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\hfill {WM-96-105}



\hfill {May 1996}

\vskip 1in   \baselineskip 24pt
{

   
   \bigskip
   \centerline{\bf $K_L\rightarrow \pi^o\nu\overline{\nu}$ in Extended Higgs Models }
 \vskip .8in
 
\centerline{Carl E. Carlson, Greg D. Dorata and Marc
Sher } 
\bigskip
\centerline {\it Physics Department, College of William and
Mary, Williamsburg, VA 23187, USA}

\vskip 1in
 
{\narrower\narrower The decay $K_L\rightarrow \pi^o\nu\overline{\nu}$ is an 
excellent probe of the nature of CP
violation.  It is almost entirely CP-violating, and hadronic uncertainties are negligible. 
Experiments which hope to detect the decay are currently being planned. We calculate the
decay rate in several extensions of the standard model Higgs sector, including the
Liu-Wolfenstein two-doublet model of spontaneous CP-violation and the Weinberg
three-doublet model. In a model with an extra doublet, with CP-violation arising from the CKM
sector, the rate can increase by up to 50\%.  However, in models in which the CP
violation arises either entirely or predominantly  from the Higgs sector, we find that the
decay rate is much smaller than that of the standard model, unless parameters of the model are
fine-tuned.}


\newpage
 


 


\section{Introduction}

One of the deepest mysteries in theoretical physics concerns the nature 
and origin of the CP violation observed in the kaon system.   Although 
it can be accommodated within the three generation standard model, most 
extensions of the standard model contain additional sources of CP 
violation\cite{jarlskog}.  A primary motivation for the construction of 
B-factories is to explore CP violation in a regime in which it is 
expected to be considerably larger.

Within the standard model, much of the effort in understanding CP 
violation has focused on finding the values of the CKM matrix, which are 
parameterized by the Wolfenstein parameterization\cite{wolf,blg}, in which one has
\begin{equation}
|V_{us}|=\lambda;\quad |V_{cb}|=A\lambda^2;\quad 
V_{ub}=A\lambda^3(\rho-i\eta);
\quad V_{td}=A\lambda^3(1-\overline{\rho}-i\overline{\eta})
\end{equation}
where $\overline{\rho}\equiv \rho(1-\lambda^2/2)$ and 
$\overline{\eta}\equiv \eta(1-\lambda^2/2)$.  From K- and B- decays, 
these parameters can be determined.
Unfortunately, the interpretations of many current measurements of CP 
violation in the kaon system, as well as of many future measurements in 
the B system, are plagued by theoretical uncertainties.  These result 
from the absence of precise non-perturbative calculations of hadronic 
matrix elements.  For example, determination of $V_{cb}$ and  $V_{ub}$ 
to an accuracy of better than 5\% and 10\% respectively may not be 
possible without a significant improvement in the determination of 
hadronic matrix elements.  In addition, loop-induced decays also contain 
significant theoretical uncertainties, which affect the predictions (and 
interpretations) for $\epsilon'/\epsilon$,
$B^0-\overline{B}^0$ mixing, etc.   As emphasized by Buras {\it et 
al.}\cite{burasa}, even with optimistic assumptions about the theoretical 
and experiment errors, it will be difficult to achieve an accuracy 
better than $\pm 0.15$ in $\rho$ and $\pm 0.05$ in $\eta$.

As also emphasized by Buras and others\cite{burasb}, there are two processes 
in which the hadronic uncertainties are significantly reduced, and two 
processes in which they are virtually absent.  The former two are 
$K^+\rightarrow \pi^+\nu\overline{\nu}$ and the ratio of 
$B^0_d-\overline{B}_d^0$ mixing to $B^0_s-\overline{B}_s^0$ mixing.  The 
gold-plated decays, in which theoretical uncertainties are extremely 
small, are the CP asymmetry in $B^0_d \rightarrow \psi K_s$ and the 
decay $K_L\rightarrow \pi^o\nu\overline{\nu}$.   Buras\cite{burasa} has 
noted that measurement of the CP asymmetry in $B^0_d \rightarrow \psi K_s$ plus a 
measurement of the branching ratio for $K_L\rightarrow 
\pi^o\nu\overline{\nu}$ would allow a determination of all of the 
elements of the CKM matrix without any significant hadronic 
uncertainties, assuming that the CP violation is entirely in the CKM matrix.
  In this paper, we will be concentrating on the mode 
$K_L\rightarrow \pi^o\nu\overline{\nu}$, which (up to $O(\epsilon)$ 
corrections) is entirely CP-violating and free of substantial hadronic 
uncertainties.

The expected branching ratio for $K_L\rightarrow \pi^o\nu\overline{\nu}$ 
in the standard model is approximately $3\times 10^{-11}$.  This is many 
orders of magnitude smaller than the current upper bound of 
$5.8\times 10^{-5}$\cite{current}.  However, upcoming experiments are expected to 
improve the bound to $10^{-8}$, and preliminary studies at CEBAF and 
BNL\cite{private} 
 claim  that, not only could $K_L\rightarrow \pi^o\nu\overline{\nu}$ be 
detected, but  as many as 100 events could be seen.  Although these 
studie
s are only in a very preliminary stage, a 10\% measurement of the 
branching ratio does not appear to be impossible within the next decade.

As discussed above, virtually all extensions of the standard model 
contain additional sources of CP violation.  One might expect the 
branching ratio for $K_L\rightarrow \pi^o\nu\overline{\nu}$ to be 
different in these models.  Although the branching ratio has been 
calculated in the standard model, including QCD 
corrections\cite{burasc}, we know of no calculations of the branching 
ratio in models in which CP violation arises from a source other than
the CKM matrix.

Given the potential precision of a measurement of $K_L\rightarrow 
\pi^o\nu\overline{\nu}$, and the likelihood of additional sources of CP 
violation in extensions of the standard model, it is important to 
calculate the branching ratio in these extensions.  Even if it is some 
time before the necessary precision is reached, one should still look at 
the branching ratio in extensions of the standard model in the hope that 
some models might have a significantly higher rate--this might motivate 
``intermediate" experiments which might not reach the standard model 
rate.  For example, the electric dipole moment of the neutron is very 
small in the standard model, and is not in reach of experiments, but 
extensions of the model can have a much larger rate, and this has 
provided strong motivation for experiments which have lowered the bound 
substantially (ruling out several models in the process).

In this paper, we will calculate the rate for $K_L\rightarrow 
\pi^o\nu\overline{\nu}$ in models with an extended Higgs sector.  Such 
models are the simplest extensions of the standard model, and have 
additional sources of CP violation.  Models with additional gauge 
groups, such as left-right models, and supersymmetric models, are 
currently under investigation.

In the next section, we will review the standard model result
for $K_L\rightarrow \pi^o\nu\overline{\nu}$, and then consider
the simplest extension of the standard model, in which a single
Higgs doublet is added to the standard model, and yet all of the
CP violation still arises from the CKM matrix.  In Section 3, the
most general two-doublet model in which CP is violated
spontaneously will be considered, along with the Weinberg three-doublet model; in one
subsection, the effects of neutral Higgs bosons will be considered and in the next
subsection, the effects of charged Higgs bosons will be included.  Finally,  in
Section IV, we present our conclusions.

\section{The standard model and simplest extension}

The calculation of $K_L\rightarrow \pi^o\nu\overline{\nu}$ amounts
to determining the coefficient of the effective Lagrangian for
$d\overline{s}
\rightarrow \nu\overline{\nu}$, and evaluating the hadronic matrix element.
The matrix element will be the same as that in semileptonic $K_L$ decay,
and thus in the ratio of the rate for $K_L\rightarrow \pi^o\nu\overline{\nu}$ to that of
the semileptonic decay, the matrix element will cancel.  There are two types of diagrams which
contribute to this effective Lagrangian.  The first are Z-penguins, generated by an induced
$d\overline{s}Z$ coupling, and are shown in Figure 1. The second consist of box diagrams, shown
in Figure 2.

Inami and Lim\cite{il} have calculated these contributions, in the limit that
external masses and momenta are much smaller than the internal masses.
The amplitudes are then described by an effective four-fermion interaction:
The effective Lagrangian is given by the form
\begin{equation}
{\cal{L}}_{eff}=-{G_F\over\sqrt{2}}{\alpha\over 4\pi\sin^2\theta_W} V^*_{ts}V_{td}
\left(
4\ D\overline{s}_L\gamma_{\mu}d_L\sum_{i=1}^3 \nu_{L_i}\gamma^{\mu}\nu_{L_i}
\right)
\end{equation}
where the sum is over the three neutrino flavors.
  Using unitarity,
Inami and Lim show that one can calculate the contribution due to the top
quark, and then (ignoring the up and charm quark masses) subtract  the
mass independent part, so that only the CKM matrix elements involving the top quark
enter.
In the CP-violating decay, $K_L\rightarrow\pi^o\nu\overline{\nu}$, the
imaginary part of ${\cal L}_{eff}$ will enter.

Including the contribution of the box diagrams (in the limit that lepton masses
are ignored compared to the top),
\begin{equation}
D(x_t) = {x_t\over 4}\bigg[{3x_t-6\over (1-x_t)^2}\ln\ x_t + {x_t+2\over x_t-1}\bigg]
\end{equation}
where $x_t\equiv m_t^2/M_W^2$.
The ratio of branching ratios is then
\begin{equation}
{B(K_L\rightarrow \pi^o\nu\overline{\nu})\over B(K^+\rightarrow \pi^oe^+\nu)}
= {3\over 4\pi^4}{\tau_{K_L}\over \tau_{K^+}}\left(G_F^2m_W^4\right)^2
A^4\lambda^8\eta^2D(x_t)^2
\end{equation}
which gives a branching ratio of $\sim 3\times 10^{-11}$ in the standard model (using
$\eta-0.35\
$(see ref.
\cite{burasa}).




We begin our consideration of extensions beyond the standard model by looking
at the simplest extension:  the two-doublet model, in which CP violation
occurs through the CKM sector.  In this case, the rate will also depend on 
the imaginary part of $V_{ts}^*V_{td}$, and the only change will be the addition
of charged Higgs loops in Figure 1.



In this simplest extension, one Higgs doublet couples to
one quark charge, and the other couples to the other
quark charge.  The detailed vertices and Lagrangian are
well-known\cite{hhg} (and can be obtained from the $\xi=0$ limit of the model
discussed in the next section).  The neutral
Higgs boson interactions are flavor-conserving, and thus will not
contribute to the diagrams of Fig. 1.  The only
difference is that we now have physical charged Higgs
bosons in the loop instead of just W and Goldstone bosons.
The charged Higgs bosons appear in diagrams (a), (b),
(d) and (h) (note that there is no ZWH vertex in the
model).  The divergences in these diagrams cancel, and we
find that the ratio of the contribution of charged Higgs
boson loops to the amplitude relative to the standard model
result, $R$ is \begin{equation} R=
-{1\over 4}\cot^2\beta{(1-x_t)^2\over(1-x)^2}\left(
{
(x(4-x)-2x^2\cos 2\theta_W)\ln\ x\ +\ x(1-x)(3-2\cos 2\theta_W)
\over
(3x_t-6)\ln\ x_t\ -\ (1-x_t)(2+x_t)}\right)
\end{equation}
where $x\equiv (m_t/m_{H^+})^2$ and 
 $\tan\beta$ is the ratio of the vacuum
expectation values of the two Higgs doublets (in most unified models it is
 greater than unity, and must be greater than
$0.5$ for perturbation theory to be valid).  For a charged Higgs mass of $150\
(250,400)$ GeV, the ratio is $R=.32\ (.20,.12)$ times $\cot^2\beta$.  Thus, for $\tan\beta$
near unity, this can increase the branching ratio by a factor of $1.74$ for a charged Higgs
mass of $150$ GeV.  It should be noted that this model has a lower bound on the charged Higgs
mass arising from $b\rightarrow s\gamma$ of $200$  GeV\cite{grant}, which gives an increase in
the branching ratio of approximately $50\%$ (for $\tan\beta\sim 1$).

Belanger et al.\cite{belanger}  have also considered the rate for $K_L
\rightarrow \pi^o\nu\overline{\nu}$ in this model .  Their results
are consistent with ours.  They note that the ratio of the rates
is given by $(1+R)^2Q$, where $Q$ is the ratio of the CKM parameters $(A^4\eta^2)$
as determined  from
experiments {\it including} the effects of the charged Higgs to the values of
these parameters as determined from experiments in the standard model (without the charged
Higgs).   The value of $Q$ is consistent with unity, since no discrepancy with the
standard model is seen.  However, by scanning parameter-space, and
requiring all experimental results to be within the 90\% confidence
level, they show that there is a region of parameter-space in which the
value of
$Q$ can be somewhat larger,  leading to a larger rate.  Our philosophy is that this
involves charged Higgs effects in  experiments other than $K_L
\rightarrow \pi^o\nu\overline{\nu}$, and that by the time the experiment
is done, the uncertainties in $(A^4\eta^2)$ will be much smaller, in the
range of 10 percent\cite{burasa}.  Nonetheless, one should be aware that
the extraction of the CKM angles in this model may give results different
from those in the standard model.

\section{Spontaneous CP-violation}

Another attractive mechanism for CP-violation is
spontaneous CP-violation\cite{lee}.  This cannot occur in
the single Higgs model, and thus requires extension of
the Higgs sector.  If one adds one more Higgs doublet,
then one can violate CP spontaneously, but at the cost of
tree level flavor-changing neutral currents(FCNC).  The
discrete symmetry that is usually implemented to
eliminate such currents will also eliminate the
spontaneous CP violation\cite{branco}.  One has two
choices:  break the discrete symmetry by parameters which
are sufficiently small that FCNC are not
phenomenologically problematic, or keep the discrete
symmetry and enlarge the Higgs sector by adding the third
doublet.  The former option was analyzed in detail by Liu
and Wolfenstein\cite{lw}, the latter is the model of
Weinberg\cite{weinberg}.  We first consider the Liu-Wolfenstein model.

The model contains two Higgs doublets, and the most
general CP-invariant Yukawa coupling and Higgs potential
is
\begin{eqnarray}
-{\cal{L}}_Y&=&\overline{\Psi}_{Li}^o(F_{ij}\tilde{\Phi}_2+
\xi F'_{ij}\tilde{\Phi}_1)U^o_{Rj}+
\overline{\Psi}^o_{Li}(G_{ij}\Phi_1+\xi
G'_{ij}\Phi_2)D^o_{Rj}+{\rm h.c.},\cr\cr
V&=&-\mu^2_1\Phi_1^\dagger\Phi_1-\mu^2_2\Phi_2^\dagger\Phi_2
+\lambda_1(\Phi_1^\dagger\Phi_1)^2+\lambda_2(\Phi_2^\dagger
\Phi_2)^2\cr
&+& \lambda_3(\Phi^\dagger_1\Phi_1)(\Phi^\dagger_2\Phi_2)
+\lambda_4(\Phi^\dagger_1\Phi_2)(\Phi^\dagger_2\Phi_1)
+{1\over 2}\lambda_5\left[(\Phi^\dagger_1\Phi_2)^2
+(\Phi^\dagger_2\Phi_1)^2\right]\cr
&+&{1\over
2}\xi'(\Phi^\dagger_1\Phi_2+\Phi^\dagger_2\Phi_1)(\lambda_6
\Phi^\dagger_1\Phi_1+\lambda_7\Phi^\dagger_2\Phi_2)
\end{eqnarray}

Here, $\xi$ and $\xi'$ are small parameters which
determine the amount by which the discrete symmetry
($\Phi_2\leftrightarrow -\Phi_2,\ D^o_R\leftrightarrow 
-D^o_R$) which eliminates FCNC is broken. The fact that
both Higgs doublets couple to all of the fermions ensures
the existence of FCNC, since diagonalizing the quark mass
matrix will not automatically diagonalize the Yukawa
coupling matrices.   Minimizing the potential yields
\begin{equation} \langle\Phi_1\rangle=\sqrt{1\over
2}\left({0\atop v_1}\right),\qquad
\langle\Phi_2\rangle=\sqrt{1\over 2}\left( {0\atop
v_2e^{i\alpha}}\right). \end{equation} The CP-violating
phase $\alpha$ is given by \begin{equation} \cos\
\alpha=-\xi'{\lambda_6v_1^2+\lambda_7v^2_2\over
4\lambda_5v_1v_2}. \end{equation} 

Liu and Wolfenstein discuss two limiting cases.  If
$\xi=0, \xi'\neq 0$, then the model becomes an earlier
model of Branco and Rebelo\cite{rebelo}.    Here,
CP-violation occurs in the Higgs sector, however, there
are no FCNC at tree level, and thus in order to obtain a
$\Delta S=2$ CP-violation one must go to two loops. 
As a result, the value of $\epsilon$ is too small.
The second case is if $\xi'=0, \xi\neq 0$, then the
CP-violating phase is $\pi/2$.  As Liu and Wolfenstein discuss, spontaneous
CP violation in this limit is the same as introducing a
purely imaginary Yukawa coupling $i\xi$ which breaks the
discrete symmetry. Although this model is certainly
viable, there is no natural mechanism for ensuring
$\xi'=0$, although they use this limit in their numerical
examples, as will we.

In this model, there will be  contributions to
the $K_L\rightarrow \pi^o\nu\overline{\nu}$ rate 
from charged Higgs loops (as in the simple model in the
last section, albeit with very different couplings), as
well as from neutral Higgs loops.  Since CP violation has
a different origin in this model, one might hope to avoid
the $V_{ts}^*V_{td}$ suppression factor present in the
standard model result.  

\subsection{Neutral Higgs bosons}

We will first consider
effects of neutral Higgs bosons.  Since the neutrinos are
very light, their interactions with Higgs bosons will be
negligible, and thus box diagrams will not contribute.  We
have only corrections to the $\overline{s}dZ$ vertex, and
the internal fermion line will be a $b$-quark, rather
than a top quark.   It is clear that we will need two
flavor-changing neutral current couplings, so the result
will be proportional to $\xi^2$.   

The flavor-changing Yukawa couplings can be found from the
Yukawa terms in Eq. 6.   The couplings of the neutral
complex fields, $\phi_1$ and $\phi_2$, to down-type quarks are given by
\begin{equation}
-{\cal
L}_Y=\overline{D'}_{Li}(G_{ij}\phi_1+\xi
G'_{ij}\phi_2)D'_{Rj}+{\rm h.c.}\end{equation}
where the primes indicate the weak eigenstate basis.
Plugging in $v_1/\sqrt{2}$ and $v_2e^{i\alpha}/\sqrt{2}$ for
the vacuum expectation values, and defining $D'_R\rightarrow
e^{-i\alpha} D'_R$ yields the mass matrix
\begin{equation}
M_d={1\over\sqrt{2}}(G+e^{-i\alpha}\xi{v_2\over
v_1}G')v_1\equiv M_d^o+e^{-i\alpha}\xi{v_2\over v_1}M_d'
\end{equation}
where flavor indices have been suppressed.
There are three neutral physical Higgs fields and one neutral Goldstone boson, which
we denote by
$H_j$, with
$j=1-4$ where $H_4$ is the Goldstone boson (the
calculation is done in the Feynman gauge, so the Goldstone
boson mass is the Z-boson mass).   To rotate to the fermion
mass eigenstate basis, we need to define
\begin{equation}
N\equiv V_LM'_{d}V_R^\dagger\end{equation}
where $V_{L,R}$ rotate $D'_L$ and $D'_R$ into their mass
eigenstates $D_L$ and $D_R$.
We then find that the general flavor-changing Yukawa coupling of
$\overline{D}_{Li}D_{Rj}H_k$ is given by
\begin{equation}
i{(\sqrt{2}G_F)^{1\over 2}\over \cos^2\beta}
\xi\overline{D_i}\left[ e^{i\alpha}N_{ij}(S_{2k}+iS_{4k})R
+e^{-i\alpha}N_{ji}^*(S_{2k}-iS_{4k})L\right]D_jH_k
\end{equation}
Here, $L$ and $R$ are ${1\over 2}(1\mp\gamma_5)$,
$\tan\beta\equiv v_2/v_1$ and $S_{ij}$ is the matrix which
diagonalizes the $4\times 4$ Higgs mass matrix.  $S_{ij}$
depends on parameters in the Higgs potential and is
essentially undetermined. Note that if $\xi'=0$, then the
$4\times 4$ matrix divides into two $2\times 2$ matrices (the
scalar and pseudoscalar matrices, respectively), and then
either $S_{2k}$ or $S_{4k}$ will vanish, greatly simplifying
the vertex.


Due to the proliferation of parameters, we will
 now greatly simplify the calculation by taking
the special case $\xi'=0$, as was done by Liu and
Wolfenstein.  There is a potential delicacy with that
limit.  If the Lagrangian is CP-invariant (i.e. all of
the CP-violation arises spontaneously), then only $\xi$
and $\xi'$ can violate CP.  Any effect proportional to
$\xi^2$ only will then not violate CP (as discussed
above, $\xi'=0$ is equivalent to multiplying $\xi$ by $i$).
However, one can certainly have a model in which there
is both explicit {\it and} spontaneous CP violation,
thus the $N$ matrices need not be real. In that case, our
results will not be significantly affected by this
assumption. Even if one assumes that the Lagrangian is
CP-invariant, and relaxes the $\xi'=0$ assumption, then there will be terms of
$O(\xi^2\xi')$, as well as $O(\xi^3)$, which do violate CP; these terms will be
$O(\xi')$ or $O(\xi)$ times terms that we will calculate.  In that particular case,
under the
assumption that the Lagrangian is CP-invariant, our numerical
results would be somewhat larger than the actual result
(note that there are no real bounds on the size of $\xi'$
other than it is ``small").  We will discuss the
implications of $\xi'\neq 0$ later.

The neutral Higgs loops contribute to diagrams (a), (b) and
(d) in Fig. 1., in which the $G^-$ is replaced by a neutral Higgs (and the
$u_i$ fermion is replaced by a $d_i$; the leading contribution will come from internal
$b$-quarks.  Note that under the assumption
$\xi'=0$, the scalars and pseudoscalars decouple, and the Z boson only
couples to a scalar plus a pseudoscalar.  As a result,
diagram (h) doesn't contribute to the vector
$\overline{s}\gamma_\mu d$ effective Lagrangian. In addition, the need for two
flavor-changing neutral current vertices implies that both fermion vertices must
involve a Higgs boson, and thus diagrams (f) and (g) will not contribute. A further
simplification, for the sake of illustration, can be made by taking all of the neutral scalars
to have the same mass as the Goldstone boson, i.e. $M_Z$--we will discuss the results of
relaxing this assumption shortly. In that case, the resulting sum over the four Higgs boson
contributions just becomes $\sum_k (S_{2k}^2+S_{4k}^2)$, which is $2$.  The effective
Lagrangian from these loops is found to be \begin{equation}
{\cal L}={G_F\over
4\sqrt{2}\cos^4\beta}{\alpha \over 4\pi\sin^2\theta_w}T_1 
\overline{s}\gamma_\mu d \overline{\nu}\gamma^\mu \nu
\end{equation}
where 
\begin{equation}
T_1=\xi^2
{N_{sb}N^*_{db}-N^*_{bs}N_{bd}\over m^2_W}\left(
{x_b(4-x_b)\ln(x_b)\over (1-x_b)^2}+{3x_b\over 1-x_b}\right)\end{equation}
and $x_b\equiv m_b^2/m^2_H$.  Note that if we relax the
assumption that the Higgs masses will be the Z mass, but
still assume that they are degenerate, then this result will
hold except for a slightly different contribution from the
Goldstone boson.  One expects, of course, the lightest of
the Higgs bosons to give the biggest contribution. 

Using this result, we can find the ratio of amplitudes, ${\cal A}$ for 
$K_L\rightarrow \pi^o\nu\overline{\nu}$ in this model to that in the standard
model. This gives

\begin{equation}
{{\cal A}_{new}\over {\cal A}_{SM}}=
{\xi^2\over\cos^4\beta}{{\rm Im}(N_{sb}N^*_{db}-N^*_{bs}N_{bd})
\over m^2_b}\quad T_2
\end{equation}
where
\begin{equation}
T_2\equiv {m^2_b\over 2m^2_W}{1\over {\rm Im} (V^*_{ts}V_{td})}
{x_b\over x_t}{[{(4-x_b)\ln(x_b)\over (1-x_b)^2}+{3\over (1-x_b)}
]\over [ {(3x_t-6)\over (1-x_t)^2}-{(2+x_t)\over (1-x_t)}]}
\end{equation}
Using the Wolfenstein parametrization, the ${\rm Im}V^*_{ts}V_{td}
$ term is $A\eta\lambda^5$, which is (for $\eta\simeq .35$) $1.8\times
10^{-4}$.  Using neutral Higgs masses of $m_Z$, as discussed
earlier, we find
\begin{equation}
{{\cal A}_{new}\over {\cal A}_{SM}}=
.06\ {\xi^2\over\cos^4\beta}{{\rm Im}(N_{sb}N^*_{db}-N^*_{bs}N_{bd}
)\over m^2_b}\end{equation}
At first sight, it appears that this ratio could be quite large. In
virtually all models, the value of $\tan\beta$ ranges from unity to
$m_t/m_b\sim 35$.  At the upper end of the range, $\cos^4\beta$ can
be as small as $10^{-6}$.  If $\xi\sim 0.1$, and the $N$ matrix elements
are the size of the largest mass scale expected ($m_b$), then the
ratio could be several hundred, leading to a rate as much as five orders
of magnitude greater than the standard model rate.

However, the value of $\xi N$ is not arbitrary.  It contributes to $\epsilon$
and thus is constrained. Liu and Wolfenstein have calculated the neutral Higgs contribution to
$\epsilon$.  In the two-generation case, they find,  taking the Higgs scalar masses to be
100 GeV,
\begin{equation}
{\xi^2\over\cos^4\beta}={2\times 10^{-3}\over\cos^{2/3}\beta\sin^{2/3}\beta}
\left({1\over(\sigma+\sigma')^{2/3}}{m_dm_s\over(N_{12}-N_{21})^{4/3}(N_{12}+N_{21})^{2/3}}\right)
\end{equation}
where one writes $N_{ij}$ in terms of its real and imaginary parts: 
$N_{ij}=N'_{ij}+i\xi\tan\beta\ n_{ij}$ and defines
\begin{equation}
\sigma\equiv-{n_{12}+n_{21}\over N'_{12}+N'_{21}}\qquad\sigma'\equiv{n_{21}-n_{12}\over 
N'_{12}+N'_{21}}.
\end{equation}
Since physical quantities can only depend on the  product $\xi N$, the expressions for
$\sigma$ and $\sigma'$ depend on a particular convention.  Liu and Wolfenstein scale $\xi$ by
assuming that
$N_{12}-N_{21}= m_s\sin\theta_c\simeq \sqrt{m_dm_s}$. With this convention, they  argue
that the natural values of
$\sigma$ and
$\sigma'$ are of
$O(1)$, and that if one assumes that the $N$ matrices have the same structure as the quark mass
matrices, then all of the terms in parentheses in Eq. (18) should be of $O(1)$.  Writing the
terms in parentheses ar $A'$, we can then write (with this convention)


\begin{equation}
{{\cal A}_{new}\over {\cal A}_{SM}}= 1.2\times 10^{-4}\ A' {1\over \cos^{2/3}\beta
\sin^{2/3}\beta}{{\rm Im}(N_{sb}N^*_{db}-N^*_{bs}N_{bd})\over m^2_b}
\end{equation}
Of course, the expression in Eq. (18) is only valid in the two-generation case.  In the
general case, the expression in parentheses will be much more complicated.  Nonetheless, the
result in Eq. (20) will be unaltered, and one still also expect the value of $A'$ to be $O(1)$.

Even if $\tan\beta\sim m_t/m_b$, this ratio will be no greater than one percent,
and thus unmeasurable.  The only way to get a large rate would be to assume that
either $A'$ is much greater than unity (which requires extensive fine-tuning)
or that the off-diagonal terms in the $N$ matrix are much larger than the
largest mass scale in the down-quark sector.
Neither of these seems likely.  In addition, the requirement that Higgs
mediated $B-\overline{B}$ mixing not be too large gives strong constraints\cite
{yao}
on $N_{bd}$, which we find to be approximately $\xi N_{bd}/m_b \leq 0.007$,
which further constrains the ratio.

\subsection{Charged Higgs Bosons}

What about the contribution of charged Higgs bosons in the Liu-Wolfenstein model?  In
this model, the coupling of the charged Higgs bosons to fermions is given by
\begin{equation}
{\cal L}=-i(2\sqrt{2}G_F)^{1/2}\left(
H^+\overline{U}(\Gamma_1L+\Gamma_2R)D+H^-\overline{D}(\Gamma_1^\dagger R
+\Gamma_2^\dagger L)U\right)
\end{equation}
where
\begin{eqnarray}
\Gamma_1&=& V_L^\dagger[\cot\beta M_u - \xi e^{i\alpha}N_u/\sin^2
\beta]\cr\cr
\Gamma_2&=& [\tan\beta M_d -
 \xi e^{-i\alpha}N_d/\cos^2
\beta]V_L^\dagger
\end{eqnarray}
Here, the matrices $M_u$ and $M_d$ are diagonal, and the matrices $N_u$
and $N_d$ are defined as in the neutral Higgs case.  $V_L$ is the CKM matrix.
Note that if $\xi=0$, the couplings reduce to the usual two-Higgs model.

The diagrams are the same as in the two-Higgs case, and only internal top
quarks are considered.  The effective Lagrangian arising from diagrams (a), (b) and
(d) is found to be
\begin{equation}{\cal L}_1=
{G_F\over
8\sqrt{2}}{\alpha \over 4\pi\sin^2\theta_W} T_1
\overline{s}\gamma_\mu d \overline{\nu}_L\gamma^\mu \nu_L
\end{equation}
where
\begin{equation}
T_1={({\Gamma_1})_{st}^\dagger(\Gamma_1)_{td}-
({\Gamma_2})_{st}^\dagger(\Gamma_2)_{td}
\over m^2_W}\left(
{x(4-x)\ln\ x\over (1-x)^2}+{3x\over 1-x}\right)\end{equation}
and  $x\equiv m^2_t/m^2_{H^+}$.  In this case, diagram (h) also contributes, and
the effective Lagrangian is 
\begin{equation}{\cal L}_2=
{G_F\over
4\sqrt{2}}{\alpha \over 4\pi\sin^2\theta_W} T_2
\overline{s}\gamma_\mu d \overline{\nu}_L\gamma^\mu \nu_L
\end{equation}
and 
\begin{equation}
T_2=-\cos 2\theta_W{({\Gamma_1})_{st}^\dagger(\Gamma_1)_{td}+
({\Gamma_2})_{st}^\dagger(\Gamma_2)_{td}
\over m^2_W}\left(
{x^2\ln\ x\over (1-x)^2}+{x\over 1-x}\right)\end{equation}

The  $\xi=0$ part of the effective Lagrangian is identical to the simplest extension
considered in the last section, in which there is no spontaneous CP-violation and
the CKM matrix is real.  What about the 
$\xi^2$ terms?  The
$\Gamma$ factors become
\begin{equation}
({\Gamma_1})_{st}^\dagger(\Gamma_1)_{td}\pm
({\Gamma_2})_{st}^\dagger(\Gamma_2)_{td}
=\xi^2\left({(V_LN_u^\dagger)_{st}(N_uV_L^\dagger)_{td}\over \sin^4\beta}
\pm{(N_d^\dagger V_L)_{st}(V_L^\dagger N_d)_{td}\over \cos^4\beta}\right)
\end{equation}
Once again, we don't know the values of the $N_u$ and $N_d$ matrix elements, but can
assume that they are not much larger than the top and bottom masses, respectively.
Consider the contribution of the $N_d$ terms.  They give an expression which is
identical to that of the neutral case except for some extra $V_L$ matrices and
replacing $x_b=m_b^2/m_H^2$ with $x=m^2_t/m^2_{H^+}$.  This latter change will reduce
the size of the final result (due to the absence of the large logarithm), and it is
unlikely that including the CKM matrices will increase the result, and thus the
contribution of the $N_d$ terms will also be very small.   The ratio of the
contribution of the $N_u$ terms to the standard model result is (choosing $m_{H^+}$
= 150 GeV and using Eq. (18))
\begin{equation}
\bigg|{{\cal A}_{new}\over {\cal A}_{SM}}\bigg| \simeq  10^{-5} A'
{\cos^{10/3}\beta\over
\sin^{14/3}\beta} {Im [(V_LN_U^\dagger)_{st}(N_uV_L^\dagger)_{td}]\over m^2_b}
\end{equation}
Even if one chose to ignore the CKM factors, and assume that $N_u$ is of order $m_t$,
then, since $\tan\beta\geq 1$, this is no more than $0.02 A'$, and thus will also
not be large (unless, as discussed earlier, one fine-tunes to make $A'$ large.
We conclude that the $\xi^2$ effects are not significant.

There is a cross-term which is linearly dependent on $\xi$  We find that 
\begin{equation}
\bigg|{{\cal A}_{new}\over {\cal A}_{SM}}\bigg| \simeq  10^{-2}\ \sqrt{A'}
{\cos^{8/3}\beta\over
\sin^{10/3}\beta} {m_t [V_{st} (N_uV_L)_{td} +V_{dt}(N_uV_L)_{ts}]\over m_b^2}
\end{equation}  Again,  if one assumes that $(N_uV_L)_{td}$ is approximately
$m_tV_{td}$, this is approximately $3\times 10^{-3} \sqrt{A'}$, which is not
measurable\cite{foo}

If one assumes that the CP violation is entirely spontaneous, i.e. that 
there is no CKM CP-violation, then this model has the ability, as shown by
Liu and Wolfenstein, to explain all observed CP-violating phenomena.  However,
as we have seen, it will generally give a much smaller rate for $K_L
\rightarrow \pi^o\nu\overline{\nu}$ than the standard model.  Note that, as discussed
earlier, if one does not assume $\xi'=0$, then the result will be $O(\xi)$ or
$O(\xi')$ times smaller than the terms that we have calculated.

Perhaps the most well-known model of spontaneous CP violation is the Weinberg
model\cite{weinberg}.  Although bounds from the neutron electric dipole moment and
$b\rightarrow s\gamma$ seem to rule out the model\cite{wein}, it might survive with some
fine-tuning and other similar models might still be viable.  This model assumes that there
are no tree-level flavor changing neutral currents, and as a result three Higgs doublets are
needed in order to violate CP spontaneously. All CP violation is to come from the Higgs sector,
and thus the CKM matrix is real.  Since there are no tree-level flavor changing neutral
currents, neutral Higgs bosons will not contribute to the
$K_L\rightarrow
\pi^o\nu\overline{\nu}$ decay at one-loop.  There are two charged Higgs bosons (in
addition to the charged Goldstone boson), whose couplings to fermions are given by
\begin{equation}
{\cal L}_Y=(2\sqrt{2}G_F)^{1/2}\sum_{i=1}^2 \left(
\alpha_i\overline{U}_LV_LM_DD_R+\beta_i\overline{U}_RM_UV_LD_L\right)H_i^++{\rm h.c.}
\end{equation}
where $V_L$ is the real CKM matrix.  The CP violation occurs in the (complex)
parameters $\alpha_i$ and $\beta_i$.  The observed CP violation parameter $\epsilon$
is proportional to $\sum_i\ {\rm Im}\ (\alpha_i\beta_i^*)/ m^2_{H_i^+}$.  Since the
neutron electric dipole moment is proportional to the same parameter, it is predicted in the
 model (modulo long-distance effects), and, as discussed above, tends to give too large a
value\cite{wein}.

In the calculation of the contribution of the charged Higgs bosons to the 
diagrams in figure 1, we find that all terms are proportional to $\alpha_i^*\alpha_i$
or to $\beta_i^*\beta_i$, and thus have no imaginary part; the one-loop penguin contributions
vanishes.  This is not surprising, since the value of $\epsilon$ and of the neutron
electric dipole moment involve the operator $\overline{d}\sigma_{\mu\nu}s$ whereas we
are here interested in $\overline{d}\gamma_\mu s$, and the extra $\gamma$ matrix is
needed to give the $\alpha_i\beta_i^*$ structure instead of $\alpha_i\alpha_i^*$.  There will
be a one-loop box contribution, but this will be suppressed by two powers of the tau-lepton
mass divided by $M_W$.   Thus the rate for
$K_L\rightarrow
\pi^o\nu\overline{\nu}$ in the Weinberg model will be much lower than that of the
standard model.

\section{Conclusions}

The process $K_L\rightarrow \pi^o\nu\overline{\nu}$ is an extremely promising probe
of the nature of CP violation.  It is almost entirely CP-violating and is free of
significant hadronic uncertainties.  The branching ratio, which is calculated quite
precisely in the standard model, is small, but within reach of currently planned
experiments, and its measurement to $10\%$ accuracy may be
possible.   In this paper, we have calculated the branching ratio in models in which
the CP violation arises either completely or partially from an extended Higgs sector.
We have concentrated on the Liu-Wolfenstein and Weinberg models, although the results
should be fairly general.  In spite of potentially large contributions, it has been
shown that when the constraints caused by fitting the value of $\epsilon$ are
included, the contribution of both neutral and charged Higgs bosons to the branching
ratio become very small.  Thus, in a model in which most or all of the CP violation
arises from the Higgs sector, the branching ratio for
$K_L\rightarrow \pi^o\nu\overline{\nu}$ will be much smaller than the standard
model result, and thus unmeasurable.  

We thank David Atwood for several useful discussions.  This work was supported by the
National Science Foundation grant No. .


\def\prd#1#2#3{{\rm Phys.~Rev.~}{\bf D#1}, #2 (19#3)}
\def\plb#1#2#3{{\rm Phys.~Lett.~}{\bf B#1}, #2 (19#3)}
\def\npb#1#2#3{{\rm Nucl.~Phys.~}{\bf B#1}, #2 (19#3)}
\def\prl#1#2#3{{\rm Phys.~Rev.~Lett.~}{\bf #1}, #2 (19#3)}

\bibliographystyle{unsrt}
\begin{thebibliography}{99}
\bibitem{jarlskog}
``CP Violation," ed. C. Jarlskog (World Scientific, 1989).
\bibitem{wolf}
L. Wolfenstein, \prl{51}{1995}{83}
\bibitem{blg}
A.J. Buras, M.E. Lautenbacher and G. Ostermaier, \prd{50}{3433}{94}.
\bibitem{burasa}
A.J. Buras, Proceedings of the International Conference on High Energy
Physics, Glasgow, July 1994.
\bibitem{burasb}
A.J. Buras, \plb{333}{476}{94}; L.S. Littenberg, \prd{39}{3322}{89}.
\bibitem{current}
E799 Collaboration, M. Weaver, \prl{72}{3758}{94}.
\bibitem{private}
L. Littenberg and M. Ito, private communication.
\bibitem{burasc}
G. Buchalla and A.J. Buras, \npb{398}{285}{93}; \npb{400}{225}{93}.
\bibitem{il}
T. Inami and C.S. Lim, {\rm Prog. Theor. Phys.~}{\bf 65}, 297 (1981).
\bibitem{hhg}
J.F. Gunion, H.E. Haber, G. Kane and S. Dawson, {\it The Higgs Hunter's Guide}
(Addison-Wesley Publishing, Reading MA, 1990).
\bibitem{grant}
A.K. Grant, \prd{51}{207}{95}.
\bibitem{belanger}
G. Belanger, C.Q. Geng and P. Turcotte, \prd{46}{2950}{92}.
\bibitem{lee}
T. D. Lee, \prd{8}{1226}{73}
\bibitem{branco}
G.C. Branco, \prd{22}{2901}{80}.
\bibitem{lw}
J. Liu and L. Wolfenstein, \npb{289}{1}{87}.
\bibitem{weinberg}
S. Weinberg, \prl{37}{657}{76}.
\bibitem{rebelo}
G.C. Branco and M.N. Rebelo,
 \plb{166}{117}{85}.
\bibitem{yao}
M. Sher and Y. Yuan,  \prd{44}{1461}{91}
\bibitem{foo}
Of course, in the unlikely event that this corner element of the $N_uV_L$
matrix is extremely large (of the order of the largest element of the matrix), then the
effect would be measurable. 
\bibitem{wein}
P. Krawczyk and S. Pokorski\npb{364}{10}{91}; Y. Grossman and Y. Nir,\plb{313}{126}{93};
Y. Grossman, \npb{426}{355}{94}.
\end{thebibliography}

\newpage

\begin{figure}

\vglue 7.5in  
\hskip 1.00in {\special{picture penguins scaled 1000}} \hfil
\vglue 0.1in

\caption{Corrections to the $\overline{s}dZ$ vertex in the standard model.  $G$ refers to the
charged Goldstone boson.}
\end{figure}


\begin{figure}

\vglue 7.7in  
\hskip 1.25in {\special{picture boxes scaled 1000}} \hfil
\vglue 0.1in

\caption{Box diagrams contributing to $K_L\rightarrow\pi^O\nu\overline{\nu}$.}
\end{figure}


\end{document}
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1194 899 :L
eofill
1202 901 -4 4 1246 897 4 1202 897 @a
np 1449 900 :M
1499 888 :L
1499 900 :L
1499 913 :L
1449 900 :L
eofill
1457 902 -4 4 1501 898 4 1457 898 @a
570 1502 -4 4 615 1498 4 570 1498 @a
np 316 1500 :M
366 1488 :L
366 1500 :L
366 1513 :L
316 1500 :L
eofill
346 1502 -4 4 368 1498 4 346 1498 @a
np 563 1500 :M
612 1485 :L
612 1498 :L
613 1510 :L
563 1500 :L
eofill
-4 -4 585 1501 4 4 610 1496 @b
306 1502 -4 4 320 1498 4 306 1498 @a
615 1501 -4 4 629 1497 4 615 1497 @a
np 1211 1499 :M
1261 1486 :L
1261 1499 :L
1261 1511 :L
1211 1499 :L
eofill
1222 1501 -4 4 1263 1497 4 1222 1497 @a
np 1475 1499 :M
1525 1486 :L
1525 1499 :L
1525 1511 :L
1475 1499 :L
eofill
1493 1501 -4 4 1527 1497 4 1493 1497 @a
np 1200 2099 :M
1250 2088 :L
1250 2100 :L
1250 2113 :L
1200 2099 :L
eofill
1246 2102 -4 4 1252 2098 4 1246 2098 @a
1251 2102 -4 4 1268 2098 4 1251 2098 @a
np 1453 2100 :M
1503 2088 :L
1503 2100 :L
1503 2113 :L
1453 2100 :L
eofill
1471 2102 -4 4 1505 2098 4 1471 2098 @a
np 291 2100 :M
340 2085 :L
341 2098 :L
341 2110 :L
291 2100 :L
eofill
-4 -4 317 2101 4 4 339 2096 @b
np 552 2098 :M
602 2085 :L
602 2098 :L
602 2110 :L
552 2098 :L
eofill
564 2100 -4 4 604 2096 4 564 2096 @a
np 1452 299 :M
1502 286 :L
1502 299 :L
1502 311 :L
1452 299 :L
eofill
1465 301 -4 4 1504 297 4 1465 297 @a
np 297 900 :M
347 888 :L
347 900 :L
347 913 :L
297 900 :L
eofill
315 902 -4 4 349 898 4 315 898 @a
np 549 899 :M
599 886 :L
599 899 :L
599 911 :L
549 899 :L
eofill
562 901 -4 4 601 897 4 562 897 @a
486 382 :M
f4_38 sf
(i)S
1362 381 :M
(i)S
460 892 :M
(i)S
1360 893 :M
(i)S
482 1377 :M
(i)S
1380 1382 :M
(i)S
457 1987 :M
(i)S
1364 1984 :M
(i)S
401 196 :M
f4_50 sf
(W)S
459 184 :M
(,)S
485 193 :M
(G)S
321 275 :M
(s)S
577 274 :M
(d)S
649 273 :M
(d)S
454 361 :M
(u)S
612 454 :M
(Z)S
1294 201 :M
(W)S
1344 199 :M
(,)S
1366 201 :M
(G)S
1129 273 :M
(s)S
1206 274 :M
(s)S
1465 270 :M
(d)S
1160 465 :M
(Z)S
1331 359 :M
(u)S
1329 799 :M
(G)S
1207 874 :M
(s)S
1464 873 :M
(d)S
1329 878 :M
(u)S
1334 1055 :M
(Z)S
1180 902 -4 4 1197 898 4 1180 898 @a
1500 902 -4 4 1516 898 4 1500 898 @a
424 799 :M
(W)S
309 874 :M
(s)S
429 879 :M
(u)S
560 876 :M
(d)S
431 1051 :M
(Z)S
450 1361 :M
(u)S
328 1475 :M
(s)S
572 1472 :M
(d)S
394 1555 :M
(W)S
490 1553 :M
(W)S
452 1651 :M
(Z)S
1350 1370 :M
(u)S
1222 1473 :M
(s)S
1489 1471 :M
(d)S
1286 1559 :M
(W)S
1392 1552 :M
(G)S
1344 1654 :M
(Z)S
1332 1974 :M
(u)S
1217 2075 :M
(s)S
1468 2075 :M
(d)S
1282 2151 :M
(G)S
1378 2151 :M
(G)S
1333 2255 :M
(Z)S
427 1974 :M
(u)S
305 2073 :M
(s)S
568 2071 :M
(d)S
384 2154 :M
(G)S
476 2157 :M
(W)S
444 2249 :M
(Z)S
gS
627 354 :T
271.001 rotate
-627 -354 :T
623.5 357.5 6.5 -90 0 @m
gR
gS
626 361 :T
271.001 rotate
-626 -361 :T
180 270 13 12 629.5 364 @n
gR
gS
632 367 :T
271.001 rotate
-632 -367 :T
629.5 363.5 6.5 0 90 @m
gR
gS
632 374 :T
271.001 rotate
-632 -374 :T
635.5 370.5 6.5 90 180 @m
gR
gS
625 330 :T
261 rotate
-625 -330 :T
-90 0 12 13 622 333.5 @n
gR
gS
626 336 :T
261 rotate
-626 -336 :T
629.5 339.5 6.5 180 270 @m
gR
gS
633 341 :T
261 rotate
-633 -341 :T
0 90 12 13 630 338.5 @n
gR
gS
634 348 :T
261 rotate
-634 -348 :T
637.5 344.5 6.5 90 180 @m
gR
gS
625 380 :T
260.999 rotate
-625 -380 :T
621.5 383.5 6.5 -90 0 @m
gR
gS
625 387 :T
260.999 rotate
-625 -387 :T
628.5 389.5 6.5 180 270 @m
gR
gS
633 392 :T
260.999 rotate
-633 -392 :T
629.5 388.5 6.5 0 90 @m
gR
gS
634 398 :T
260.999 rotate
-634 -398 :T
636.5 394.5 6.5 90 180 @m
gR
gS
628 303 :T
276.001 rotate
-628 -303 :T
625.5 306.5 6.5 -90 0 @m
gR
gS
628 309 :T
276.001 rotate
-628 -309 :T
180 270 12 13 631 312.5 @n
gR
gS
633 316 :T
276.001 rotate
-633 -316 :T
0 90 12 13 630 313.5 @n
gR
gS
632 323 :T
276.001 rotate
-632 -323 :T
635.5 319.5 6.5 90 180 @m
gR
gS
1182 354 :T
270.001 rotate
-1182 -354 :T
-90 0 12 13 1179 357.5 @n
gR
gS
1182 361 :T
270.001 rotate
-1182 -361 :T
1185 364 6 180 270 @m
gR
gS
1188 367 :T
270.001 rotate
-1188 -367 :T
0 90 12 13 1185 363.5 @n
gR
gS
1188 373 :T
270.001 rotate
-1188 -373 :T
90 180 12 13 1191 370.5 @n
gR
gS
1180 330 :T
260 rotate
-1180 -330 :T
-90 0 12 13 1177 333.5 @n
gR
gS
1181 336 :T
260 rotate
-1181 -336 :T
1184.5 339.5 6.5 180 270 @m
gR
gS
1188 341 :T
260 rotate
-1188 -341 :T
0 90 13 12 1185.5 338 @n
gR
gS
1189 347 :T
260 rotate
-1189 -347 :T
1192.5 344.5 6.5 90 180 @m
gR
gS
1180 380 :T
259.999 rotate
-1180 -380 :T
-90 0 12 13 1177 383.5 @n
gR
gS
1181 387 :T
259.999 rotate
-1181 -387 :T
1184.5 389.5 6.5 180 270 @m
gR
gS
1189 392 :T
259.999 rotate
-1189 -392 :T
1185.5 388.5 6.5 0 90 @m
gR
gS
1190 398 :T
259.999 rotate
-1190 -398 :T
1192.5 394.5 6.5 90 180 @m
gR
gS
1183 303 :T
275.001 rotate
-1183 -303 :T
1179.5 306.5 6.5 -90 0 @m
gR
gS
1182 309 :T
275.001 rotate
-1182 -309 :T
1185.5 312.5 6.5 180 270 @m
gR
gS
1188 316 :T
275.001 rotate
-1188 -316 :T
1184.5 312.5 6.5 0 90 @m
gR
gS
1187 323 :T
275.001 rotate
-1187 -323 :T
1190.5 319.5 6.5 90 180 @m
gR
gS
1349 955 :T
270.001 rotate
-1349 -955 :T
-90 0 12 13 1346 958.5 @n
gR
gS
1349 962 :T
270.001 rotate
-1349 -962 :T
180 270 12 13 1352 965.5 @n
gR
gS
1355 968 :T
270.001 rotate
-1355 -968 :T
0 90 12 13 1352 964.5 @n
gR
gS
1355 975 :T
270.001 rotate
-1355 -975 :T
90 180 12 13 1358 971.5 @n
gR
gS
1347 931 :T
260 rotate
-1347 -931 :T
1344.5 934.5 6.5 -90 0 @m
gR
gS
1348 937 :T
260 rotate
-1348 -937 :T
1351.5 940.5 6.5 180 270 @m
gR
gS
1356 942 :T
260 rotate
-1356 -942 :T
1352.5 939.5 6.5 0 90 @m
gR
gS
1357 949 :T
260 rotate
-1357 -949 :T
90 180 12 13 1360 945.5 @n
gR
gS
1348 982 :T
259.999 rotate
-1348 -982 :T
1344.5 984.5 6.5 -90 0 @m
gR
gS
1349 988 :T
259.999 rotate
-1349 -988 :T
1351.5 991.5 6.5 180 270 @m
gR
gS
1356 993 :T
259.999 rotate
-1356 -993 :T
1352.5 989.5 6.5 0 90 @m
gR
gS
1357 999 :T
259.999 rotate
-1357 -999 :T
90 180 12 13 1360 995.5 @n
gR
gS
1350 904 :T
275.001 rotate
-1350 -904 :T
-90 0 12 13 1347 907.5 @n
gR
gS
1349 911 :T
275.001 rotate
-1349 -911 :T
1352.5 913.5 6.5 180 270 @m
gR
gS
1355 917 :T
275.001 rotate
-1355 -917 :T
0 90 12 13 1352 914.5 @n
gR
gS
1354 924 :T
275.001 rotate
-1354 -924 :T
1357.5 920.5 6.5 90 180 @m
gR
gS
444 955 :T
270.002 rotate
-444 -955 :T
441.5 958.5 6.5 -90 0 @m
gR
gS
444 961 :T
270.002 rotate
-444 -961 :T
447.5 964.5 6.5 180 270 @m
gR
gS
450 967 :T
270.002 rotate
-450 -967 :T
447.5 964.5 6.5 0 90 @m
gR
gS
450 974 :T
270.002 rotate
-450 -974 :T
453.5 970.5 6.5 90 180 @m
gR
gS
443 930 :T
260.001 rotate
-443 -930 :T
439.5 933.5 6.5 -90 0 @m
gR
gS
444 937 :T
260.001 rotate
-444 -937 :T
180 270 12 13 447 939.5 @n
gR
gS
451 942 :T
260.001 rotate
-451 -942 :T
0 90 12 13 448 938.5 @n
gR
gS
452 948 :T
260.001 rotate
-452 -948 :T
90 180 12 13 455 944.5 @n
gR
gS
443 981 :T
260 rotate
-443 -981 :T
439.5 984.5 6.5 -90 0 @m
gR
gS
444 987 :T
260 rotate
-444 -987 :T
180 270 12 13 447 990.5 @n
gR
gS
451 992 :T
260 rotate
-451 -992 :T
0 90 12 13 448 988.5 @n
gR
gS
452 998 :T
260 rotate
-452 -998 :T
90 180 12 13 455 995.5 @n
gR
gS
445 903 :T
275.002 rotate
-445 -903 :T
-90 0 12 13 442 906.5 @n
gR
gS
445 910 :T
275.002 rotate
-445 -910 :T
180 270 12 13 448 913.5 @n
gR
gS
450 917 :T
275.002 rotate
-450 -917 :T
0 90 12 13 447 913.5 @n
gR
gS
450 923 :T
275.002 rotate
-450 -923 :T
90 180 12 13 453 919.5 @n
gR
gS
424 1495 :T
359.002 rotate
-424 -1495 :T
-90 0 12 13 421 1497.5 @n
gR
gS
418 1495 :T
359.002 rotate
-418 -1495 :T
180 270 12 13 421 1497.5 @n
gR
gS
412 1501 :T
359.002 rotate
-412 -1501 :T
408.5 1497.5 6.5 0 90 @m
gR
gS
405 1501 :T
359.002 rotate
-405 -1501 :T
408.5 1497.5 6.5 90 180 @m
gR
gS
449 1492 :T
349.001 rotate
-449 -1492 :T
445.5 1495.5 6.5 -90 0 @m
gR
gS
442 1494 :T
349.001 rotate
-442 -1494 :T
445.5 1496.5 6.5 180 270 @m
gR
gS
437 1501 :T
349.001 rotate
-437 -1501 :T
0 90 12 13 434 1497.5 @n
gR
gS
431 1502 :T
349.001 rotate
-431 -1502 :T
434.5 1498.5 6.5 90 180 @m
gR
gS
398 1493 :T
349 rotate
-398 -1493 :T
-90 0 12 13 395 1496.5 @n
gR
gS
392 1495 :T
349 rotate
-392 -1495 :T
180 270 12 13 395 1497.5 @n
gR
gS
387 1502 :T
349 rotate
-387 -1502 :T
0 90 12 13 384 1498.5 @n
gR
gS
381 1503 :T
349 rotate
-381 -1503 :T
90 180 12 13 384 1499.5 @n
gR
gS
476 1495 :T
4.001 rotate
-476 -1495 :T
472.5 1497.5 6.5 -90 0 @m
gR
gS
469 1494 :T
4.001 rotate
-469 -1494 :T
180 270 12 13 472 1497.5 @n
gR
gS
462 1500 :T
4.001 rotate
-462 -1500 :T
0 90 12 13 459 1496.5 @n
gR
gS
456 1499 :T
4.001 rotate
-456 -1499 :T
459 1496 6 90 180 @m
gR
gS
523 1492 :T
359.002 rotate
-523 -1492 :T
520.5 1495.5 6.5 -90 0 @m
gR
gS
517 1492 :T
359.002 rotate
-517 -1492 :T
180 270 12 13 520 1495.5 @n
gR
gS
511 1499 :T
359.002 rotate
-511 -1499 :T
507.5 1495.5 6.5 0 90 @m
gR
gS
504 1498 :T
359.002 rotate
-504 -1498 :T
507.5 1495.5 6.5 90 180 @m
gR
gS
548 1490 :T
349.001 rotate
-548 -1490 :T
544.5 1493.5 6.5 -90 0 @m
gR
gS
542 1491 :T
349.001 rotate
-542 -1491 :T
544.5 1494.5 6.5 180 270 @m
gR
gS
537 1498 :T
349.001 rotate
-537 -1498 :T
533.5 1495.5 6.5 0 90 @m
gR
gS
530 1500 :T
349.001 rotate
-530 -1500 :T
533.5 1496.5 6.5 90 180 @m
gR
gS
497 1491 :T
349 rotate
-497 -1491 :T
-90 0 12 13 494 1494.5 @n
gR
gS
491 1492 :T
349 rotate
-491 -1492 :T
180 270 12 13 494 1495.5 @n
gR
gS
486 1500 :T
349 rotate
-486 -1500 :T
0 90 12 13 483 1496.5 @n
gR
gS
480 1501 :T
349 rotate
-480 -1501 :T
90 180 12 13 483 1497.5 @n
gR
90 180 12 13 559 1492.5 @n
557 1501 -4 4 563 1497 4 557 1497 @a
gS
464 1549 :T
270.001 rotate
-464 -1549 :T
-90 0 12 13 461 1552.5 @n
gR
gS
464 1556 :T
270.001 rotate
-464 -1556 :T
180 270 12 13 467 1558.5 @n
gR
gS
470 1562 :T
270.001 rotate
-470 -1562 :T
0 90 12 13 467 1558.5 @n
gR
gS
470 1568 :T
270.001 rotate
-470 -1568 :T
90 180 12 13 473 1564.5 @n
gR
gS
462 1525 :T
260 rotate
-462 -1525 :T
459 1528 6 -90 0 @m
gR
gS
463 1531 :T
260 rotate
-463 -1531 :T
466.5 1534.5 6.5 180 270 @m
gR
gS
471 1536 :T
260 rotate
-471 -1536 :T
467.5 1532.5 6.5 0 90 @m
gR
gS
472 1542 :T
260 rotate
-472 -1542 :T
474.5 1539.5 6.5 90 180 @m
gR
gS
462 1575 :T
259.999 rotate
-462 -1575 :T
459.5 1578.5 6.5 -90 0 @m
gR
gS
463 1581 :T
259.999 rotate
-463 -1581 :T
466.5 1584.5 6.5 180 270 @m
gR
gS
471 1587 :T
259.999 rotate
-471 -1587 :T
467.5 1583.5 6.5 0 90 @m
gR
gS
472 1593 :T
259.999 rotate
-472 -1593 :T
90 180 12 13 475 1589.5 @n
gR
gS
465 1498 :T
275.001 rotate
-465 -1498 :T
-90 0 12 13 462 1501.5 @n
gR
gS
464 1504 :T
275.001 rotate
-464 -1504 :T
467.5 1507.5 6.5 180 270 @m
gR
gS
470 1511 :T
275.001 rotate
-470 -1511 :T
0 90 12 13 467 1507.5 @n
gR
gS
469 1517 :T
275.001 rotate
-469 -1517 :T
472.5 1514.5 6.5 90 180 @m
gR
gS
1306 1497 :T
359.002 rotate
-1306 -1497 :T
-90 0 12 13 1303 1499.5 @n
gR
gS
1300 1497 :T
359.002 rotate
-1300 -1497 :T
180 270 12 13 1303 1499.5 @n
gR
gS
1294 1503 :T
359.002 rotate
-1294 -1503 :T
1290.5 1499.5 6.5 0 90 @m
gR
gS
1287 1503 :T
359.002 rotate
-1287 -1503 :T
1290.5 1499.5 6.5 90 180 @m
gR
gS
1331 1495 :T
349.001 rotate
-1331 -1495 :T
1327.5 1497.5 6.5 -90 0 @m
gR
gS
1324 1496 :T
349.001 rotate
-1324 -1496 :T
1327.5 1498.5 6.5 180 270 @m
gR
gS
1320 1503 :T
349.001 rotate
-1320 -1503 :T
1316.5 1499.5 6.5 0 90 @m
gR
gS
1313 1504 :T
349.001 rotate
-1313 -1504 :T
1316.5 1500.5 6.5 90 180 @m
gR
gS
1358 1497 :T
4.001 rotate
-1358 -1497 :T
1354.5 1499.5 6.5 -90 0 @m
gR
gS
1351 1496 :T
4.001 rotate
-1351 -1496 :T
1354.5 1499.5 6.5 180 270 @m
gR
gS
1344 1502 :T
4.001 rotate
-1344 -1502 :T
1341.5 1498.5 6.5 0 90 @m
gR
gS
1338 1501 :T
4.001 rotate
-1338 -1501 :T
90 180 12 13 1341 1498.5 @n
gR
1264 1501 -4 4 1279 1497 4 1264 1497 @a
1271 1501 -4 4 1284 1497 4 1271 1497 @a
gS
1357 1555 :T
270.002 rotate
-1357 -1555 :T
1353.5 1557.5 6.5 -90 0 @m
gR
gS
1357 1561 :T
270.002 rotate
-1357 -1561 :T
180 270 12 13 1360 1564.5 @n
gR
gS
1363 1567 :T
270.002 rotate
-1363 -1567 :T
0 90 12 13 1360 1563.5 @n
gR
gS
1363 1574 :T
270.002 rotate
-1363 -1574 :T
90 180 12 13 1366 1570.5 @n
gR
gS
1355 1530 :T
260.001 rotate
-1355 -1530 :T
-90 0 12 13 1352 1533.5 @n
gR
gS
1356 1536 :T
260.001 rotate
-1356 -1536 :T
180 270 12 13 1359 1539.5 @n
gR
gS
1363 1541 :T
260.001 rotate
-1363 -1541 :T
0 90 12 13 1360 1538.5 @n
gR
gS
1364 1548 :T
260.001 rotate
-1364 -1548 :T
1367.5 1544.5 6.5 90 180 @m
gR
gS
1355 1581 :T
260 rotate
-1355 -1581 :T
-90 0 12 13 1352 1583.5 @n
gR
gS
1356 1587 :T
260 rotate
-1356 -1587 :T
1359.5 1590.5 6.5 180 270 @m
gR
gS
1364 1592 :T
260 rotate
-1364 -1592 :T
1360.5 1588.5 6.5 0 90 @m
gR
gS
1365 1598 :T
260 rotate
-1365 -1598 :T
1367.5 1594.5 6.5 90 180 @m
gR
gS
1358 1503 :T
275.002 rotate
-1358 -1503 :T
1354.5 1506.5 6.5 -90 0 @m
gR
gS
1357 1510 :T
275.002 rotate
-1357 -1510 :T
180 270 12 13 1360 1512.5 @n
gR
gS
1363 1516 :T
275.002 rotate
-1363 -1516 :T
1359.5 1513.5 6.5 0 90 @m
gR
gS
1362 1523 :T
275.002 rotate
-1362 -1523 :T
90 180 12 13 1365 1519.5 @n
gR
gS
1349 2156 :T
271.002 rotate
-1349 -2156 :T
1345.5 2158.5 6.5 -90 0 @m
gR
gS
1349 2162 :T
271.002 rotate
-1349 -2162 :T
1351.5 2165.5 6.5 180 270 @m
gR
gS
1355 2168 :T
271.002 rotate
-1355 -2168 :T
1351.5 2165.5 6.5 0 90 @m
gR
gS
1354 2175 :T
271.002 rotate
-1354 -2175 :T
1357.5 2171.5 6.5 90 180 @m
gR
gS
1348 2131 :T
261.001 rotate
-1348 -2131 :T
1344.5 2134.5 6.5 -90 0 @m
gR
gS
1348 2137 :T
261.001 rotate
-1348 -2137 :T
1351.5 2140.5 6.5 180 270 @m
gR
gS
1356 2143 :T
261.001 rotate
-1356 -2143 :T
1352.5 2139.5 6.5 0 90 @m
gR
gS
1356 2149 :T
261.001 rotate
-1356 -2149 :T
1359.5 2145.5 6.5 90 180 @m
gR
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%%Trailer
end
%%EOF
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%%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. All Rights Reserved.
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%%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. All Rights Reserved.
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/ps Z
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makeblendedfont
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def
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%%EndFile
%%BeginFile: adobe_psp_derived_styles
%%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. All Rights Reserved.
/wi
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{
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{
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/dxd Z
/dsdx2 Z
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gl
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gR
basefonto setfont
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gS
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{
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setcharwidth
dup 0 0 ms
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0 dxd ms
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/italicmtx[1 0 -.212557 1 0 0]def
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italicmtx :mf def
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{
/FontInfo get
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}{
.1
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3 1 roll
dup/UnderlineThickness known
{
/UnderlineThickness get
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abs
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pop pop .067
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/$t Z
/$p Z
/$s Z
/:p
{
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1 index mul/$p xs
.012 mul/$s xs
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/:m
{gS
0 $p rm
$t lw
0 rl stroke
gR
}bd
/:n
{
gS
0 $p rm
$t lw
0 rl
gS
gl
stroke
gR
strokepath
$s lw
/setstrokeadjust where{pop
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stroke
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gR
}bd
/:o
{gS
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$t lw
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stroke
gR
:n
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%%EndFile
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%%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. All Rights Reserved.
/:q/setdash ld
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np
:M
:L
stroke
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(sher)setjob
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:mre
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5.001 rotate
-684 -1800 :T
681.5 1796.5 6.5 0 90 @m
gR
gS
678 1799 :T
5.001 rotate
-678 -1799 :T
90 180 12 13 681 1795.5 @n
gR
gS
747 1795 :T
359.001 rotate
-747 -1795 :T
-90 0 12 13 744 1798.5 @n
gR
gS
741 1795 :T
359.001 rotate
-741 -1795 :T
743.5 1798.5 6.5 180 270 @m
gR
gS
734 1801 :T
359.001 rotate
-734 -1801 :T
731.5 1798.5 6.5 0 90 @m
gR
gS
728 1801 :T
359.001 rotate
-728 -1801 :T
90 180 12 13 731 1798.5 @n
gR
gS
771 1793 :T
349 rotate
-771 -1793 :T
-90 0 12 13 768 1796.5 @n
gR
gS
765 1794 :T
349 rotate
-765 -1794 :T
180 270 12 13 768 1797.5 @n
gR
gS
760 1801 :T
349 rotate
-760 -1801 :T
0 90 12 13 757 1798.5 @n
gR
gS
754 1803 :T
349 rotate
-754 -1803 :T
90 180 12 13 757 1799.5 @n
gR
gS
721 1794 :T
348.999 rotate
-721 -1794 :T
-90 0 12 13 718 1797.5 @n
gR
gS
715 1795 :T
348.999 rotate
-715 -1795 :T
180 270 12 13 718 1798.5 @n
gR
gS
710 1803 :T
348.999 rotate
-710 -1803 :T
706.5 1799.5 6.5 0 90 @m
gR
gS
703 1804 :T
348.999 rotate
-703 -1804 :T
706.5 1800.5 6.5 90 180 @m
gR
gS
798 1795 :T
4.001 rotate
-798 -1795 :T
-90 0 12 13 795 1798.5 @n
gR
gS
792 1795 :T
4.001 rotate
-792 -1795 :T
180 270 12 13 795 1797.5 @n
gR
gS
785 1800 :T
4.001 rotate
-785 -1800 :T
0 90 12 13 782 1797.5 @n
gR
gS
779 1800 :T
4.001 rotate
-779 -1800 :T
90 180 12 13 782 1796.5 @n
gR
gS
848 1795 :T
359.001 rotate
-848 -1795 :T
-90 0 12 13 845 1797.5 @n
gR
gS
842 1795 :T
359.001 rotate
-842 -1795 :T
180 270 12 13 845 1797.5 @n
gR
gS
836 1801 :T
359.001 rotate
-836 -1801 :T
832.5 1797.5 6.5 0 90 @m
gR
gS
829 1801 :T
359.001 rotate
-829 -1801 :T
832.5 1797.5 6.5 90 180 @m
gR
gS
873 1792 :T
349 rotate
-873 -1792 :T
869.5 1795.5 6.5 -90 0 @m
gR
gS
866 1794 :T
349 rotate
-866 -1794 :T
869.5 1796.5 6.5 180 270 @m
gR
gS
862 1801 :T
349 rotate
-862 -1801 :T
858.5 1797.5 6.5 0 90 @m
gR
gS
855 1802 :T
349 rotate
-855 -1802 :T
858.5 1798.5 6.5 90 180 @m
gR
gS
822 1794 :T
348.999 rotate
-822 -1794 :T
-90 0 12 13 819 1796.5 @n
gR
gS
816 1795 :T
348.999 rotate
-816 -1795 :T
180 270 12 13 819 1797.5 @n
gR
gS
811 1802 :T
348.999 rotate
-811 -1802 :T
0 90 12 13 808 1798.5 @n
gR
gS
805 1803 :T
348.999 rotate
-805 -1803 :T
90 180 12 13 808 1799.5 @n
gR
gS
898 1794 :T
4.001 rotate
-898 -1794 :T
895 1797 6 -90 0 @m
gR
gS
892 1794 :T
4.001 rotate
-892 -1794 :T
895 1797 6 180 270 @m
gR
gS
886 1799 :T
4.001 rotate
-886 -1799 :T
883 1796 6 0 90 @m
gR
gS
880 1799 :T
4.001 rotate
-880 -1799 :T
883 1796 6 90 180 @m
gR
[25
16.667
] 0 :q
900 1498 600 1498 :r
298 1500 -4 4 602 1496 4 298 1496 @a
298 1800 -4 4 600 1796 4 298 1796 @a
-4 -4 602 1800 4 4 598 1496 @b
898 1500 -4 4 1202 1496 4 898 1496 @a
-4 -4 902 1800 4 4 898 1496 @b
898 1800 -4 4 1202 1796 4 898 1796 @a
729 1479 :M
f2_50 sf
(G)S
711 1772 :M
(W)S
np 479 1498 :M
429 1510 :L
429 1498 :L
429 1485 :L
479 1498 :L
[] 0 :q
eofill
408 1500 -4 4 431 1496 4 408 1496 @a
np 600 1675 :M
588 1625 :L
600 1625 :L
613 1625 :L
600 1675 :L
eofill
-4 -4 602 1650 4 4 598 1623 @b
np 419 1798 :M
469 1785 :L
469 1798 :L
469 1810 :L
419 1798 :L
eofill
445 1800 -4 4 471 1796 4 445 1796 @a
np 1024 1498 :M
1074 1485 :L
1074 1498 :L
1074 1510 :L
1024 1498 :L
eofill
1053 1500 -4 4 1076 1496 4 1053 1496 @a
np 1092 1798 :M
1042 1810 :L
1042 1798 :L
1042 1785 :L
1092 1798 :L
eofill
1040 1800 -4 4 1072 1796 4 1040 1796 @a
np 900 1676 :M
888 1626 :L
900 1626 :L
913 1626 :L
900 1676 :L
eofill
-4 -4 902 1665 4 4 898 1624 @b
[25
16.667
] 0 :q
900 2098 600 2098 :r
298 2100 -4 4 602 2096 4 298 2096 @a
298 2400 -4 4 600 2396 4 298 2396 @a
-4 -4 602 2400 4 4 598 2096 @b
898 2100 -4 4 1202 2096 4 898 2096 @a
-4 -4 902 2399 4 4 898 2096 @b
898 2400 -4 4 1202 2396 4 898 2396 @a
729 2079 :M
(G)S
np 479 2098 :M
429 2110 :L
429 2098 :L
429 2085 :L
479 2098 :L
[] 0 :q
eofill
408 2100 -4 4 431 2096 4 408 2096 @a
np 600 2275 :M
588 2225 :L
600 2225 :L
613 2225 :L
600 2275 :L
eofill
-4 -4 602 2250 4 4 598 2223 @b
np 419 2398 :M
469 2385 :L
469 2398 :L
469 2410 :L
419 2398 :L
eofill
445 2400 -4 4 471 2396 4 445 2396 @a
np 1024 2098 :M
1074 2085 :L
1074 2098 :L
1074 2110 :L
1024 2098 :L
eofill
1053 2100 -4 4 1076 2096 4 1053 2096 @a
np 1092 2398 :M
1042 2410 :L
1042 2398 :L
1042 2385 :L
1092 2398 :L
eofill
1040 2400 -4 4 1072 2396 4 1040 2396 @a
np 900 2276 :M
888 2226 :L
900 2226 :L
913 2226 :L
900 2276 :L
eofill
-4 -4 902 2265 4 4 898 2224 @b
707 1300 :M
f4_50 sf
(\(b\))S
706 2509 :M
(\(d\))S
[25
16.667
] 0 :q
901 2398 600 2398 :r
723 2378 :M
f2_50 sf
(G)S
695 301 -4 4 700 300 4 695 297 @a
795 301 -4 4 802 299 4 795 297 @a
696 599 -4 4 701 598 4 696 595 @a
720 600 -4 4 726 598 4 720 596 @a
698 902 -4 4 704 900 4 698 898 @a
722 900 -4 4 729 898 4 722 896 @a
697 1800 -4 4 702 1799 4 697 1796 @a
722 1798 -4 4 728 1796 4 722 1794 @a
1227 310 :M
f6_50 sf
(n)S
endp
%%Trailer
end
%%EOF

