\label{87a}
\end{equation}
However this constraint is valid for neutrinos reaching the Earth's
orbit, and no mixing in the
leptonic sector. In fact, we have seen in Sec.~\ref{subsec:cosmo} that for a
mass of 165 MeV the mixing angles are such that the lifetime of the
heavy neutrino is of the order of $10^{-6}-10^{-5}$ s. It means that
this neutrino can appear only through the mixing effect as it is
kinematically forbidden at the characteristic energies of the
supernova and must decay before reaching the Earth's orbit.
We see that only the lightest neutrinos $\nu_{1,2}$ are
constrained from these data.

On the other hand, since neutrinos are massive Dirac particles the
right-handed components can be produced in the hot supernova via
helicity flip. This produce an extra source of cooling in the
supernova since this right-handed neutrinos are almost sterile.
The relevant spin-flip processes are $NN\to NN\bar\nu_R\nu_R$
(bremsstrahlung), $N\nu_L\to N\nu_R$ and $e^+e^-\to\bar\nu_R\nu_R$.
However bounds coming from this  processes in supernova are valid
only for small enough neutrino masses and do not apply to our
heavy neutrino. It is not thermally emitted from
the supernova~\cite{gri,ra}.


\section{conclusions}%sec.4
\label{sec:con}
We have
examined some weak processes which are modified by a three generation
mixing in the leptonic sector and
found compatibility regions for $m_{\nu_3}$ and mixing angles.
These parameters are likely to have consequences elsewhere.

The main result of this paper is that the current upper limit on the
tau-neutrino mass ``$m_{\nu_\tau}$'' does not seem to exclude a
heavy neutrino $155\,\,\mbox{MeV}<m_{\nu_3}
<225\,\,\mbox{MeV}$ mixed up
with the light ones, and solving at the same time a possible deviation of the
$\tau-\mu$ universality. However, our analysis
will still be valid if no deviation from $\tau-\mu$ universality is confirmed
by data. Only the value for $m_{\nu_3}$ and mixing angles will be different.
If the new data imply lowest values for the $\nu_3$ mass
and, as in $\nu_3$ decays photons are ultimately
produced~\cite{dolgov} it will be interesting to study
astrophysical bounds more in detail.

Of course, there are other experiments we could have considered as the
$\nu_\mu e\to\nu_\mu e$ cross section ~\cite{mg1}, measurements of
the Michel parameter in muon decay ~\cite{msd}, etc., but we do not
expect they to produce strongest constraints than the ones we have analyzed.

%%%kaons
The $\mu-e$ universality has been also verified in the kaon
decays~\cite{kaons1}. In fact, in the mass range (\ref{m3b})
the decays $K\to e(\mu)\nu_i$ with
$i=1,2,3$ occur.
In particular, the branching
ratio of the kaon decays, $\Gamma(K\to e\nu)/\Gamma(K\to \mu\nu)$
have been measured~\cite{kaons2} and, in principle, provide another
confirmation of the $e\mu$ universality, but in this case the
structure--dependent radiative correction is expected to be
$\sim1000$ times larger than that for the pion decays partial widths
{}~\cite{b2}. For this reason we will not analyze these decays
in this work.
However we recall  that in the literature a similar expression to
(\ref{expe}) was also assumed.

Notice that in the extension of the standard model we are considering
there are no flavor changing neutral currents at tree level, since
they are GIM suppressed~\cite{gim}, and for this reason it is not necessary
to consider processes producing $\mu e$ events.

We recall that the final allowed values for masses
and mixing angles must be obtained taken into account radiative
corrections~\cite{pk2}.
Any way, radiative corrections
imply a reduction of about $\sim4\,\%$~\cite{b2} and will
not modify qualitatively our results.

%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%

\acknowledgments
We thank Funda\c c\~ao de Amparo \`a Pesquisa do Estado de
S\~ao Paulo (FAPESP) (O.L.G.P.) for full financial support and
Con\-se\-lho Na\-cio\-nal de De\-sen\-vol\-vi\-men\-to Cien\-t\'\i
\-fi\-co e Tec\-no\-l\'o\-gi\-co (CNPq) (V.P.) for partial and full (R.Z.F.)
financial support. We are very grateful to C.O. Escobar for stimulating
discussi
   ons
at the early stage of this work. We also thank M.M. Guzzo for very useful
comments concerning solar neutrino data.

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\begin{figure}
\setepsfscale{0.78}
\begin{picture}(700,400)(0,0)
\postscript {fig1.ps}
\end{picture}
\protect
\caption[Fig.1]{Allowed region (within $1\sigma$) in the plane $s^2_\beta
\times s^2_\gamma$ for the ratios $G_\tau/G_\mu$
(Eq.~(\ref{tmu})),$B^{\tau \mu}/B^{\tau e}$ (Eq.~(\ref{tmu2})) and $B^{\pi
e}/B^{\pi\mu}$ (Eq.~(\ref{pi})) for $m_{\nu_3}=165 \mbox{ MeV}$. The region
 for $B^{\tau \mu}/B^{\tau e}$  is larger than the area shown in the
picture. We will also show the contour plot of (Eq.~(\ref{width})).
The black area is the allowed region combining all ratios.}
\label{f1}
\end{figure}
\begin{figure}
\setepsfscale{0.77}
\begin{picture}(700,554)(0,0)
\postscript {fig2.ps}
\end{picture}
\caption[Fig.2]{Spectrum for $m_{\nu_3}=0$ (dashed line), $m_{\nu_3}=165
\mbox{ MeV}$ (dotted line) and $m_{\nu_3}=215 \mbox{ MeV}$ (continuous line).
Here $d\Gamma^*/dx=(8\pi m_\tau^3/G^2V_{KM})^{-1}d\Gamma/dx$, $G^2=
G^2_\mu/(s_\beta^2s_\gamma^2+c_\gamma^2)c_\beta^2$ and $x=m_{h}/m_{\tau}$.
The coefficient
$(s_\beta^2s_\gamma^2+c_\gamma^2)c_\beta^2$
has been absorbed in the definition of $\Gamma$.
We also shown the blow-up of region near of end point.}
\label{f2}
\end{figure}
\begin{figure}
\setepsfscale{0.87}
\begin{picture}(700,554)(0,0)
\postscript {fig3.ps}
\end{picture}
\caption[Fig.3]{Contour plot of Eq.~(\ref{solarav}) for Sage, Gallex
and Kamiokande--II using the values on Table~\ref{tab1} (within 1 $\sigma$).
 The arrows shown the allowed contours for each experiment.}
\label{f3}
\end{figure}
\begin{figure}
\begin{picture}(700,554)(0,0)
\postscript {fig4.ps}
\end{picture}
\caption[Fig.4]{Contour plot for Gallex, Kamiokande--II and Sage data
in the restricted region showed in black
on Figure~\ref{f1}. The shaded areas are the allowed regions for these
experiments.}
\label{f4}
\end{figure}

\begin{table}[h]
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
\bf Experiment & Process & $E_{\mbox{threshold}}$ (MeV) & Expt./SSM \\
\hline
\bf Davis et al. & $\nu_e+^{37}\mbox{Cl} \rightarrow e +^{37}\mbox{Ar}$ & 0.81
&
    $0.27\pm0.04$ \\
\bf Kamiokande--II & $\nu + e \rightarrow \nu + e$ & 7.5 & $0.49\pm0.05\pm0.06$
   \\
\bf SAGE & $\nu_e+^{71}\mbox{Ga} \rightarrow e+^{71}\mbox{Ge}$ &0.24 &
$0.44^{+0
   .13}_{-0.18}\pm0.11$ \\
\bf GALLEX & $\nu_e +^{71}\mbox{Ga} \rightarrow e+^{71}\mbox{Ge}$ &0.24 &
$0.66\
   pm0.11\pm0.05$ \\
\hline
\end{tabular}
\end{center}
\caption[Tab.1]{Results of solar neutrino experiments. The flux is given as a
function of the Solar Standard Model prediction~\cite{fn1}.}
\label{tab1}
\end{table}

\end{document}


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l
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1782
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1880
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725
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 1665 m 1205 1665 l s [] 0 sd 1212 1690 m 1205 1690 l s 1212 1714 m 1205 1714 l
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s
 [4 8] 0 sd 1952 1787 m 1205 1787 l s [] 0 sd 1212 1811 m 1205 1811 l s 1212
 1836 m 1205 1836 l s 1212 1860 m 1205 1860 l s 1212 1884 m 1205 1884 l s 1219
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s
 [4 8] 0 sd 1952 2273 m 1205 2273 l s [] 0 sd 1212 2298 m 1205 2298 l s 1212
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 1783 l 1166 1786 l 1166 1788 l 1165 1791 l 1164 1793 l 1162 1794 l 1161 1794 l
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1792
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1913
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2037
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 2025 l 1176 2026 l s 1161 2158 m 1159 2158 l 1158 2156 l 1157 2153 l 1157 2151
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 [4 8] 0 sd 1326 2412 m 1326 1665 l s [] 0 sd 1344 1672 m 1344 1665 l s 1362
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 1679 m 1416 1665 l s [4 8] 0 sd 1416 2412 m 1416 1665 l s [] 0 sd 1434 1672 m
 1434 1665 l s 1451 1672 m 1451 1665 l s 1469 1672 m 1469 1665 l s 1487 1672 m
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 1665 l s 1577 1672 m 1577 1665 l s 1594 1679 m 1594 1665 l s [4 8] 0 sd 1594
 2412 m 1594 1665 l s [] 0 sd 1612 1672 m 1612 1665 l s 1630 1672 m 1630 1665 l
 s 1648 1672 m 1648 1665 l s 1666 1672 m 1666 1665 l s 1684 1679 m 1684 1665 l
s
 [4 8] 0 sd 1684 2412 m 1684 1665 l s [] 0 sd 1702 1672 m 1702 1665 l s 1720
 1672 m 1720 1665 l s 1737 1672 m 1737 1665 l s 1755 1672 m 1755 1665 l s 1773
 1679 m 1773 1665 l s [4 8] 0 sd 1773 2412 m 1773 1665 l s [] 0 sd 1791 1672 m
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1652
 m 1237 1650 l 1236 1649 l 1234 1648 l 1233 1648 l 1231 1649 l 1230 1650 l 1230
 1654 l 1231 1655 l 1233 1656 l 1234 1656 l 1236 1655 l 1237 1654 l 1238 1652 l
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 1645 l 1599 1647 l 1600 1649 l 1601 1650 l 1603 1650 l 1606 1651 l 1608 1653 l
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 1649 l 1748 1645 l 1749 1644 l 1751 1643 l 1752 1643 l 1754 1644 l 1755 1645 l
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%%%%%%%%%%%%%%%%%%
%%% FIGURE 3 %%%%%
%%%%%%%%%%%%%%%%%%

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326
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391
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2245
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534
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605
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s
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636
 2340 l s 643 2346 m 640 2343 l s 1466 2340 m 1465 2346 l s 646 2349 m 643 2346
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s
 652 2354 m 650 2352 l s 657 2357 m 652 2354 l s 1464 2352 m 1463 2357 l s 658
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1461
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 1452 2409 m 1452 2411 l s 1453 2405 m 1452 2409 l s 723 2413 m 721 2411 l s
728
 2417 m 723 2413 l s 1452 2411 m 1451 2417 l s 729 2418 m 728 2417 l s 735 2423
 m 729 2418 l s 735 2423 m 735 2423 l s 1451 2417 m 1449 2423 l s 741 2427 m
735
 2423 l s 742 2429 m 741 2427 l s 1449 2423 m 1448 2429 l s 747 2432 m 742 2429
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 757 2440 m 753 2437 l s 1446 2435 m 1445 2440 l s 1447 2435 m 1446 2435 l s
759
 2442 m 757 2440 l s 765 2446 m 759 2442 l s 1445 2440 m 1444 2446 l s 765 2446
 m 765 2446 l s 771 2451 m 765 2446 l s 772 2452 m 771 2451 l s 1444 2446 m
1442
 2452 l s 777 2456 m 772 2452 l s 780 2458 m 777 2456 l s 1441 2458 m 1440 2458
 l s 1442 2452 m 1441 2458 l s 918 1012 m 919 1012 l s 919 1012 m 925 1011 l s
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836
 1035 m 842 1032 l s 842 1032 m 848 1030 l s 848 1030 m 849 1030 l s 1043 1030
m
 1043 1030 l s 1043 1030 m 1049 1033 l s 1049 1033 m 1053 1036 l s 819 1041 m
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 1049 m 1078 1053 l s 782 1059 m 783 1059 l s 783 1059 m 788 1056 l s 788 1056
m
 794 1053 l s 1078 1053 m 1079 1054 l s 1079 1054 m 1085 1059 l s 771 1065 m
777
 1062 l s 777 1062 m 782 1059 l s 1085 1059 m 1085 1059 l s 1085 1059 m 1091
 1065 l s 761 1071 m 765 1069 l s 765 1069 m 771 1065 l s 771 1065 m 771 1065 l
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 1071 m 1102 1077 l s 741 1083 m 741 1083 l s 741 1083 m 747 1079 l s 747 1079
m
 750 1077 l s 1102 1077 m 1103 1078 l s 1103 1078 m 1107 1083 l s 731 1089 m
735
 1086 l s 735 1086 m 741 1083 l s 1107 1083 m 1109 1085 l s 1109 1085 m 1111
 1089 l s 722 1095 m 723 1094 l s 723 1094 m 729 1090 l s 729 1090 m 731 1089 l
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 1106 m 711 1102 l s 711 1102 m 713 1101 l s 1120 1101 m 1120 1101 l s 1120
1101
 m 1124 1107 l s 696 1113 m 700 1110 l s 700 1110 m 704 1107 l s 1124 1107 m
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s
 672 1130 m 676 1127 l s 676 1127 m 680 1124 l s 1135 1124 m 1138 1130 l s 1138
 1130 m 1138 1130 l s 664 1136 m 670 1132 l s 670 1132 m 672 1130 l s 1138 1130
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 1136 l s 1142 1136 m 1144 1141 l s 1144 1141 m 1145 1142 l s 650 1148 m 652
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s
 635 1160 m 640 1156 l s 640 1156 m 642 1154 l s 1150 1154 m 1153 1160 l s 629
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1156
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605
 1188 m 609 1184 l s 1163 1184 m 1165 1190 l s 596 1196 m 599 1193 l s 599 1193
 m 603 1190 l s 1165 1190 m 1167 1196 l s 590 1201 m 593 1199 l s 593 1199 m
596
 1196 l s 1167 1196 m 1168 1196 l s 1168 1196 m 1170 1201 l s 584 1207 m 587
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 575 1217 m 579 1213 l s 1174 1213 m 1174 1214 l s 1174 1214 m 1176 1219 l s
567
 1225 m 569 1223 l s 569 1223 m 573 1219 l s 1176 1219 m 1177 1225 l s 562 1231
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557
 1236 l s 557 1236 m 562 1231 l s 1179 1231 m 1180 1233 l s 1180 1233 m 1181
 1237 l s 551 1243 m 551 1243 l s 551 1243 m 556 1237 l s 1181 1237 m 1183 1243
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 545 1249 m 546 1249 l s 1184 1249 m 1186 1254 l s 1186 1254 m 1186 1255 l s
536
 1261 m 539 1256 l s 539 1256 m 541 1255 l s 1186 1255 m 1187 1261 l s 531 1267
 m 534 1264 l s 534 1264 m 536 1261 l s 1187 1261 m 1189 1267 l s 526 1273 m
528
 1271 l s 528 1271 m 531 1267 l s 1189 1267 m 1190 1273 l s 521 1279 m 522 1278
 l s 522 1278 m 526 1273 l s 1190 1273 m 1192 1277 l s 1192 1277 m 1192 1279 l
s
 517 1284 m 521 1279 l s 1192 1279 m 1193 1284 l s 512 1290 m 516 1286 l s 516
 1286 m 517 1284 l s 1193 1284 m 1194 1290 l s 508 1296 m 510 1294 l s 510 1294
 m 512 1290 l s 1194 1290 m 1196 1296 l s 504 1302 m 504 1302 l s 504 1302 m
508
 1296 l s 1196 1296 m 1197 1302 l s 500 1308 m 504 1302 l s 1197 1302 m 1197
 1305 l s 1197 1305 m 1198 1308 l s 496 1314 m 498 1311 l s 498 1311 m 500 1308
 l s 1198 1308 m 1199 1314 l s 492 1320 m 492 1319 l s 492 1319 m 496 1314 l s
 1199 1314 m 1200 1320 l s 488 1326 m 492 1320 l s 1200 1320 m 1202 1326 l s
484
 1332 m 486 1328 l s 486 1328 m 488 1326 l s 1202 1326 m 1203 1332 l s 480 1338
 m 480 1338 l s 480 1338 m 484 1332 l s 1203 1332 m 1203 1336 l s 1203 1336 m
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 1348 l s 474 1348 m 477 1344 l s 1205 1344 m 1206 1350 l s 470 1356 m 473 1350
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 1207 1356 m 1207 1362 l s 463 1367 m 466 1362 l s 1207 1362 m 1208 1367 l s
460
 1373 m 462 1369 l s 462 1369 m 463 1367 l s 1208 1367 m 1209 1373 l s 457 1379
 m 460 1373 l s 1209 1373 m 1209 1375 l s 1209 1375 m 1210 1379 l s 454 1385 m
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 1385 l s 1211 1385 m 1211 1391 l s 448 1397 m 451 1392 l s 451 1392 m 451 1391
 l s 1211 1391 m 1212 1397 l s 446 1403 m 448 1397 l s 1212 1397 m 1213 1403 l
s
 443 1409 m 445 1405 l s 445 1405 m 446 1403 l s 1213 1403 m 1213 1409 l s 441
 1415 m 443 1409 l s 1213 1409 m 1214 1415 l s 438 1421 m 439 1419 l s 439 1419
 m 441 1415 l s 1214 1415 m 1215 1421 l s 436 1427 m 438 1421 l s 1215 1421 m
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 1216 1433 l s 431 1439 m 433 1435 l s 433 1435 m 434 1433 l s 1216 1433 m 1216
 1439 l s 429 1445 m 431 1439 l s 1216 1439 m 1217 1445 l s 427 1450 m 429 1445
 l s 1217 1445 m 1217 1450 l s 426 1456 m 427 1453 l s 427 1453 m 427 1450 l s
 1217 1450 m 1218 1456 l s 424 1462 m 426 1456 l s 1218 1456 m 1218 1462 l s
422
 1468 m 424 1462 l s 1218 1462 m 1219 1468 l s 421 1474 m 421 1473 l s 421 1473
 m 422 1468 l s 1219 1468 m 1219 1474 l s 419 1480 m 421 1474 l s 1219 1474 m
 1219 1480 l s 418 1486 m 419 1480 l s 1219 1480 m 1220 1486 l s 417 1492 m 418
 1486 l s 1220 1486 m 1220 1492 l s 415 1498 m 417 1492 l s 1220 1492 m 1220
 1498 l s 414 1504 m 415 1500 l s 415 1500 m 415 1498 l s 1220 1498 m 1221 1504
 l s 413 1510 m 414 1504 l s 1221 1504 m 1221 1510 l s 412 1516 m 413 1510 l s
 1221 1510 m 1221 1516 l s 411 1522 m 412 1516 l s 1221 1516 m 1221 1518 l s
 1221 1518 m 1221 1522 l s 411 1528 m 411 1522 l s 1221 1522 m 1221 1528 l s
410
 1533 m 411 1528 l s 1221 1528 m 1222 1533 l s 410 1539 m 410 1533 l s 1222
1533
 m 1222 1539 l s 409 1545 m 410 1539 l s 1222 1539 m 1222 1545 l s 409 1551 m
 409 1545 l s 409 1545 m 409 1545 l s 1222 1545 m 1222 1551 l s 408 1557 m 409
 1551 l s 1222 1551 m 1222 1557 l s 408 1563 m 408 1557 l s 1222 1557 m 1222
 1563 l s 408 1569 m 408 1563 l s 1222 1563 m 1222 1569 l s 408 1575 m 408 1569
 l s 1222 1569 m 1222 1575 l s 408 1581 m 408 1575 l s 1222 1575 m 1222 1581 l
s
 408 1587 m 408 1581 l s 1222 1581 m 1222 1587 l s 409 1593 m 408 1587 l s 1222
 1587 m 1222 1593 l s 409 1599 m 409 1593 l s 409 1599 m 409 1599 l s 1222 1593
 m 1222 1599 l s 410 1605 m 409 1599 l s 1222 1599 m 1222 1605 l s 410 1611 m
 410 1605 l s 1222 1605 m 1222 1611 l s 411 1616 m 410 1611 l s 1222 1611 m
1221
 1616 l s 411 1622 m 411 1616 l s 1221 1616 m 1221 1622 l s 412 1628 m 411 1622
 l s 1221 1626 m 1221 1628 l s 1221 1622 m 1221 1626 l s 413 1634 m 412 1628 l
s
 1221 1628 m 1221 1634 l s 414 1640 m 413 1634 l s 1221 1634 m 1221 1640 l s
415
 1644 m 414 1640 l s 415 1646 m 415 1644 l s 1221 1640 m 1220 1646 l s 417 1652
 m 415 1646 l s 1220 1646 m 1220 1652 l s 418 1658 m 417 1652 l s 1220 1652 m
 1220 1658 l s 419 1664 m 418 1658 l s 1220 1658 m 1219 1664 l s 421 1670 m 419
 1664 l s 1219 1664 m 1219 1670 l s 421 1671 m 421 1670 l s 422 1676 m 421 1671
 l s 1219 1670 m 1219 1676 l s 424 1682 m 422 1676 l s 1219 1676 m 1218 1682 l
s
 426 1688 m 424 1682 l s 1218 1682 m 1218 1688 l s 427 1691 m 426 1688 l s 427
 1694 m 427 1691 l s 1218 1688 m 1217 1694 l s 429 1699 m 427 1694 l s 1217
1694
 m 1217 1699 l s 431 1705 m 429 1699 l s 1217 1699 m 1216 1705 l s 433 1709 m
 431 1705 l s 434 1711 m 433 1709 l s 1216 1705 m 1216 1711 l s 436 1717 m 434
 1711 l s 1216 1711 m 1215 1717 l s 438 1723 m 436 1717 l s 1215 1718 m 1215
 1723 l s 1215 1717 m 1215 1718 l s 439 1725 m 438 1723 l s 441 1729 m 439 1725
 l s 1215 1723 m 1214 1729 l s 443 1735 m 441 1729 l s 1214 1729 m 1213 1735 l
s
 445 1739 m 443 1735 l s 446 1741 m 445 1739 l s 1213 1735 m 1213 1741 l s 448
 1747 m 446 1741 l s 1213 1741 m 1212 1747 l s 451 1752 m 448 1747 l s 451 1753
 m 451 1752 l s 1212 1747 m 1211 1753 l s 454 1759 m 451 1753 l s 1211 1753 m
 1211 1759 l s 456 1764 m 454 1759 l s 457 1765 m 456 1764 l s 1211 1759 m 1210
 1765 l s 460 1771 m 457 1765 l s 1209 1769 m 1209 1771 l s 1210 1765 m 1209
 1769 l s 462 1775 m 460 1771 l s 463 1777 m 462 1775 l s 1209 1771 m 1208 1777
 l s 466 1782 m 463 1777 l s 1208 1777 m 1207 1782 l s 468 1786 m 466 1782 l s
 470 1788 m 468 1786 l s 1207 1782 m 1207 1788 l s 473 1794 m 470 1788 l s 1207
 1788 m 1206 1794 l s 474 1796 m 473 1794 l s 477 1800 m 474 1796 l s 1206 1794
 m 1205 1800 l s 480 1806 m 477 1800 l s 1205 1800 m 1204 1806 l s 480 1806 m
 480 1806 l s 484 1812 m 480 1806 l s 1203 1808 m 1203 1812 l s 1204 1806 m
1203
 1808 l s 486 1816 m 484 1812 l s 488 1818 m 486 1816 l s 1203 1812 m 1202 1818
 l s 492 1824 m 488 1818 l s 1202 1818 m 1200 1824 l s 492 1825 m 492 1824 l s
 496 1830 m 492 1825 l s 1200 1824 m 1199 1830 l s 498 1833 m 496 1830 l s 500
 1836 m 498 1833 l s 1199 1830 m 1198 1836 l s 504 1842 m 500 1836 l s 1197
1839
 m 1197 1842 l s 1198 1836 m 1197 1839 l s 504 1842 m 504 1842 l s 508 1848 m
 504 1842 l s 1197 1842 m 1196 1848 l s 510 1850 m 508 1848 l s 512 1854 m 510
 1850 l s 1196 1848 m 1194 1854 l s 516 1858 m 512 1854 l s 517 1860 m 516 1858
 l s 1194 1854 m 1193 1860 l s 521 1865 m 517 1860 l s 1193 1860 m 1192 1865 l
s
 522 1866 m 521 1865 l s 526 1871 m 522 1866 l s 1192 1867 m 1190 1871 l s 1192
 1865 m 1192 1867 l s 528 1873 m 526 1871 l s 531 1877 m 528 1873 l s 1190 1871
 m 1189 1877 l s 534 1880 m 531 1877 l s 536 1883 m 534 1880 l s 1189 1877 m
 1187 1883 l s 539 1888 m 536 1883 l s 541 1889 m 539 1888 l s 1187 1883 m 1186
 1889 l s 545 1895 m 541 1889 l s 546 1895 m 545 1895 l s 1186 1890 m 1184 1895
 l s 1186 1889 m 1186 1890 l s 551 1901 m 546 1895 l s 1184 1895 m 1183 1901 l
s
 551 1901 m 551 1901 l s 556 1907 m 551 1901 l s 1183 1901 m 1181 1907 l s 557
 1908 m 556 1907 l s 562 1913 m 557 1908 l s 1180 1911 m 1179 1913 l s 1181
1907
 m 1180 1911 l s 563 1914 m 562 1913 l s 567 1919 m 563 1914 l s 1179 1913 m
 1177 1919 l s 569 1921 m 567 1919 l s 573 1925 m 569 1921 l s 1177 1919 m 1176
 1925 l s 575 1927 m 573 1925 l s 579 1931 m 575 1927 l s 1174 1930 m 1174 1931
 l s 1176 1925 m 1174 1930 l s 581 1933 m 579 1931 l s 584 1937 m 581 1933 l s
 1174 1931 m 1172 1937 l s 587 1939 m 584 1937 l s 590 1943 m 587 1939 l s 1172
 1937 m 1170 1943 l s 593 1945 m 590 1943 l s 596 1948 m 593 1945 l s 1168 1948
 m 1167 1948 l s 1170 1943 m 1168 1948 l s 599 1951 m 596 1948 l s 603 1954 m
 599 1951 l s 1167 1948 m 1165 1954 l s 605 1956 m 603 1954 l s 609 1960 m 605
 1956 l s 1165 1954 m 1163 1960 l s 611 1962 m 609 1960 l s 615 1966 m 611 1962
 l s 1162 1963 m 1161 1966 l s 1163 1960 m 1162 1963 l s 617 1967 m 615 1966 l
s
 622 1972 m 617 1967 l s 1161 1966 m 1158 1972 l s 622 1973 m 622 1972 l s 628
 1978 m 622 1973 l s 629 1978 m 628 1978 l s 1156 1977 m 1156 1978 l s 1158
1972
 m 1156 1977 l s 634 1983 m 629 1978 l s 635 1984 m 634 1983 l s 1156 1978 m
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%%%%%%%%%%%%%%%%%%
%%% FIGURE 4 %%%%%
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/m23 {mp x y w2 sub m w2 w d w neg 0 d cl f} def
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 /m28 {mp x w2 sub y w2 sub w3 add m w3 0 d 0 w3 neg d w3 0 d 0 w3 d w3 0 d
 0 w3 d w3 neg 0 d 0 w3 d w3 neg 0 d 0 w3 neg d w3 neg 0 d cl s } def
 /m29 {mp gsave x w2 sub y w2 add w3 sub m currentpoint t
 4 {side} repeat cl fill gr} def
 /m30 {mp gsave x w2 sub y w2 add w3 sub m currentpoint t
 5 {side} repeat s gr} def /m31 {mp x y w2 sub m 0 w d x w2 sub y m w 0 d
 x w2 sub y w2 add m w w neg d x w2 sub y w2
 sub m w w d s} def
/m2 {mp x y w2 sub m 0 w d x w2 sub y m w 0 d s} def
/m5 {mp x w2 sub y w2 sub m w w d x w2 sub y w2 add m w w neg d s} def
/cl {closepath} def /sf {scalefont setfont} def
/box {m dup 0 exch d exch 0 d 0 exch neg d cl} def
/bl {box s} def /bf {box f} def
/oshow {gsave [] 0 sd true charpath stroke gr} def
/p {gsave 4 2 rl t exch r dup sw pop neg} def
/rs {p 0 m show gr} def /ors {p 0 m oshow gr} def
/cs {p 2 div 0 m show gr} def /ocs {p 2 div 0 m oshow gr} def
/reencsmalldict 24 dict def /ReEncodeSmall
{reencsmalldict begin /newcodesandnames exch def /newfontname exch def
/basefontname exch def /basefontdict basefontname findfont def
 /newfont basefontdict maxlength dict def basefontdict
{exch dup /FID ne {dup /Encoding eq {exch dup length array copy
newfont 3 1 roll put} {exch newfont 3 1 roll put} ifelse}
{pop pop} ifelse } forall
newfont /FontName newfontname put newcodesandnames aload pop
newcodesandnames length 2 idiv {newfont /Encoding get 3 1 roll put} repeat
newfontname newfont definefont pop end } def /accvec [
8#260 /agrave 8#265 /Agrave 8#276 /acircumflex 8#300 /Acircumflex
8#311 /adieresis 8#314 /Adieresis 8#321 /ccedilla 8#322 /Ccedilla
8#323 /eacute 8#324 /Eacute 8#325 /egrave 8#326 /Egrave
8#327 /ecircumflex 8#330 /Ecircumflex 8#331 /edieresis 8#332 /Edieresis
8#333 /icircumflex 8#334 /Icircumflex 8#335 /idieresis 8#336 /Idieresis
8#337 /ntilde 8#340 /Ntilde 8#342 /ocircumflex 8#344 /Ocircumflex
8#345 /odieresis 8#346 /Odieresis 8#347 /ucircumflex 8#354 /Ucircumflex
8#355 /udieresis 8#356 /Udieresis 8#357 /aring 8#362 /Aring
8#363 /ydieresis 8#364 /Ydieresis 8#366 /aacute 8#367 /Aacute
8#374 /ugrave 8#375 /Ugrave] def
/Times-Roman /Times-Roman accvec ReEncodeSmall
/Times-Italic /Times-Italic accvec ReEncodeSmall
/Times-Bold /Times-Bold accvec ReEncodeSmall
/Times-BoldItalic /Times-BoldItalic accvec ReEncodeSmall
/Helvetica /Helvetica accvec ReEncodeSmall
/Helvetica-Oblique /Helvetica-Oblique accvec ReEncodeSmall
/Helvetica-Bold /Helvetica-Bold accvec ReEncodeSmall
/Helvetica-BoldOblique /Helvetica-BoldOblique  accvec ReEncodeSmall
/Courier /Courier accvec ReEncodeSmall
/Courier-Oblique /Courier-Oblique accvec ReEncodeSmall
/Courier-Bold /Courier-Bold accvec ReEncodeSmall
/Courier-BoldOblique /Courier-BoldOblique accvec ReEncodeSmall
/Zone {/iy exch def /ix exch def gsave ix 1 sub 2224 mul 1 iy sub 3144
 mul t} def
gsave 20 28 t .25 .25 scale gsave
%%EndProlog
%%Page: number 1
 1 1 Zone
 gsave 0 0 t 0 0 0 c [] 0 sd 1 lw 1779 1778 222 683 bl 1616 1130 m 1618 1130 l
s
 1618 1130 m 1624 1130 l s 1624 1130 m 1630 1130 l s 1630 1130 m 1636 1130 l s
 1636 1130 m 1642 1130 l s 1642 1130 m 1648 1130 l s 1648 1130 m 1654 1130 l s
 1654 1130 m 1660 1130 l s 1660 1130 m 1666 1130 l s 1666 1130 m 1672 1130 l s
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 1707 1129 m 1713 1129 l s 1713 1129 m 1719 1129 l s 1719 1129 m 1725 1129 l s
 1725 1129 m 1731 1129 l s 1731 1129 m 1737 1129 l s 1737 1129 m 1743 1129 l s
 1743 1129 m 1749 1129 l s 1749 1129 m 1755 1129 l s 1755 1129 m 1761 1129 l s
 1761 1129 m 1767 1129 l s 1767 1129 m 1772 1129 l s 1772 1129 m 1778 1129 l s
 1778 1129 m 1784 1128 l s 1784 1128 m 1790 1128 l s 1790 1128 m 1796 1128 l s
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 1814 1128 m 1820 1128 l s 1820 1128 m 1826 1128 l s 1826 1128 m 1832 1128 l s
 1832 1128 m 1838 1128 l s 1838 1128 m 1844 1128 l s 1844 1128 m 1850 1128 l s
 1850 1128 m 1855 1128 l s 1855 1128 m 1861 1128 l s 1861 1128 m 1867 1128 l s
 1867 1128 m 1873 1128 l s 1873 1128 m 1879 1127 l s 1879 1127 m 1885 1127 l s
 1885 1127 m 1891 1127 l s 1891 1127 m 1897 1127 l s 1897 1127 m 1903 1127 l s
 1903 1127 m 1909 1127 l s 1909 1127 m 1915 1127 l s 1915 1127 m 1921 1127 l s
 1921 1127 m 1927 1127 l s 1927 1127 m 1933 1127 l s 1933 1127 m 1938 1127 l s
 1938 1127 m 1944 1127 l s 1944 1127 m 1950 1127 l s 1950 1127 m 1956 1127 l s
 1956 1127 m 1962 1127 l s 1962 1127 m 1968 1126 l s 1968 1126 m 1974 1126 l s
 1974 1126 m 1980 1126 l s 1980 1126 m 1986 1126 l s 1986 1126 m 1992 1126 l s
 1992 1126 m 1998 1126 l s 1087 1136 m 1091 1136 l s 1091 1136 m 1097 1136 l s
 1097 1136 m 1103 1136 l s 1103 1136 m 1109 1136 l s 1109 1136 m 1114 1136 l s
 1114 1136 m 1120 1136 l s 1120 1136 m 1126 1136 l s 1126 1136 m 1132 1136 l s
 1132 1136 m 1138 1136 l s 1138 1136 m 1144 1136 l s 1144 1136 m 1150 1136 l s
 1150 1136 m 1156 1136 l s 1156 1136 m 1162 1135 l s 1162 1135 m 1168 1135 l s
 1168 1135 m 1174 1135 l s 1174 1135 m 1180 1135 l s 1180 1135 m 1186 1135 l s
 1186 1135 m 1192 1135 l s 1192 1135 m 1197 1135 l s 1197 1135 m 1203 1135 l s
 1203 1135 m 1209 1135 l s 1209 1135 m 1215 1135 l s 1215 1135 m 1221 1135 l s
 1221 1135 m 1227 1135 l s 1227 1135 m 1233 1135 l s 1233 1135 m 1239 1135 l s
 1239 1135 m 1245 1135 l s 1245 1135 m 1251 1134 l s 1251 1134 m 1257 1134 l s
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 1363 1133 m 1369 1133 l s 1369 1133 m 1375 1133 l s 1375 1133 m 1381 1133 l s
 1381 1133 m 1387 1133 l s 1387 1133 m 1393 1133 l s 1393 1133 m 1399 1133 l s
 1399 1133 m 1405 1133 l s 1405 1133 m 1411 1133 l s 1411 1133 m 1417 1133 l s
 1417 1133 m 1423 1133 l s 1423 1133 m 1429 1132 l s 1429 1132 m 1435 1132 l s
 1435 1132 m 1441 1132 l s 1441 1132 m 1446 1132 l s 1446 1132 m 1452 1132 l s
 1452 1132 m 1458 1132 l s 1458 1132 m 1464 1132 l s 1464 1132 m 1470 1132 l s
 1470 1132 m 1476 1132 l s 1476 1132 m 1482 1132 l s 1482 1132 m 1488 1132 l s
 1488 1132 m 1494 1132 l s 1494 1132 m 1500 1132 l s 1500 1132 m 1506 1132 l s
 1506 1132 m 1512 1132 l s 1512 1132 m 1518 1131 l s 1518 1131 m 1524 1131 l s
 1524 1131 m 1529 1131 l s 1529 1131 m 1535 1131 l s 1535 1131 m 1541 1131 l s
 1541 1131 m 1547 1131 l s 1547 1131 m 1553 1131 l s 1553 1131 m 1559 1131 l s
 1559 1131 m 1565 1131 l s 1565 1131 m 1571 1131 l s 1571 1131 m 1577 1131 l s
 1577 1131 m 1583 1131 l s 1583 1131 m 1589 1131 l s 1589 1131 m 1595 1131 l s
 1595 1131 m 1601 1131 l s 1601 1131 m 1607 1130 l s 1607 1130 m 1612 1130 l s
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 599 1142 l s 599 1142 m 605 1142 l s 605 1142 m 611 1142 l s 611 1142 m 617
 1142 l s 617 1142 m 622 1142 l s 622 1142 m 628 1141 l s 628 1141 m 634 1141 l
 s 634 1141 m 640 1141 l s 640 1141 m 646 1141 l s 646 1141 m 652 1141 l s 652
 1141 m 658 1141 l s 658 1141 m 664 1141 l s 664 1141 m 670 1141 l s 670 1141 m
 676 1141 l s 676 1141 m 682 1141 l s 682 1141 m 688 1141 l s 688 1141 m 694
 1141 l s 694 1141 m 700 1141 l s 700 1141 m 705 1141 l s 705 1141 m 711 1141 l
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 1140 m 735 1140 l s 735 1140 m 741 1140 l s 741 1140 m 747 1140 l s 747 1140 m
 753 1140 l s 753 1140 m 759 1140 l s 759 1140 m 765 1140 l s 765 1140 m 771
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 1139 m 889 1139 l s 889 1139 m 895 1138 l s 895 1138 m 901 1138 l s 901 1138 m
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 1138 l s 925 1138 m 931 1138 l s 931 1138 m 937 1138 l s 937 1138 m 943 1138 l
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 984 1137 l s 984 1137 m 990 1137 l s 990 1137 m 996 1137 l s 996 1137 m 1002
 1137 l s 1002 1137 m 1008 1137 l s 1008 1137 m 1014 1137 l s 1014 1137 m 1020
 1137 l s 1020 1137 m 1026 1137 l s 1026 1137 m 1031 1137 l s 1031 1137 m 1037
 1137 l s 1037 1137 m 1043 1137 l s 1043 1137 m 1049 1137 l s 1049 1137 m 1055
 1137 l s 1055 1137 m 1061 1137 l s 1061 1137 m 1067 1137 l s 1067 1137 m 1073
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 1136 l s 225 1146 m 231 1146 l s 231 1146 m 237 1146 l s 237 1146 m 243 1146 l
 s 243 1146 m 249 1146 l s 249 1146 m 255 1146 l s 255 1146 m 261 1146 l s 261
 1146 m 267 1146 l s 267 1146 m 273 1146 l s 273 1146 m 279 1145 l s 279 1145 m
 285 1145 l s 285 1145 m 290 1145 l s 290 1145 m 296 1145 l s 296 1145 m 302
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 s 320 1145 m 326 1145 l s 326 1145 m 332 1145 l s 332 1145 m 338 1145 l s 338
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 362 1145 l s 362 1145 m 368 1144 l s 368 1144 m 373 1144 l s 373 1144 m 379
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 s 397 1144 m 403 1144 l s 403 1144 m 409 1144 l s 409 1144 m 415 1144 l s 415
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 439 1144 l s 439 1144 m 445 1144 l s 445 1144 m 451 1144 l s 451 1144 m 456
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 s 474 1143 m 480 1143 l s 480 1143 m 486 1143 l s 486 1143 m 492 1143 l s 492
 1143 m 498 1143 l s 498 1143 m 504 1143 l s 504 1143 m 510 1143 l s 510 1143 m
 516 1143 l s 516 1143 m 522 1143 l s 522 1143 m 528 1143 l s 528 1143 m 534
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 s 551 1142 m 557 1142 l s 557 1142 m 563 1142 l s 563 1142 m 564 1142 l s 231
 1998 m 225 1998 l s 237 1998 m 231 1998 l s 243 1998 m 237 1998 l s 249 1998 m
 243 1998 l s 255 1998 m 249 1998 l s 261 1998 m 255 1998 l s 267 1998 m 261
 1998 l s 273 1998 m 267 1998 l s 279 1999 m 273 1998 l s 285 1999 m 279 1999 l
 s 290 1999 m 285 1999 l s 296 1999 m 290 1999 l s 302 1999 m 296 1999 l s 308
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 320 1999 l s 332 1999 m 326 1999 l s 338 1999 m 332 1999 l s 344 1999 m 338
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 2000 m 379 2000 l s 391 2000 m 385 2000 l s 397 2000 m 391 2000 l s 403 2000 m
 397 2000 l s 409 2000 m 403 2000 l s 415 2000 m 409 2000 l s 421 2000 m 415
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 2001 m 456 2001 l s 468 2001 m 462 2001 l s 474 2001 m 468 2001 l s 480 2001 m
 474 2001 l s 486 2001 m 480 2001 l s 492 2001 m 486 2001 l s 498 2001 m 492
 2001 l s 504 2001 m 498 2001 l s 510 2001 m 504 2001 l s 516 2001 m 510 2001 l
 s 522 2001 m 516 2001 l s 528 2001 m 522 2001 l s 534 2001 m 528 2001 l s 539
 2002 m 534 2001 l s 545 2002 m 539 2002 l s 551 2002 m 545 2002 l s 557 2002 m
 551 2002 l s 563 2002 m 557 2002 l s 564 2002 m 563 2002 l s 569 2002 m 564
 2002 l s 575 2002 m 569 2002 l s 581 2002 m 575 2002 l s 587 2002 m 581 2002 l
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 2002 m 605 2002 l s 617 2002 m 611 2002 l s 622 2002 m 617 2002 l s 628 2003 m
 622 2002 l s 634 2003 m 628 2003 l s 640 2003 m 634 2003 l s 646 2003 m 640
 2003 l s 652 2003 m 646 2003 l s 658 2003 m 652 2003 l s 664 2003 m 658 2003 l
 s 670 2003 m 664 2003 l s 676 2003 m 670 2003 l s 682 2003 m 676 2003 l s 688
 2003 m 682 2003 l s 694 2003 m 688 2003 l s 700 2003 m 694 2003 l s 705 2003 m
 700 2003 l s 711 2003 m 705 2003 l s 717 2004 m 711 2003 l s 723 2004 m 717
 2004 l s 729 2004 m 723 2004 l s 735 2004 m 729 2004 l s 741 2004 m 735 2004 l
 s 747 2004 m 741 2004 l s 753 2004 m 747 2004 l s 759 2004 m 753 2004 l s 765
 2004 m 759 2004 l s 771 2004 m 765 2004 l s 777 2004 m 771 2004 l s 783 2004 m
 777 2004 l s 788 2004 m 783 2004 l s 794 2004 m 788 2004 l s 800 2004 m 794
 2004 l s 806 2005 m 800 2004 l s 812 2005 m 806 2005 l s 818 2005 m 812 2005 l
 s 824 2005 m 818 2005 l s 830 2005 m 824 2005 l s 836 2005 m 830 2005 l s 842
 2005 m 836 2005 l s 848 2005 m 842 2005 l s 854 2005 m 848 2005 l s 860 2005 m
 854 2005 l s 865 2005 m 860 2005 l s 871 2005 m 865 2005 l s 877 2005 m 871
 2005 l s 883 2005 m 877 2005 l s 889 2005 m 883 2005 l s 895 2006 m 889 2005 l
 s 901 2006 m 895 2006 l s 907 2006 m 901 2006 l s 913 2006 m 907 2006 l s 919
 2006 m 913 2006 l s 925 2006 m 919 2006 l s 931 2006 m 925 2006 l s 937 2006 m
 931 2006 l s 943 2006 m 937 2006 l s 948 2006 m 943 2006 l s 954 2006 m 948
 2006 l s 960 2006 m 954 2006 l s 966 2006 m 960 2006 l s 972 2006 m 966 2006 l
 s 978 2006 m 972 2006 l s 984 2007 m 978 2006 l s 990 2007 m 984 2007 l s 996
 2007 m 990 2007 l s 1002 2007 m 996 2007 l s 1008 2007 m 1002 2007 l s 1014
 2007 m 1008 2007 l s 1020 2007 m 1014 2007 l s 1026 2007 m 1020 2007 l s 1031
 2007 m 1026 2007 l s 1037 2007 m 1031 2007 l s 1043 2007 m 1037 2007 l s 1049
 2007 m 1043 2007 l s 1055 2007 m 1049 2007 l s 1061 2007 m 1055 2007 l s 1067
 2007 m 1061 2007 l s 1073 2008 m 1067 2007 l s 1079 2008 m 1073 2008 l s 1085
 2008 m 1079 2008 l s 1087 2008 m 1085 2008 l s 1091 2008 m 1087 2008 l s 1097
 2008 m 1091 2008 l s 1103 2008 m 1097 2008 l s 1109 2008 m 1103 2008 l s 1114
 2008 m 1109 2008 l s 1120 2008 m 1114 2008 l s 1126 2008 m 1120 2008 l s 1132
 2008 m 1126 2008 l s 1138 2008 m 1132 2008 l s 1144 2008 m 1138 2008 l s 1150
 2008 m 1144 2008 l s 1156 2008 m 1150 2008 l s 1162 2009 m 1156 2008 l s 1168
 2009 m 1162 2009 l s 1174 2009 m 1168 2009 l s 1180 2009 m 1174 2009 l s 1186
 2009 m 1180 2009 l s 1192 2009 m 1186 2009 l s 1197 2009 m 1192 2009 l s 1203
 2009 m 1197 2009 l s 1209 2009 m 1203 2009 l s 1215 2009 m 1209 2009 l s 1221
 2009 m 1215 2009 l s 1227 2009 m 1221 2009 l s 1233 2009 m 1227 2009 l s 1239
 2009 m 1233 2009 l s 1245 2009 m 1239 2009 l s 1251 2010 m 1245 2009 l s 1257
 2010 m 1251 2010 l s 1263 2010 m 1257 2010 l s 1269 2010 m 1263 2010 l s 1275
 2010 m 1269 2010 l s 1280 2010 m 1275 2010 l s 1286 2010 m 1280 2010 l s 1292
 2010 m 1286 2010 l s 1298 2010 m 1292 2010 l s 1304 2010 m 1298 2010 l s 1310
 2010 m 1304 2010 l s 1316 2010 m 1310 2010 l s 1322 2010 m 1316 2010 l s 1328
 2010 m 1322 2010 l s 1334 2010 m 1328 2010 l s 1340 2011 m 1334 2010 l s 1346
 2011 m 1340 2011 l s 1352 2011 m 1346 2011 l s 1358 2011 m 1352 2011 l s 1363
 2011 m 1358 2011 l s 1369 2011 m 1363 2011 l s 1375 2011 m 1369 2011 l s 1381
 2011 m 1375 2011 l s 1387 2011 m 1381 2011 l s 1393 2011 m 1387 2011 l s 1399
 2011 m 1393 2011 l s 1405 2011 m 1399 2011 l s 1411 2011 m 1405 2011 l s 1417
 2011 m 1411 2011 l s 1423 2011 m 1417 2011 l s 1429 2012 m 1423 2011 l s 1435
 2012 m 1429 2012 l s 1441 2012 m 1435 2012 l s 1446 2012 m 1441 2012 l s 1452
 2012 m 1446 2012 l s 1458 2012 m 1452 2012 l s 1464 2012 m 1458 2012 l s 1470
 2012 m 1464 2012 l s 1476 2012 m 1470 2012 l s 1482 2012 m 1476 2012 l s 1488
 2012 m 1482 2012 l s 1494 2012 m 1488 2012 l s 1500 2012 m 1494 2012 l s 1506
 2012 m 1500 2012 l s 1512 2012 m 1506 2012 l s 1518 2013 m 1512 2012 l s 1524
 2013 m 1518 2013 l s 1529 2013 m 1524 2013 l s 1535 2013 m 1529 2013 l s 1541
 2013 m 1535 2013 l s 1547 2013 m 1541 2013 l s 1553 2013 m 1547 2013 l s 1559
 2013 m 1553 2013 l s 1565 2013 m 1559 2013 l s 1571 2013 m 1565 2013 l s 1577
 2013 m 1571 2013 l s 1583 2013 m 1577 2013 l s 1589 2013 m 1583 2013 l s 1595
 2013 m 1589 2013 l s 1601 2013 m 1595 2013 l s 1607 2014 m 1601 2013 l s 1612
 2014 m 1607 2014 l s 1616 2014 m 1612 2014 l s 1618 2014 m 1616 2014 l s 1624
 2014 m 1618 2014 l s 1630 2014 m 1624 2014 l s 1636 2014 m 1630 2014 l s 1642
 2014 m 1636 2014 l s 1648 2014 m 1642 2014 l s 1654 2014 m 1648 2014 l s 1660
 2014 m 1654 2014 l s 1666 2014 m 1660 2014 l s 1672 2014 m 1666 2014 l s 1678
 2014 m 1672 2014 l s 1684 2014 m 1678 2014 l s 1689 2014 m 1684 2014 l s 1695
 2015 m 1689 2014 l s 1701 2015 m 1695 2015 l s 1707 2015 m 1701 2015 l s 1713
 2015 m 1707 2015 l s 1719 2015 m 1713 2015 l s 1725 2015 m 1719 2015 l s 1731
 2015 m 1725 2015 l s 1737 2015 m 1731 2015 l s 1743 2015 m 1737 2015 l s 1749
 2015 m 1743 2015 l s 1755 2015 m 1749 2015 l s 1761 2015 m 1755 2015 l s 1767
 2015 m 1761 2015 l s 1772 2015 m 1767 2015 l s 1778 2015 m 1772 2015 l s 1784
 2016 m 1778 2015 l s 1790 2016 m 1784 2016 l s 1796 2016 m 1790 2016 l s 1802
 2016 m 1796 2016 l s 1808 2016 m 1802 2016 l s 1814 2016 m 1808 2016 l s 1820
 2016 m 1814 2016 l s 1826 2016 m 1820 2016 l s 1832 2016 m 1826 2016 l s 1838
 2016 m 1832 2016 l s 1844 2016 m 1838 2016 l s 1850 2016 m 1844 2016 l s 1855
 2016 m 1850 2016 l s 1861 2016 m 1855 2016 l s 1867 2016 m 1861 2016 l s 1873
 2016 m 1867 2016 l s 1879 2017 m 1873 2016 l s 1885 2017 m 1879 2017 l s 1891
 2017 m 1885 2017 l s 1897 2017 m 1891 2017 l s 1903 2017 m 1897 2017 l s 1909
 2017 m 1903 2017 l s 1915 2017 m 1909 2017 l s 1921 2017 m 1915 2017 l s 1927
 2017 m 1921 2017 l s 1933 2017 m 1927 2017 l s 1938 2017 m 1933 2017 l s 1944
 2017 m 1938 2017 l s 1950 2017 m 1944 2017 l s 1956 2017 m 1950 2017 l s 1962
 2017 m 1956 2017 l s 1968 2018 m 1962 2017 l s 1974 2018 m 1968 2018 l s 1980
 2018 m 1974 2018 l s 1986 2018 m 1980 2018 l s 1992 2018 m 1986 2018 l s 1998
 2018 m 1992 2018 l s 222 683 m 222 2461 l s 256 683 m 222 683 l s 239 718 m
222
 718 l s 239 754 m 222 754 l s 239 789 m 222 789 l s 239 825 m 222 825 l s 256
 861 m 222 861 l s 239 896 m 222 896 l s 239 932 m 222 932 l s 239 967 m 222
967
 l s 239 1003 m 222 1003 l s 256 1038 m 222 1038 l s 239 1074 m 222 1074 l s
239
 1110 m 222 1110 l s 239 1145 m 222 1145 l s 239 1181 m 222 1181 l s 256 1216 m
 222 1216 l s 239 1252 m 222 1252 l s 239 1287 m 222 1287 l s 239 1323 m 222
 1323 l s 239 1359 m 222 1359 l s 256 1394 m 222 1394 l s 239 1430 m 222 1430 l
 s 239 1465 m 222 1465 l s 239 1501 m 222 1501 l s 239 1536 m 222 1536 l s 256
 1572 m 222 1572 l s 239 1608 m 222 1608 l s 239 1643 m 222 1643 l s 239 1679 m
 222 1679 l s 239 1714 m 222 1714 l s 256 1750 m 222 1750 l s 239 1785 m 222
 1785 l s 239 1821 m 222 1821 l s 239 1857 m 222 1857 l s 239 1892 m 222 1892 l
 s 256 1928 m 222 1928 l s 239 1963 m 222 1963 l s 239 1999 m 222 1999 l s 239
 2034 m 222 2034 l s 239 2070 m 222 2070 l s 256 2106 m 222 2106 l s 239 2141 m
 222 2141 l s 239 2177 m 222 2177 l s 239 2212 m 222 2212 l s 239 2248 m 222
 2248 l s 256 2283 m 222 2283 l s 239 2319 m 222 2319 l s 239 2355 m 222 2355 l
 s 239 2390 m 222 2390 l s 239 2426 m 222 2426 l s 256 2461 m 222 2461 l s
 /Times-Italic findfont 48 sf 178 667 0
(0) rs
 178 845 0
(0.1) rs
 178 1023 0
(0.2) rs
 178 1201 0
(0.3) rs
 178 1379 0
(0.4) rs
 178 1556 0
(0.5) rs
 178 1734 0
(0.6) rs
 178 1912 0
(0.7) rs
 178 2090 0
(0.8) rs
 178 2268 0
(0.9) rs
 178 2446 0
(1) rs 222 683 m 2001 683 l s 222 716 m 222 683 l s 261 699 m 261 683 l s 300
 699 m 300 683 l s 338 699 m 338 683 l s 377 699 m 377 683 l s 416 716 m 416
683
 l s 454 699 m 454 683 l s 493 699 m 493 683 l s 532 699 m 532 683 l s 570 699
m
 570 683 l s 609 716 m 609 683 l s 648 699 m 648 683 l s 686 699 m 686 683 l s
 725 699 m 725 683 l s 764 699 m 764 683 l s 802 716 m 802 683 l s 841 699 m
841
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m
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 1941 m 676 1941 l s 688 1941 m 682 1941 l s 694 1941 m 688 1941 l s 700 1941 m
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 1941 l s 723 1941 m 717 1941 l s 729 1941 m 723 1941 l s 735 1941 m 729 1941 l
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 1942 l s 800 1942 m 794 1942 l s 806 1942 m 800 1942 l s 812 1942 m 806 1942 l
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 1943 m 824 1943 l s 836 1943 m 830 1943 l s 842 1943 m 836 1943 l s 848 1943 m
 842 1943 l s 854 1943 m 848 1943 l s 860 1943 m 854 1943 l s 865 1943 m 860
 1943 l s 871 1943 m 865 1943 l s 877 1943 m 871 1943 l s 883 1943 m 877 1943 l
 s 889 1943 m 883 1943 l s 895 1943 m 889 1943 l s 901 1943 m 895 1943 l s 907
 1944 m 901 1943 l s 913 1944 m 907 1944 l s 919 1944 m 913 1944 l s 925 1944 m
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 1944 l s 948 1944 m 943 1944 l s 954 1944 m 948 1944 l s 960 1944 m 954 1944 l
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m
 996 1945 l s 1008 1945 m 1002 1945 l s 1014 1945 m 1008 1945 l s 1020 1945 m
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 1802 1955 l s 1814 1955 m 1808 1955 l s 1820 1955 m 1814 1955 l s 1826 1955 m
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s
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s
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s
 1571 820 m 1577 820 l s 1577 820 m 1583 820 l s 1583 820 m 1589 820 l s 1589
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s
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s
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s
 1802 818 m 1808 818 l s 1808 818 m 1814 818 l s 1814 818 m 1820 818 l s 1820
 818 m 1826 818 l s 1826 818 m 1832 818 l s 1832 818 m 1838 818 l s 1838 818 m
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s
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s
 1956 817 m 1962 817 l s 1962 817 m 1968 817 l s 1968 817 m 1974 817 l s 1974
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 753 828 m 759 828 l s 759 828 m 765 828 l s 765 828 m 771 828 l s 771 828 m
777
 828 l s 777 828 m 783 828 l s 783 828 m 788 828 l s 788 828 m 794 828 l s 794
 828 m 800 827 l s 800 827 m 806 827 l s 806 827 m 812 827 l s 812 827 m 818
827
 l s 818 827 m 824 827 l s 824 827 m 830 827 l s 830 827 m 836 827 l s 836 827
m
 842 827 l s 842 827 m 848 827 l s 848 827 m 854 827 l s 854 827 m 860 827 l s
 860 827 m 865 827 l s 865 827 m 871 827 l s 871 827 m 877 827 l s 877 827 m
883
 827 l s 883 827 m 889 827 l s 889 827 m 895 827 l s 895 827 m 901 827 l s 901
 827 m 907 826 l s 907 826 m 913 826 l s 913 826 m 919 826 l s 919 826 m 925
826
 l s 925 826 m 931 826 l s 931 826 m 937 826 l s 937 826 m 943 826 l s 943 826
m
 948 826 l s 948 826 m 954 826 l s 954 826 m 960 826 l s 960 826 m 966 826 l s
 966 826 m 972 826 l s 972 826 m 978 826 l s 978 826 m 984 826 l s 984 826 m
990
 826 l s 990 826 m 996 826 l s 996 826 m 1002 826 l s 1002 826 m 1008 826 l s
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s
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s
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s
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s
 1316 823 m 1322 823 l s 1322 823 m 1328 823 l s 1328 823 m 1334 823 l s 1334
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237
 833 m 243 833 l s 243 833 m 249 833 l s 249 833 m 255 833 l s 255 833 m 261
832
 l s 261 832 m 267 832 l s 267 832 m 273 832 l s 273 832 m 279 832 l s 279 832
m
 285 832 l s 285 832 m 290 832 l s 290 832 m 296 832 l s 296 832 m 302 832 l s
 302 832 m 308 832 l s 308 832 m 314 832 l s 314 832 m 320 832 l s 320 832 m
326
 832 l s 326 832 m 332 832 l s 332 832 m 338 832 l s 338 832 m 344 832 l s 344
 832 m 350 832 l s 350 832 m 356 832 l s 356 832 m 362 832 l s 362 832 m 368
831
 l s 368 831 m 373 831 l s 373 831 m 379 831 l s 379 831 m 385 831 l s 385 831
m
 391 831 l s 391 831 m 397 831 l s 397 831 m 403 831 l s 403 831 m 409 831 l s
 409 831 m 415 831 l s 415 831 m 421 831 l s 421 831 m 427 831 l s 427 831 m
433
 831 l s 433 831 m 439 831 l s 439 831 m 445 831 l s 445 831 m 451 831 l s 451
 831 m 456 831 l s 456 831 m 462 831 l s 462 831 m 468 831 l s 468 831 m 474
830
 l s 474 830 m 480 830 l s 480 830 m 486 830 l s 486 830 m 492 830 l s 492 830
m
 498 830 l s 498 830 m 504 830 l s 504 830 m 510 830 l s 510 830 m 516 830 l s
 516 830 m 522 830 l s 522 830 m 528 830 l s 528 830 m 534 830 l s 534 830 m
539
 830 l s 539 830 m 545 830 l s 545 830 m 551 830 l s 551 830 m 557 830 l s 557
 830 m 563 830 l s 563 830 m 569 830 l s 569 830 m 575 830 l s 575 830 m 581
829
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