%Paper: 
%From: ph@ttpux7.physik.uni-karlsruhe.de
%Date: Thu, 03 Feb 94 10:24:44 +0100




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\makebox[14cm][r]{TTP 93-39}\par
\makebox[14cm][r]{PITHA 93/44} \par
\makebox[14cm][r]{December 1993}\par
\vspace{.7cm}
\centerline{\Large \bf $K^0 $ Decay into Three Photons
\footnote{\it in celebration of the S.N. Bose birth centenary}}
\vspace{1.cm}
\centerline{P. Heiliger$^{a)} $, B. McKellar$^{b)} $  and L.M. Sehgal$^{c)} $}
\par
\centerline{$^{a)} $Institut f\"ur Theoretische Teilchenphysik} \par
\centerline{Universit\"at Karlsruhe}\par
\centerline{D--76128 Karlsruhe, Germany}\par
\centerline{$^{b)} $Research Center for High Energy Physics, School of
Physics,} \par
\centerline{University of Melbourne, Parkville, Victoria 3052, Australia} \par
\centerline{$^{c)} $III. Physikalisches Institut (A), RWTH Aachen}\par
\centerline{D--52056 Aachen, Germany}\par
\normalsize
\vspace{1.cm}

\begin{abstract}
The decays $K_{L,S} \to 3 \gamma $ are not forbidden by any selection rules or
symmetry principles. However, gauge invariance and Bose statistics dictate
that every photon pair in these transitions has at least two units of angular
momentum. This gives rise to an extraordinary suppression. Using a simple
model, we obtain the branching ratios $B(K_L \to  3 \gamma) \sim
3 \cdot 10^{-19}$, $B(K_S \to 3 \gamma) \sim 5 \cdot 10^{-22} $.
\end{abstract}

\newpage
The decays $K_L \to 2 \gamma $ and $K_S \to 2 \gamma $ have been observed with
branching ratios of $5.7 \cdot 10^{-4} $ and $2.4 \cdot 10^{-6} $,
respectively. What is the expected rate of the decays $K_{L,S} \to 3 \gamma $ ?
\par
\bigskip
First of all, one should observe that both $K_L \to 3 \gamma $ and $K_S \to
3 \gamma $ are possible without violating $CP $ invariance or any general
symmetry principle. Since the $3 \gamma $ system has $C = -1 $, the decay
$K_L \to 3 \gamma $ can proceed via the $C $--violating, $P $--violating
part of the $|\Delta S| = 1 $ nonleptonic weak interaction, while $K_S \to
3 \gamma $ proceeds via the $C $--conserving, $P $--conserving part.
Naively, one would imagine that these decays would occur at rates that are
roughly a factor $\alpha_{em} $ times the two--photon decay rates. \par
\bigskip
This naive expectation, however, disregards the constraints of gauge invariance
and Bose statistics. Gauge invariance dictates that in the decay $K^0 \to 3
\gamma $, no pair of photons can have angular momentum zero, since that would
correspond to a $0 \to 0 $ radiative transition, which is forbidden for a
real photon. Similarly, no pair of photons can have $J = 1 $, since that
conflicts with Bose statistics (Yang's theorem). It follows that the
decays $K_{L,S} \to 3 \gamma $ can only occur if each pair of photons in the
final state has at least two units of angular momentum. The matrix element
thus inevitably has a large number of angular momentum suppression factors.
Using a simple model, we show below that the decays $K_{L,S} \to 3 \gamma $
have rates that are {\bf 15 orders of magnitude lower} than the corresponding
rates of $K_{L,S} \to 2 \gamma $ ! \par
\bigskip
The model we employ is illustrated in Fig. 1. We assume that the $K_{L,S}
\to 3 \gamma $ transition is mediated by the decay $K_{L,S} \to \pi^0
\pi^0 \gamma $, with the two $\pi^0 $'s converting into two photons. The use
of this particular channel is motivated by the fact that the decays
$K_{L,S} \to \pi^0 \pi^0 \gamma $ are necessarily quadrupole transitions
(E2 and M2 respectively \cite{own}), so that the pion pair has $J = 2$, which
is the minimum angular momentum required for the photon pairs in $K_{L,S}
\to 3 \gamma $. Other intermediate states are undoubtedly possible
(including e.g. $K_{L,S} \to \pi^+ \pi^- \gamma $, with pions in a D--wave).
Our aim, however, is to expose the symmetry structure of the $K^0 \to 3
\gamma $ amplitude and to obtain a rough estimate of its magnitude. The
$\pi^0 \pi^0 \gamma $ intermediate state turns out to be adequate to this
purpose. \par
\bigskip
To determine the matrix element of $K_L \to 3 \gamma $ from the model shown
in Fig. 1, we require the amplitudes for $K_L \to \pi^0 \pi^0 \gamma $ and
$\pi^0 \pi^0 \to \gamma \gamma $, which we parameterise as follows:
\begin{eqnarray}
& & {\cal M} \left ( K_L(p_K) \to \pi^0(p_1) \; \pi^0(p_2) \;
\gamma(k_3,\epsilon_3) \right )
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\;\;\;\;\;\;\;\;\;\;\; \nonumber  \\
& = & {h_L \over M_K^5} \left [\epsilon_3 \cdot p_1 k_3 \cdot p_2 - \epsilon_3
\cdot p_2 k_3 \cdot p_1 \right ] \; k_3 \cdot (p_1 - p_2)
\end{eqnarray}
\begin{eqnarray}
& & {\cal M} \left ( \gamma (k_1,\epsilon_1) + \gamma (k_2, \epsilon_2)
\to \pi^0 (p_1) \; \pi^0(p_2) \right )  \nonumber \\
& = & {\tilde{G} s_{12} \over M_V^2} \cdot \left [ {p_1 \cdot k_1 \; p_1 \cdot
k_2 \over k_1 \cdot k_2} \; g_{\mu \nu} + p_{1 \mu} p_{1 \nu} - {p_1 \cdot k_1
\over k_1 \cdot k_2} \; k_{2 \mu} p_{1 \nu} - {p_1 \cdot k_2 \over k_1 \cdot
k_2} \; k_{1 \nu} p_{1 \mu} \right ] \epsilon_1^{\mu} \; \epsilon_2^{\nu}
\;\;\;\;\;\;
\end{eqnarray}
The structure in Eq. (1) is appropriate to an E2 transition, while that in
Eq. (2) is obtained using a vector meson exchange model for $\pi^0 \pi^0
\to \gamma \gamma $ (see, e.g., Ref. \cite{ko}), keeping only the leading
term in $s_{12}/M_V^2 $ ($s_{12} \equiv (k_1 + k_2)^2 $). (Numerically, the
constant $\tilde{G} $ is given by $\tilde{G} = \tilde{G}_{\rho} +
\tilde{G}_{\omega} $ with $\tilde{G}_{\rho} = {1\over 9} \;
g_{\omega \pi \gamma}^2, \; \tilde{G}_{\omega} = g_{\omega \pi \gamma}^2 $ and
$g_{\omega \pi \gamma} = 7.7 \cdot 10^{-4} \; \mbox{MeV}^{-1} $.) \par
\bigskip
We now employ unitarity to obtain the absorptive part of $K_L \to 3 \gamma $:
\begin{eqnarray}
& & Im \; {\cal M} \left [ K_L(p_K) \to \gamma(\epsilon_1, k_1) + \gamma
(\epsilon_2, k_2) + \gamma(\epsilon_3, k_3) \right ] \nonumber \\
& = & {1\over 2} \int {d^3p_1 \over 2 p_{1\, 0} (2\pi )^3} \; {d^3p_2 \over
2 p_{2\, 0} (2\pi )^3} \; (2\pi )^4 \delta^{(4)} (p_K - p_1 - p_2 - k_3)
\nonumber \\
& & \cdot \left \{ {h_L \over M_K^2} (\epsilon_3 \cdot p_1 \; k_3 \cdot p_2
- \epsilon_3 \cdot p_2 \; k_3 \cdot p_1) \; k_3 \cdot (p_1 - p_2) \right \}
\nonumber \\
& & \cdot {\tilde{G} s_{12} \over M_V^2} \; \epsilon_1^{\mu} \epsilon_2^{\nu}
\left \{ {p_1 \cdot k_1 \; p_2 \cdot k_2 \over k_1 \cdot k_2}\; g_{\mu \nu}
+ p_{1 \mu} p_{2 \nu} - {p_1 \cdot k_1 \over k_1 \cdot k_2}\; k_{2 \mu}
p_{1 \nu} - {p_1 \cdot k_2 \over k_1 \cdot k_2} \; k_{1 \nu} p_{1 \mu}
\right \} \nonumber \\
& & \cdot \Theta (s_{12} - 4 m_{\pi}^2) \nonumber \\
& & + \; \mbox{permutations of} \; (\epsilon_1, k_1), (\epsilon_2, k_2),
(\epsilon_3,k_3)
\end{eqnarray}
To evaluate this, we need to calculate integrals of the form
\begin{eqnarray}
K^{\mu \nu \rho \sigma} & = & \int {d^3p_1 \over 2 p_{1\, 0}} \;
{d^3p_2 \over 2 p_{2\, 0}} \; \delta^{(4)} (P - p_1 -p_2) f(p_1 \cdot p_2)\;
p_1^{\mu} p_1^{\nu} p_1^{\rho} p_1^{\sigma} \; , \nonumber \\
L^{\mu \nu \rho \sigma} & = & \int {d^3p_1 \over 2 p_{1\, 0}} \;
{d^3p_2 \over 2 p_{2\, 0}} \; \delta^{(4)} (P - p_1 -p_2) f(p_1 \cdot p_2)\;
p_1^{\mu} p_1^{\nu} p_1^{\rho} p_2^{\sigma}
\end{eqnarray}
These are given in the appendix. The resulting expression for ${\cal M}_{abs}
\equiv Im \; {\cal M} (K_L \to \gamma \gamma \gamma) $ is then squared, and
the polarizations of the photons summed over, using the symbolic computation
program FORM \cite{form}, with the result
\begin{eqnarray}
\sum_{pol} |{\cal M}_{abs}|^2 & = & V_{12}^2 F_{12}^2 \{s_{12} s_{23}^3 s_{13}
                                    +s_{12} s_{23} s_{13}^3 \} \nonumber \\
                              & + & V_{23}^2 F_{23}^2 \{s_{12}^3 s_{23} s_{13}
                                    +s_{12} s_{23} s_{13}^3 \} \nonumber \\
                              & + & V_{13}^2 F_{13}^2 \{s_{12}^3 s_{23} s_{13}
                                    +s_{12} s_{23}^3 s_{13} \} \nonumber \\
& - & 2 V_{12} V_{23} F_{12} F_{23} s_{12} s_{23} s_{13}^3 \nonumber \\
& - & 2 V_{12} V_{13} F_{12} F_{13} s_{12} s_{23}^3 s_{13} \nonumber \\
& - & 2 V_{13} V_{23} F_{13} F_{23} s_{12}^3 s_{23} s_{13}
\end{eqnarray}
where the functions $V_{ij} $ and $F_{ij} $ are defined by
\begin{eqnarray}
V_{ij} & = & {1\over 16 \pi} {h_L \over M_K^5} {\tilde{G} \over M_V^2}
{1 \over s_{ij}^2} \sqrt{ \lambda (s_{ij}, m_{\pi}^2, m_{\pi}^2) } \nonumber \\
F_{ij} & = & {1\over 5}\, s_{ij} \left \{ {1\over 8}\, s_{ij}^2 - {29 \over
24} \, s_{ij} m_{\pi}^2 + 2 m_{\pi}^4 \right \} \nonumber \\
\mbox{with} & & \lambda (x,y,z) = x^2 + y^2 + z^2 - 2xy - 2yz - 2xz
\end{eqnarray}
The following features of the above result (5) should be noted:
\begin{itemize}
\item[(i)] The Dalitz plot density, given by $\sum |{\cal M}_{abs}|^2 $, is
manifestly symmetric in $s_{12}, s_{23} $ and $s_{13} $, as required by
Bose statistics.
\item[(ii)] The density vanishes at the centre of the Dalitz plot ($s_{12}
= s_{23} = s_{13} $), i.e. for the configuration in which the photons have
equal energy.
\item[(iii)] Because of the overall factor $s_{12} s_{23} s_{13} $, the
density vanishes along the boundaries of the Dalitz plot, defined by
$s_{12} = 0, s_{23} = 0 $ and $s_{13} = 0 $. These correspond to the
configurations in which two of the three photons are collinear.
\item[(iv)] In the limit $m_{\pi} \to 0 $, the factors $V_{ij} F_{ij} $ become
proportional to $s_{ij}^2 $, and the density simplifies to
\begin{eqnarray}
\sum |{\cal M}_{abs}|^2 \stackrel{m_{\pi}= 0}{\sim} s_{12} s_{23} s_{13}
& & \left \{ s_{12}^2 \; (s_{12}^2 s_{23}^2 + s_{12}^2 s_{13}^2 - 2 s_{23}^2
s_{13}^2) \right. \nonumber \\
& & + s_{23}^2 \; (s_{23}^2 s_{12}^2 + s_{23}^2 s_{13}^2 - 2 s_{12}^2 s_{13}^2
) \nonumber \\
& & \left. + s_{13}^2 \; (s_{13}^2 s_{12}^2 + s_{13}^2 s_{23}^2 - 2 s_{12}^2
s_{23}^2 ) \right \}
\end{eqnarray}
\item[(v)] The result (7) has some resemblance to the matrix element for
$\pi^0 \to 3 \gamma $ obtained by Dicus \cite{dic}:
\newpage
\begin{eqnarray}
\sum |{\cal M}|^2\,|_{\mbox{Dicus}}  & \sim & s_{12} s_{13} s_{23} \;
\{ s_{12}^2 s_{13}^2 + s_{12}^2 s_{23}^2 + s_{23}^2 s_{13}^2 \nonumber \\
& & - s_{13} s_{23} s_{12}^2 - s_{13}^2 s_{23} s_{12} - s_{13} s_{23}^2
s_{12} \}
\end{eqnarray}
In concordance with Ref. \cite{dic}, we find that the $3 \gamma $ matrix
element contains a large number of momentum factors, which are ultimately
responsible for an enormous suppression of the decay rate. (The result given
by our model has two extra powers of $s_{ij} $ compared to the expression
in Eq. (8).)
\end{itemize}
Finally, we obtain the rate of $K_L \to 3 \gamma $ using
\begin{equation}
{d \Gamma \over ds_{12}\, ds_{23} } = {1\over (2 \pi )^3} { 1\over 32 M_K^3}
{1\over 3!} \sum |{\cal M}|^2
\end{equation}
and assuming that $\sum |{\cal M}|^2 $ is reasonably approximated by the
absorptive part given in Eq. (5). The parameter $h_L $ is determined by the
decay rate of $K_L \to \pi^0 \pi^0 \gamma $, for which we use the theoretical
estimate $B(K_L \to \pi^0 \pi^0 \gamma ) \sim 1 \cdot 10^{-8} $ obtained in
\cite{own}. The resulting branching ratio for $K_L \to 3 \gamma $ is
\begin{equation}
B(K_L \to 3 \gamma ) \sim 3 \cdot 10^{-19}
\end{equation}
The above considerations can be repeated for the decay $K_S \to 3 \gamma $, the
only difference being that the E2 matrix element given in Eq. (1) has to be
replaced by the M2 matrix element for $K_S \to \pi^0 \pi^0 \gamma $:
\begin{equation}
{\cal M} (K_S (p_K) \to \pi^0(p_1) \pi^0 (p_2) \gamma (\epsilon, k)
= {h_S \over M_K^5} \, (p_1 - p_2) \cdot k \, \epsilon_{\mu \nu \rho \sigma}
\epsilon^{\mu} k^{\nu} p_1^{\rho} p_2^{\sigma}
\end{equation}
The Dalitz plot density turns out to have exactly the same functional
dependence on $s_{12}, s_{23} $ and $s_{13} $ as in Eq. (5). The branching
ratio is estimated to be
\begin{equation}
B(K_S \to 3 \gamma ) \sim 5 \cdot 10^{-22}
\end{equation}
The exceedingly low rates given by Eqs. (10) and (12) imply that the three
photon decay of the neutral $K $ meson cannot be a significant background to
decays of the type $K^0 \to 2 \pi^0 \to 4 \gamma $, in which one photon is
undetected. \par
\bigskip
\bigskip

{\bf \large Acknowledgement} \par
\bigskip
This work was initiated during a visit by one of us (L.M.S.) to the University
of Melbourne. The hospitality of the School of Physics is gratefully
acknowledged. The research has been supported by the German Ministry of
Research and Technology.


\newpage
{\large \bf Appendix} \par

\bigskip
\noindent
We give here the integrals defined in Eq. (4), for the general case $p_1^2
= m_1^2, \; p_2^2 = m_2^2 $.
\begin{eqnarray}
K^{\mu \nu \rho \sigma} & = & \int {d^3 p_1 \over 2 p_{1 \, 0} }
{d^3 p_2 \over 2 p_{2 \, 0} } \delta^{(4)} \; (P - p_1 -p_2) \;
f(p_1\cdot p_2) \; p_1^{\mu} p_1^{\nu} p_1^{\rho} p_1^{\sigma} \nonumber
\\
& = & {\pi \over 2} {1\over s^3} \sqrt{\lambda (s, m_1^2, m_2^2)}
f \left [ {1\over 2} (s - m_1^2 -m_2^2) \right ] \; \cdot {1\over 5} \nonumber
\\
& & \cdot \left \{ {1\over s^2} \left [ (s + m_1^2 - m_2^2)^4 - 3 s m_1^2
(s+ m_1^2 - m_2^2)^2 + s^2 m_1^4 \right ] P^{\mu} P^{\nu} P^{\rho} P^{\sigma}
\right. \nonumber \\
& & - \left [ {1\over 8} {1\over s} (s + m_1^2 - m_2^2)^4 - {7 \over 12} m_1^2
(s + m_1^2 - m_2^2)^2 + {1\over 3} s m_1^4 \right ] \nonumber \\
& & \cdot \left (P^{\mu} P^{\nu} g^{\rho \sigma} + P^{\nu} P^{\rho}
g^{\mu \sigma} + P^{\mu} P^{\rho} g^{\nu \sigma} + P^{\nu} P^{\sigma}
g^{\mu \rho} + P^{\rho} P^{\sigma} g^{\mu \nu} + P^{\mu} P^{\sigma}
g^{\nu \rho} \right ) \nonumber \\
& & + {1\over 3} \left [ {1\over 16} (s + m_1^2 - m_2^2)^4 - {1\over 2} s
m_1^2 (s+ m_1^2-m_2^2)^2 + s^2 m_1^4 \right ] \nonumber \\
& &  \cdot \left ( g^{\mu \nu} g^{\rho \sigma} + g^{\nu \rho}
g^{\mu \sigma} + g^{\mu \rho} g^{\nu \sigma } \right ) {\Large \}} \nonumber
\end{eqnarray}

\begin{eqnarray}
L^{\mu \nu \rho \sigma} & = & \int {d^3 p_1 \over 2 p_{1 \, 0} }
{d^3 p_2 \over 2 p_{2 \, 0} } \delta^{(4)} \; (P - p_1 - p_2) \;
f(p_1 \cdot p_2) \; p_1^{\mu } p_1^{\nu } p_1^{\rho } p_2^{\sigma} \nonumber
\\
& = & {\pi \over 2} {1 \over s^3} \sqrt{\lambda (s, m_1^2, m_2^2)}
f \left [ {1\over 2} (s - m_1^2 - m_2^2) \right ] \; \cdot {1 \over 5}
\nonumber \\
& & \cdot \left \{ {1\over s^2} \left [ (s + m_1^2 - m_2^2)^3
(s - m_1^2 + m_2^2)^2 - {3\over 4} s \left \{ (s+ m_1^2 - m_2^2)^2 \right.
\right. \right. \nonumber \\
& & \;\;\;\;\;\;\;\; \left. \cdot (s - m_1^2 -m_2^2) + m_1^2
(s + m_1^2 -m_2^2) (s - m_1^2 + m_2^2) \right \} \nonumber \\
& & \;\;\;\;\;\;\;\; \left. + {1\over 2} s^2 m_1^2 (s - m_1^2 - m_2^2) \right ]
\; P^{\mu} P^{\nu} P^{\rho} P^{\sigma} \nonumber \\
& & \;\;\; - \left [ {1\over 8} {1\over s} (s + m_1^2 - m_2^2)^3 (s - m_1^2
+ m_2^2) - {7\over 48} \left \{ (s + m_1^2 - m_2^2)^2 \right. \right.
\nonumber \\
& & \;\;\;\;\;\;\; \left. \cdot (s - m_1^2 -m_2^2) + m_1^2 (s + m_1^2 - m_2^2)
(s - m_1^2 + m_2^2) \right \} \nonumber \\
& & \;\;\;\;\;\;\; \left. + {1\over 6} s m_1^2 (s - m_1^2 - m_2^2) \right ]
\nonumber \\
& & \cdot (P^{\mu} P^{\nu} g^{\rho \sigma} + P^{\nu} P^{\rho} g^{\mu \sigma}
+ P^{\mu} P^{\rho} g^{\nu \sigma} + P^{\nu} P^{\sigma} g^{\mu \rho}
+ P^{\rho} P^{\nu} g^{\mu \nu} + P^{\mu} P^{\sigma} g^{\nu \rho} )
\nonumber \\
& & + {1\over 3} \left [ {1\over 16} ( s+ m_1^2 - m_2^2)^3 (s - m_1^2 + m_2^2)
- {1\over 8} s \left \{ (s + m_1^2 - m_2^2)^2 \right. \right. \nonumber \\
& & \;\;\;\;\;\;\; \left. \cdot (s - m_1^2 - m_2^2) + m_1^2 (s + m_1^2
- m_2^2) (s - m_1^2 + m_2^2) \right \} \nonumber \\
& & \;\;\;\;\;\;\; \left. \left. + {1\over 2} s^2 m_1^2 (s - m_1^2 - m_2^2)
\right ] \; \cdot (g^{\mu \nu} g^{\rho \sigma} + g^{\nu \rho} g^{\mu \sigma}
+ g^{\mu \rho} g^{\nu \sigma} ) \right \} \nonumber
\end{eqnarray}





\newpage
\begin{thebibliography}{99}
%
\bibitem{own} P. Heiliger and L.M. Sehgal, Phys. Lett. B 307 (1993) 182.
\bibitem{ko} P. Ko, Phys. Rev. D 41 (1990) 1531.
\bibitem{form} J.A.M. Vermaseren, Symobolic Manipulation with FORM,
               CAN (1991).
\bibitem{dic} D.A. Dicus, Phys. Rev. D 12 (1975) 2133.
\end{thebibliography}

\end{document}





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/Courier /Courier accvec ReEncode
/Courier-Oblique /Courier-Oblique accvec ReEncode
/Courier-Bold /Courier-Bold accvec ReEncode
/Courier-BoldOblique /Courier-BoldOblique accvec ReEncode
/oshow {gsave [] 0 sd true charpath stroke gr} def
/stwn { /fs exch def /fn exch def /text exch def fn findfont fs sf
 text sw pop xs add /xs exch def} def
/stwb { /fs exch def /fn exch def /nbas exch def /textf exch def
textf length /tlen exch def nbas tlen gt {/nbas tlendef} if
fn findfont fs sf textf dup length nbas sub nbas getinterval sw
pop neg xs add /xs exch def} def
/accspe [ 65 /plusminus 66 /bar 67 /existential 68 /universal
69 /exclam 70 /numbersign 71 /greater 72 /question 73 /integral
74 /colon 75 /semicolon 76 /less 77 /bracketleft 78 /bracketright
79 /greaterequal 80 /braceleft 81 /braceright 82 /radical
83 /spade 84 /heart 85 /diamond 86 /club 87 /lessequal
88 /multiply 89 /percent 90 /infinity 48 /circlemultiply 49 /circleplus
50 /emptyset 51 /lozenge 52 /bullet 53 /arrowright 54 /arrowup
55 /arrowleft 56 /arrowdown 57 /arrowboth 48 /degree 44 /comma 43 /plus
 45 /angle 42 /angleleft 47 /divide 61 /notequal 40 /equivalence 41 /second
 97 /approxequal 98 /congruent 99 /perpendicular 100 /partialdiff 101 /florin
 102 /intersection 103 /union 104 /propersuperset 105 /reflexsuperset
 106 /notsubset 107 /propersubset 108 /reflexsubset 109 /element 110
/notelement
 111 /gradient 112 /logicaland 113 /logicalor 114 /arrowdblboth
 115 /arrowdblleft 116 /arrowdblup 117 /arrowdblright 118 /arrowdbldown
 119 /ampersand 120 /omega1 121 /similar 122 /aleph ] def
/Symbol /Special accspe ReEncode
/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 setgray [] 0 sd 1 lw 6 lw 445 1850 m 878 1850 l s 934 1850 m 934
 1852 l 933 1853 l 933 1855 l 933 1857 l 932 1859 l 932 1860 l 931 1862 l 930
 1863 l 929 1865 l 928 1866 l 927 1868 l 926 1869 l 925 1870 l 923 1871 l 922
 1873 l 921 1873 l 919 1874 l 917 1875 l 916 1876 l 914 1876 l 912 1877 l 911
 1877 l 909 1877 l 907 1878 l 905 1878 l 904 1878 l 902 1877 l 900 1877 l 898
 1877 l 897 1876 l 895 1876 l 894 1875 l 892 1874 l 890 1873 l 889 1872 l 888
 1871 l 886 1870 l 885 1868 l 884 1867 l 883 1866 l 882 1864 l 881 1863 l 880
 1861 l 880 1859 l 879 1858 l 879 1856 l 878 1854 l 878 1853 l 878 1851 l 878
 1849 l 878 1847 l 878 1845 l 879 1844 l 879 1842 l 880 1840 l 880 1839 l 881
 1837 l 882 1836 l 883 1834 l 884 1833 l 885 1831 l 886 1830 l 888 1829 l 889
 1828 l 890 1827 l 892 1826 l 894 1825 l 895 1824 l 897 1824 l 898 1823 l 900
 1823 l 902 1822 l 904 1822 l 905 1822 l 907 1822 l 909 1822 l 911 1823 l 912
 1823 l 914 1823 l 916 1824 l 917 1825 l 919 1825 l 921 1826 l 922 1827 l 923
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1751
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 1245 1934 l 1244 1945 l 1243 1957 l 1240 1968 l s 1256 1989 m 1256 1991 l 1256
 1992 l 1256 1994 l 1255 1996 l 1255 1997 l 1254 1999 l 1253 2001 l 1253 2002 l
 1252 2004 l 1251 2005 l 1250 2007 l 1248 2008 l 1247 2009 l 1246 2010 l 1244
 2011 l 1243 2012 l 1241 2013 l 1240 2014 l 1238 2015 l 1236 2015 l 1235 2016 l
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 2016 l 1221 2016 l 1219 2015 l 1217 2014 l 1216 2014 l 1214 2013 l 1213 2012 l
 1211 2011 l 1210 2010 l 1209 2009 l 1208 2007 l 1206 2006 l 1205 2005 l 1204
 2003 l 1204 2002 l 1203 2000 l 1202 1998 l 1202 1997 l 1201 1995 l 1201 1993 l
 1201 1991 l 1200 1990 l 1200 1988 l 1201 1986 l 1201 1984 l 1201 1983 l 1202
 1981 l 1202 1979 l 1203 1978 l 1204 1976 l 1204 1975 l 1205 1973 l 1206 1972 l
 1208 1970 l 1209 1969 l 1210 1968 l 1211 1967 l 1213 1966 l 1214 1965 l 1216
 1964 l 1217 1963 l 1219 1963 l 1221 1962 l 1223 1962 l 1224 1961 l 1226 1961 l
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 1963 l 1240 1964 l 1241 1964 l 1243 1965 l 1244 1966 l 1246 1967 l 1247 1968 l
 1248 1970 l 1250 1971 l 1251 1972 l 1252 1974 l 1253 1975 l 1253 1977 l 1254
 1978 l 1255 1980 l 1255 1982 l 1256 1984 l 1256 1985 l 1256 1987 l 1256 1989 l
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1604
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 1524 l 1041 1522 l 1042 1521 l 1043 1519 l 1044 1518 l 1045 1516 l s 1255 1993
 m 1257 1996 l 1259 1997 l 1260 1998 l 1262 1999 l 1264 1999 l 1265 2000 l 1267
 2001 l 1269 2001 l 1272 2002 l 1274 2002 l 1276 2002 l 1278 2003 l 1280 2003 l
 1283 2003 l 1285 2004 l 1287 2004 l 1289 2005 l 1291 2005 l 1293 2006 l 1295
 2006 l 1297 2007 l 1298 2008 l 1300 2009 l 1301 2010 l 1302 2011 l 1304 2012 l
 1305 2014 l 1306 2015 l 1306 2017 l 1307 2019 l 1308 2021 l 1308 2023 l 1309
 2025 l 1309 2027 l 1309 2029 l 1310 2031 l 1310 2033 l 1310 2036 l 1310 2038 l
 1311 2040 l 1311 2042 l 1311 2044 l 1312 2046 l 1312 2048 l 1313 2050 l 1314
 2052 l 1315 2054 l 1316 2055 l 1317 2057 l 1318 2058 l s 1318 2058 m 1319 2059
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 2203 l 1511 2205 l 1512 2207 l 1512 2209 l 1513 2211 l 1513 2213 l 1514 2215 l
 1515 2217 l 1516 2219 l 1517 2221 l 1518 2223 l 1519 2224 l 1520 2225 l 1521
 2227 l 1523 2228 l s 1255 1984 m 1257 1982 l 1259 1981 l 1260 1980 l 1261 1978
 l 1262 1976 l 1263 1975 l 1264 1973 l 1265 1971 l 1265 1969 l 1266 1967 l 1267
 1965 l 1267 1962 l 1268 1960 l 1268 1958 l 1269 1956 l 1270 1954 l 1270 1952 l
 1271 1950 l 1272 1948 l 1273 1946 l 1274 1945 l 1275 1943 l 1276 1942 l 1277
 1941 l 1279 1939 l 1280 1938 l 1282 1938 l 1283 1937 l 1285 1936 l 1287 1936 l
 1289 1936 l 1291 1935 l 1293 1935 l 1295 1935 l 1297 1935 l 1300 1935 l 1302
 1935 l 1304 1935 l 1306 1935 l 1309 1935 l 1311 1935 l 1313 1935 l 1315 1935 l
 1317 1934 l 1319 1934 l 1321 1934 l 1323 1933 l 1324 1932 l 1326 1931 l 1327
 1930 l s 1327 1930 m 1329 1929 l 1330 1928 l 1331 1926 l 1332 1925 l 1333 1923
 l 1334 1921 l 1335 1919 l 1335 1917 l 1336 1915 l 1336 1913 l 1337 1911 l 1337
 1909 l 1338 1907 l 1338 1905 l 1339 1902 l 1339 1900 l 1340 1898 l 1341 1896 l
 1341 1894 l 1342 1893 l 1343 1891 l 1344 1889 l 1345 1888 l 1346 1887 l 1348
 1885 l 1349 1884 l 1351 1883 l 1352 1883 l 1354 1882 l 1356 1881 l 1358 1881 l
 1360 1881 l 1362 1880 l 1364 1880 l 1366 1880 l 1368 1880 l 1370 1880 l 1373
 1880 l 1375 1880 l 1377 1879 l 1379 1879 l 1382 1879 l 1384 1879 l 1386 1878 l
 1388 1878 l 1389 1877 l 1391 1877 l 1393 1876 l 1394 1875 l 1396 1874 l s 1396
 1874 m 1397 1872 l 1398 1871 l 1399 1870 l 1400 1868 l 1401 1866 l 1402 1864 l
 1402 1862 l 1403 1860 l 1403 1858 l 1404 1856 l 1404 1854 l 1405 1852 l 1405
 1850 l 1405 1847 l 1406 1845 l 1406 1843 l 1407 1841 l 1407 1839 l 1408 1837 l
 1409 1835 l 1409 1833 l 1410 1832 l 1411 1830 l 1412 1829 l 1414 1828 l 1415
 1827 l 1416 1826 l 1418 1825 l 1420 1824 l 1421 1823 l 1423 1823 l 1425 1822 l
 1427 1822 l 1429 1822 l 1432 1821 l 1434 1821 l 1436 1821 l 1438 1821 l 1441
 1820 l 1443 1820 l 1445 1820 l 1447 1820 l 1449 1819 l 1451 1819 l 1453 1818 l
 1455 1817 l 1456 1817 l 1458 1816 l 1460 1815 l 1461 1813 l s 1461 1813 m 1462
 1812 l 1463 1811 l 1464 1809 l 1465 1807 l 1466 1806 l 1466 1804 l 1467 1802 l
 1467 1800 l 1468 1798 l 1468 1796 l 1468 1793 l 1469 1791 l 1469 1789 l 1469
 1787 l 1469 1784 l 1470 1782 l 1470 1780 l 1471 1778 l 1471 1776 l 1472 1774 l
 1472 1773 l 1473 1771 l 1474 1769 l 1475 1768 l 1476 1767 l 1478 1765 l 1479
 1764 l 1481 1763 l 1482 1762 l 1484 1762 l 1486 1761 l 1488 1761 l 1490 1760 l
 1492 1760 l 1494 1759 l 1496 1759 l 1498 1759 l 1501 1758 l 1503 1758 l 1505
 1758 l 1507 1757 l 1509 1757 l 1511 1756 l 1513 1756 l 1515 1755 l 1517 1754 l
 1519 1753 l 1520 1752 l 1522 1751 l 1523 1750 l s 1 lw 630 1944 m 630 1905 l s
 656 1944 m 630 1918 l s 639 1928 m 656 1905 l s 665 1915 m 665 1896 l s 665
 1896 m 676 1896 l s 683 1897 m 682 1896 l 681 1897 l 682 1898 l 683 1897 l 683
 1895 l 682 1893 l 681 1892 l s 702 1912 m 700 1914 l 697 1915 l 694 1915 l 691
 1914 l 689 1912 l 689 1911 l 690 1909 l 691 1908 l 693 1907 l 698 1905 l 700
 1904 l 701 1903 l 702 1901 l 702 1899 l 700 1897 l 697 1896 l 694 1896 l 691
 1897 l 689 1899 l s 1113 1537 m 1117 1541 l 1121 1542 l 1123 1542 l 1126 1541
l
 1128 1539 l 1130 1533 l 1130 1526 l 1128 1516 l s 1143 1542 m 1141 1537 l 1139
 1533 l 1128 1516 l 1124 1509 l 1123 1503 l s 1569 1759 m 1573 1763 l 1576 1765
 l 1578 1765 l 1582 1763 l 1584 1761 l 1586 1755 l 1586 1748 l 1584 1739 l s
 1599 1765 m 1597 1759 l 1595 1755 l 1584 1739 l 1580 1731 l 1578 1726 l s 1569
 2204 m 1573 2207 l 1576 2209 l 1578 2209 l 1582 2207 l 1584 2206 l 1586 2200 l
 1586 2193 l 1584 2183 l s 1599 2209 m 1597 2204 l 1595 2200 l 1584 2183 l 1580
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 1722 l 1176 1716 l s 1149 1737 m 1152 1741 l 1158 1742 l 1182 1742 l s 1190
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 1193 1753 l 1193 1756 l 1192 1758 l 1190 1759 l cl s 961 2165 m 954 2139 l s
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 982 2165 l s 990 2181 m 988 2180 l 987 2178 l 987 2175 l 988 2174 l 990 2173 l
 992 2174 l 993 2175 l 993 2178 l 992 2180 l 990 2181 l cl s 620 1005 m 620 961
 l s 620 1005 m 647 1005 l s 620 984 m 637 984 l s 656 1005 m 658 1003 l 660
 1005 l 658 1007 l 656 1005 l cl s 658 990 m 658 961 l s 698 990 m 698 956 l
696
 950 l 694 948 l 690 946 l 683 946 l 679 948 l s 698 984 m 694 988 l 690 990 l
 683 990 l 679 988 l 675 984 l 673 978 l 673 973 l 675 967 l 679 963 l 683 961
l
 690 961 l 694 963 l 698 967 l s 717 965 m 715 963 l 717 961 l 719 963 l 717
965
 l cl s 762 997 m 766 999 l 772 1005 l 772 961 l s 842 1005 m 842 961 l s 842
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 988 l 893 984 l 891 978 l 891 973 l 893 967 l 897 963 l 901 961 l 908 961 l
912
 963 l 916 967 l 918 973 l 918 978 l 916 984 l 912 988 l 908 990 l 901 990 l cl
 s 956 1005 m 956 961 l s 956 984 m 952 988 l 948 990 l 942 990 l 937 988 l 933
 984 l 931 978 l 931 973 l 933 967 l 937 963 l 942 961 l 948 961 l 952 963 l
956
 967 l s 971 978 m 997 978 l 997 982 l 995 986 l 992 988 l 988 990 l 982 990 l
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l
 992 963 l 997 967 l s 1011 1005 m 1011 961 l s 1062 1005 m 1058 1005 l 1054
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m
 1194 961 l s 1204 972 m 1204 950 l s 1204 950 m 1217 950 l s 1224 951 m 1223
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 1465 952 l 1463 946 l s gr
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