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%Date: Tue, 20 Jul 93 16:54 EDT


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\begin{document}

\begin{titlepage}
\begin{flushright}
{\sc SSCL-Preprint-482}\\
{\sc WIS-93/67/July-PH}\\
{\sc CMU-HEP93-10; DOE-ER/40682-35}\\
{\sc \today}\\ [.5in]
\end{flushright}
\begin{center}
{\LARGE Constraints on Extended Technicolor Models
from $B \rightarrow \mu^+ \mu^- X$}\\ [.3in]
{\large Benjam\'{\i}n Grinstein}\\[.1in]
{\small SSC Laboratory, 2550 Beckleymeade Ave.,
Dallas TX 75237} \\[.1in]
{\large Yosef Nir} \\[.1in]
{\small Physics Department, Weizmann Institute of Science, Rehovot 76100,
Israel} \\[.1in]
{\large and}\\ [.1in]
{\large Jo\~{a}o M. Soares}\\ [.1in]
{\small Department of Physics, Carnegie Mellon
University, Pittsburgh, PA 15213} \\[.5in]

{\normalsize\bf Abstract}\\ [.2in]
\end{center}
{\small A recent study by Randall and Sundrum shows that models
of Extended Technicolor (ETC) have interesting implications on rare
$B$ decays. We extend their study to the decay $B\rightarrow\mu^+\mu^-X$.
ETC models with a GIM mechanism predict a decay rate that is a
factor of order 30 above the Standard Model, violating the experimental
upper bound by a factor of 2--4. ``Traditional" ETC models predict
a decay rate that is a factor of order 4 above the Standard Model,
and will be probed when an improvement in the sensitivity of
experiments by a factor of order 2--4 is achieved.}
\end{titlepage}

\section*{}

In a recent paper, Randall and Sundrum \cite{Lisa:93}
studied the contributions to the decays
$b\rightarrow s\gamma$ and $B_s \rightarrow \mu^+ \mu^-$ from various
classes of Extended Technicolor (ETC) models.
Models with and without a GIM mechanism were considered.
In both cases, the radiative decay has roughly the same rate as
in the Standard Model (SM). However, ETC contributions enhance the rate
for $B_s \to \mu ^+\mu ^-$ by one to two
orders of magnitude.  In this paper, we point out that a similar
enhancement occurs for the decay $b \rightarrow s \mu^+ \mu^-$. This is
particularly interesting because there exists an upper bound on this
rate \cite{UA1:91} which is only an order of magnitude above the SM rate
and, furthermore, near term experiments are expected to further improve
this bound \cite{Snowmass}. In this brief note,
we estimate $BR(b\rightarrow s\mu^+\mu^-)$
in the two classes of ETC models discussed in ref.~\cite{Lisa:93},
and compare it to the present experimental upper bound.

The first class
of models considered in ref.~\cite{Lisa:93}, or ``traditional" ETC
models, contains the minimal set of interactions necessary for quark mass
generation. (The study is restricted to models where neither composite
nor fundamental scalars are involved in the quark mass generation.)
It is this very minimal set of interactions that are considered in
the generation of flavor changing neutral currents (FCNC). As such,
it is safe to assume that predictions based on these can be
considered as lower bounds. The caveat, of course, is the possibility of
cancellations once additional interactions are incorporated.  The rates
that we find strongly deviate from the SM ones and, therefore, unless
these additional cancellations are rather precise, one may conceivably
rule out this class of models in the near future.

The second class of models, those with a ``techni-GIM'' mechanism,
incorporate interactions beyond the minimal set of traditional models.
They are theoretically appealing since, unlike traditional ETC,
the massive vector bosons of the ETC interactions
are nearly degenerate, with masses of a few TeV. Although
these are rather light by
traditional ETC standards, this class of models avoids many dangerous low
energy FCNC interactions by incorporating an automatic
techni-GIM cancellation mechanism. In the absence of masses,
the quark sector has a large $SU(3)^3$ flavor symmetry. If the only
parameters that break this symmetry are the quark mass matrices (in the
weak eigenstate basis), all of
the flavor changing interactions must be proportional to them. Techni-GIM
cancellations occur when one rotates to the mass eigenstate basis.
However, as we show below, this class of models seems to violate
the existing bound on the rate for $b \rightarrow s \mu^+ \mu^-$.

The flavor changing interactions that are induced at low energies
by these classes of ETC models, and that are relevant for
$b \rightarrow s \mu^+ \mu^-$, are \cite{Lisa:93}
($i$) the neutral current $Z$-boson coupling
\begin{equation}
\xi^{(i)} \frac{m_t}{16 \pi v}\frac{e}{\cos\theta_W \sin\theta_W}
Y_u^{\dag 33} Y_u^{32} \bar{s}_L \gamma_\mu b_L Z^\mu ,
\label{eq:1}
\end{equation}
which is present and of similar strength in traditional models and in
models with a techni-GIM mechanism, and ($ii$) the 4-fermion interaction
\begin{equation} \xi^{(ii)} \frac{m_t}{4 \pi v^3}Y_u^{\dag 33}
Y_u^{32} \bar{s}_L \gamma_\mu b_L \bar{\mu}_L \gamma^\mu \mu_L,
\label{eq:2}
\end{equation}
which is highly suppressed in traditional models, but not in models with
a techni-GIM mechanism.
Here, $v=246\ {\rm GeV}$ is the electroweak symmetry breaking scale.
The coefficients $\xi^{(i)}$ and $\xi^{(ii)}$ are model-specific. The
normalization of the matrix $Y_u$ is such that $Y_u^{33} =1$.
Following ref. \cite{Lisa:93}, we take for our computations, as
a rough estimate of the couplings, $Y_u^{32} \sim V_{ts}$, $\xi^{(i)}
\sim 1$, and $\xi^{(ii)}\sim 1$ in models with a techni-GIM, while
$\xi^{(ii)}\ll1$ in traditional models. The dominant ETC contributions
to the decay Hamiltonian
\begin{equation}
{\cal H}_{\rm eff}=2\sqrt{2}G_F\sum_{j=8,9}C_j{\cal O}_j, \label{eq:3}
\end{equation}
where
\begin{eqnarray}
{\cal O}_8 &=& V_{ts} (e^2/16\pi ^2)
\bar{s}_L \gamma_\mu b_L \bar{\mu} \gamma^\mu \mu,\nonumber \\
{\cal O}_9 &=& V_{ts} (e^2/16\pi ^2)
\bar{s}_L \gamma_\mu b_L \bar{\mu} \gamma^\mu\gamma_5\mu,  \label{eq:4}
\end{eqnarray}
are then
\begin{eqnarray}
C_8^{\rm (i)} = C_9^{\rm (i)} (1- 4 \sin^2 \theta_W ) &\makebox[.25in]{}&
C_9^{\rm (i)} =\xi^{(i)} m_t / 8 v \alpha_{QED}, \label{eq:5} \\
C_8^{\rm (ii)} = - C_9^{\rm (ii)} &\makebox[.25in]{}&
C_9^{\rm (ii)} =\xi^{(ii)} m_t / 4 v \alpha_{QED}, \label{eq:6}
\end{eqnarray}
for the interactions ($i$) and ($ii$), respectively. There are no
short distance QCD corrections to this effective Hamiltonian. More
precisely, there is no multiplicative correction to $C_8$ since the FCNC
is partially conserved.  There are additive corrections \cite{Grin:89}
to $C_8$ that arise from mixing of the SM four-quark interaction into the
operator ${\cal O}_8$. Since the ETC contributions are much larger than
the SM ones, there are effectively no short distance QCD corrections.

The differential rate $d\Gamma(b \to s \mu^+ \mu^-)/dx$, where $x =
(p_{\mu^+}+p_{\mu^-})^2 /m_B^2$, is usually expressed in units $\Gamma
(b \to c e \bar{\nu}_e)$, in which case the dependence on CKM parameters
drops out. The rate is given by \cite{Grin:89}
\begin{eqnarray}
\lefteqn{\frac{1}{\Gamma(b\to ce\bar{\nu}_e)}d\Gamma(b\to s\mu^+\mu^-)/dx
=}\nonumber \\ & & \frac{\alpha_{QED}^2}{4 \pi^2 f(m_c/m_b)} (1-x)^2
(1+2x)(|C_8|^2 + |C_9|^2)
\end{eqnarray}
($f(0.3)= 0.52$ is a phase space factor)
and is plotted in fig.~1, for $m_t=160\
{\rm GeV}$, for both type ($i$) and ($ii$) contributions. The SM
result \cite{Grin:89} is also given in the figure for comparison.

Similar to the results of ref.~\cite{Lisa:93} for the decay $B_s\to\mu^+
\mu^-$, we find that the rate for $b\to s\mu^+ \mu^-$ is enhanced,
compared to the SM, by a factor of approximately 4 for
type~($i$) interactions, and by a factor of approximately 30
for type~($ii$) interactions. The corresponding integrated rates are given
in table 1 for various values of $m_t$. They should be compared to
the experimental upper bound \cite{UA1:91}:
$BR(B \rightarrow \mu^+ \mu^- X)\leq 5\times10^{-5}$.
We stress that these estimates are very rough since the precise values
of the coefficients $\xi^{(i)}$ and $\xi^{(ii)}$  are not known.
Moreover, the various contributions may interfere destructively.
Fig.~1 shows that destructive interference could significantly reduce
the rate. Nevertheless, it is unlikely
that the resulting cancellations would be so fine-tuned as to
reduce the rate down to a level close to that of the SM. Note that
even the SM contribution could significantly reduce the total rate
through destructive interference with ETC amplitudes.

In ETC models that have a GIM mechanism, the 4-fermion interaction
($ii$) gives the dominant effect in $b\rightarrow s\mu^+\mu^-$.
Our results, given in table 1, imply that this type of ETC models are
ruled out by experiment, unless the $Z$-exchange contribution from ($i$)
adds destructively or if the various couplings of order one are
somewhat suppressed compared to our naive estimates. An
improvement of the experimental bound by a factor of a few would be
sufficient to rule out that case too -- barring delicate cancellations
or fine-tuning. For traditional models, the type ($ii$) interaction is
highly suppressed and the rate is dominated by the type ($i$)
interaction. The rate is lower than in the previous case by an order of
magnitude, and is marginally allowed by the current experimental upper
limit. Again, it should be probed in the near future when on-going
experiments at CLEO and the Tevatron
derive a bound stronger than the present one by a factor of a few.

\section*{Acknowledgements}
This work was initiated at the Workshop on B Physics at Hadron
Accelerators (Snowmass, Colorado, June 1993). The authors are
grateful to the organization for partial financial support.
This work was also partly supported by the U. S. Department of Energy
under Contracts No. DE-FG02-91ER40682 and DE--AC35--89ER40486.
YN is grateful to the theory group at CERN for their hospitality.
YN is an incumbent of the Ruth E. Recu Career Development Chair
and is supported, in part, by the Israel Commission for Basic Research
and by the Minerva Foundation.

\begin{thebibliography}{Abcd 99a}
\bibitem{Lisa:93}L. Randall and R. Sundrum, preprint MIT-CTP-2211 (1993).
\bibitem{UA1:91}C. Albajar et al. (UA1 Collaboration),
 Phys. Lett. {\bf B262} (1991) 163.
\bibitem{Snowmass}See, for example, report of the $\delta$-working group
in the proceedings of the Workshop on B-Physics in Hadron Accelerators
(Snowmass 1993), to be published.
\bibitem{Grin:89}B. Grinstein, M. Savage and M. Wise,
 Nucl. Phys. {\bf B319} (1989) 271.
\end{thebibliography}


\pagebreak
\pagestyle{empty}
\section*{Figure Captions}
\begin{tabbing}
\=Figure 1: \=The differential decay rate $d\Gamma(b\to s\mu^+\mu^-)/dx$
($x = (p_{\mu^+}+p_{\mu^-})^2 /m_B^2$)\\
\>          \>due to the ETC interactions ($i$) and ($ii$) and in the SM,
\\ \>           \>for $m_t = 160\ {\rm GeV}$.\\
\>          \>    \\
\end{tabbing}

\section*{Table Captions}
\begin{tabbing}
\=Table 1: \=The branching ratio for $B \to  \mu^+ \mu^- X$
in the SM, and that due\\
\>         \>to the ETC interactions
($i$) and ($ii$).\\
\>         \>     \\
\end{tabbing}
\vspace{.8in}

\pagebreak
\pagestyle{empty}
\section*{}
\begin{center}
\begin{tabular}{c|ccc}
\multicolumn{4}{c}{{\bf Table 1}}\\ [.2in]
   $m_t$/GeV    &  SM  &  ($i$)  & ($ii$) \\ [.1in] \hline \\
$130$ &  $3.5 \times 10^{-6}$  & $1.5 \times 10^{-5}$  & $1 \times 10^{-4}$
\\ [.1in]
$160$ &  $5.1 \times 10^{-6}$  & $2 \times 10^{-5}$ & $1.5 \times 10^{-4}$
\\ [.1in]
$190$ &  $7.4 \times 10^{-6}$  & $3 \times 10^{-5}$ &  $2 \times 10^{-4}$
\end{tabular}
\end{center}

\end{document}

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  779 2350 D 781 2350 D 784 2349 D 787 2349 D 790 2349 D 793 2348 D 796 2348 D
  798 2347 D 801 2347 D 804 2347 D 807 2346 D 810 2346 D 812 2345 D 815 2345 D
  818 2345 D 821 2344 D 824 2344 D 826 2343 D 829 2343 D 832 2342 D 835 2342 D
  838 2341 D 841 2341 D 843 2340 D 846 2340 D 849 2339 D 852 2339 D 855 2338 D
  857 2338 D 860 2337 D 863 2337 D 866 2336 D 869 2336 D 871 2335 D 874 2335 D
  877 2334 D 880 2334 D 883 2333 D 886 2333 D 888 2332 D 891 2331 D 894 2331 D
  897 2330 D 900 2330 D 902 2329 D 905 2328 D 908 2328 D 911 2327 D 914 2327 D
  916 2326 D 919 2325 D 922 2325 D 925 2324 D 928 2323 D 931 2323 D 933 2322 D
  936 2321 D 939 2321 D 942 2320 D 945 2319 D 947 2319 D 950 2318 D 953 2317 D
  956 2316 D 959 2316 D 961 2315 D 964 2314 D 967 2314 D 970 2313 D 973 2312 D
  975 2311 D 978 2311 D 981 2310 D 984 2309 D 987 2308 D 990 2307 D 992 2307 D
  995 2306 D 998 2305 D 1001 2304 D 1004 2303 D 1006 2302 D 1009 2302 D
  1012 2301 D 1015 2300 D 1018 2299 D 1020 2298 D 1023 2297 D 1026 2297 D
  1029 2296 D 1032 2295 D 1035 2294 D 1037 2293 D 1040 2292 D 1043 2291 D
  1046 2290 D 1049 2289 D 1051 2288 D 1054 2287 D 1057 2286 D 1060 2286 D
  1063 2285 D 1065 2284 D 1068 2283 D 1071 2282 D 1074 2281 D 1077 2280 D
  1080 2279 D 1082 2278 D 1085 2277 D 1088 2276 D 1091 2275 D 1094 2274 D
  1096 2273 D 1099 2271 D 1102 2270 D 1105 2269 D 1108 2268 D 1110 2267 D
  1113 2266 D 1116 2265 D 1119 2264 D 1122 2263 D 1125 2262 D 1127 2261 D
  1130 2260 D 1133 2258 D 1136 2257 D 1139 2256 D 1141 2255 D 1144 2254 D
  1147 2253 D 1150 2251 D 1153 2250 D 1155 2249 D 1158 2248 D 1161 2247 D
  1164 2245 D 1167 2244 D 1169 2243 D 1172 2242 D 1175 2241 D 1178 2239 D
  1181 2238 D 1184 2237 D 1186 2236 D 1189 2234 D 1192 2233 D 1195 2232 D
  1198 2230 D 1200 2229 D 1203 2228 D 1206 2226 D 1209 2225 D 1212 2224 D
  1214 2222 D 1217 2221 D 1220 2220 D 1223 2218 D 1226 2217 D
  T 1226 2217 M 1229 2215 D 1231 2214 D 1234 2213 D 1237 2211 D 1240 2210 D
  1243 2208 D 1245 2207 D 1248 2205 D 1251 2204 D 1254 2203 D 1257 2201 D
  1259 2200 D 1262 2198 D 1265 2197 D 1268 2195 D 1271 2194 D 1274 2192 D
  1276 2190 D 1279 2189 D 1282 2187 D 1285 2186 D 1288 2184 D 1290 2183 D
  1293 2181 D 1296 2179 D 1299 2178 D 1302 2176 D 1304 2174 D 1307 2173 D
  1310 2171 D 1313 2170 D 1316 2168 D 1319 2166 D 1321 2164 D 1324 2163 D
  1327 2161 D 1330 2159 D 1333 2158 D 1335 2156 D 1338 2154 D 1341 2152 D
  1344 2150 D 1347 2149 D 1349 2147 D 1352 2145 D 1355 2143 D 1358 2141 D
  1361 2140 D 1363 2138 D 1366 2136 D 1369 2134 D 1372 2132 D 1375 2130 D
  1378 2128 D 1380 2126 D 1383 2124 D 1386 2122 D 1389 2120 D 1392 2119 D
  1394 2117 D 1397 2115 D 1400 2113 D 1403 2110 D 1406 2108 D 1408 2106 D
  1411 2104 D 1414 2102 D 1417 2100 D 1420 2098 D 1423 2096 D 1425 2094 D
  1428 2092 D 1431 2090 D 1434 2087 D 1437 2085 D 1439 2083 D 1442 2081 D
  1445 2079 D 1448 2076 D 1451 2074 D 1453 2072 D 1456 2070 D 1459 2067 D
  1462 2065 D 1465 2063 D 1468 2060 D 1470 2058 D 1473 2056 D 1476 2053 D
  1479 2051 D 1482 2048 D 1484 2046 D 1487 2044 D 1490 2041 D 1493 2039 D
  1496 2036 D 1498 2034 D 1501 2031 D 1504 2028 D 1507 2026 D 1510 2023 D
  1512 2021 D 1515 2018 D 1518 2016 D 1521 2013 D 1524 2010 D 1527 2007 D
  1529 2005 D 1532 2002 D 1535 1999 D 1538 1997 D 1541 1994 D 1543 1991 D
  1546 1988 D 1549 1985 D 1552 1982 D 1555 1980 D 1557 1977 D 1560 1974 D
  1563 1971 D 1566 1968 D 1569 1965 D 1572 1962 D 1574 1959 D 1577 1956 D
  1580 1953 D 1583 1950 D 1586 1946 D 1588 1943 D 1591 1940 D 1594 1937 D
  1597 1934 D 1600 1930 D 1602 1927 D 1605 1924 D 1608 1921 D 1611 1917 D
  1614 1914 D 1617 1911 D 1619 1907 D 1622 1904 D 1625 1900 D 1628 1897 D
  1631 1893 D 1633 1890 D 1636 1886 D 1639 1882 D 1642 1879 D 1645 1875 D
  1647 1871 D 1650 1868 D 1653 1864 D 1656 1860 D 1659 1856 D 1662 1852 D
  1664 1848 D 1667 1844 D 1670 1840 D 1673 1836 D 1676 1832 D 1678 1828 D
  1681 1824 D 1684 1820 D 1687 1816 D 1690 1812 D 1692 1807 D 1695 1803 D
  1698 1799 D 1701 1794 D 1704 1790 D 1706 1785 D 1709 1781 D 1712 1776 D
  1715 1772 D 1718 1767 D 1721 1762 D 1723 1758 D 1726 1753 D 1729 1748 D
  1732 1743 D 1735 1738 D 1737 1733 D 1740 1728 D 1743 1723 D 1746 1718 D
  1749 1713 D 1751 1708 D 1754 1702 D 1757 1697 D 1760 1692 D 1763 1686 D
  1766 1681 D 1768 1675 D 1771 1669 D 1774 1664 D 1777 1658 D 1780 1652 D
  1782 1646 D 1785 1640 D 1788 1634 D 1791 1628 D 1794 1621 D 1796 1615 D
  1799 1609 D 1802 1602 D 1805 1596 D 1808 1589 D 1811 1582 D 1813 1575 D
  1816 1569 D 1819 1562 D 1822 1554 D 1825 1547 D 1827 1540 D 1830 1533 D
  1833 1525 D 1836 1517 D 1839 1510 D 1841 1502 D 1844 1494 D 1847 1486 D
  1850 1477 D 1853 1469 D 1856 1461 D 1858 1452 D 1861 1443 D 1864 1434 D
  1867 1425 D 1870 1416 D 1872 1407 D 1875 1397 D 1878 1387 D 1881 1377 D
  1884 1367 D 1886 1357 D 1889 1346 D 1892 1336 D 1895 1325 D 1898 1314 D
  1900 1302 D 1903 1291 D 1906 1279 D 1909 1267 D 1912 1254 D 1915 1242 D
  1917 1228 D 1920 1215 D 1923 1201 D 1926 1187 D 1929 1173 D 1931 1158 D
  1934 1143 D 1937 1128 D 1940 1112 D 1943 1095 D 1945 1078 D 1948 1060 D
  1951 1042 D 1954 1024 D 1957 1004 D 1960 984 D 1962 963 D 1965 942 D
  1968 919 D 1971 896 D 1974 871 D 1976 846 D 1979 819 D 1982 791 D 1985 761 D
  1988 730 D 1990 697 D 1991 694 D
  632 1784 M 635 1784 D 638 1784 D 641 1784 D 644 1784 D 647 1784 D 649 1784 D
  652 1784 D 655 1784 D 658 1784 D 661 1784 D 663 1784 D 666 1784 D 669 1783 D
  672 1783 D 675 1783 D 677 1783 D 680 1783 D 683 1783 D 686 1783 D 689 1783 D
  692 1782 D 694 1782 D 697 1782 D 700 1782 D 703 1782 D 706 1782 D 708 1781 D
  711 1781 D 714 1781 D 717 1781 D 720 1781 D 722 1780 D 725 1780 D 728 1780 D
  731 1780 D 734 1780 D 737 1779 D 739 1779 D 742 1779 D 745 1779 D 748 1778 D
  751 1778 D 753 1778 D 756 1777 D 759 1777 D 762 1777 D 765 1776 D 767 1776 D
  770 1776 D 773 1776 D 776 1775 D 779 1775 D 781 1775 D 784 1774 D 787 1774 D
  790 1773 D 793 1773 D 796 1773 D 798 1772 D 801 1772 D 804 1772 D 807 1771 D
  810 1771 D 812 1770 D 815 1770 D 818 1769 D 821 1769 D 824 1769 D 826 1768 D
  829 1768 D 832 1767 D 835 1767 D 838 1766 D 841 1766 D 843 1765 D 846 1765 D
  849 1764 D 852 1764 D 855 1763 D 857 1763 D 860 1762 D 863 1762 D 866 1761 D
  869 1761 D 871 1760 D 874 1760 D 877 1759 D 880 1759 D 883 1758 D 886 1758 D
  888 1757 D 891 1756 D 894 1756 D 897 1755 D 900 1755 D 902 1754 D 905 1753 D
  908 1753 D 911 1752 D 914 1752 D 916 1751 D 919 1750 D 922 1750 D 925 1749 D
  928 1748 D 931 1748 D 933 1747 D 936 1746 D 939 1746 D 942 1745 D 945 1744 D
  947 1744 D 950 1743 D 953 1742 D 956 1741 D 959 1741 D 961 1740 D 964 1739 D
  967 1738 D 970 1738 D 973 1737 D 975 1736 D 978 1735 D 981 1735 D 984 1734 D
  987 1733 D 990 1732 D 992 1732 D 995 1731 D 998 1730 D 1001 1729 D 1004 1728
D
  1006 1727 D 1009 1727 D 1012 1726 D 1015 1725 D 1018 1724 D 1020 1723 D
  1023 1722 D 1026 1721 D 1029 1721 D 1032 1720 D 1035 1719 D 1037 1718 D
  1040 1717 D 1043 1716 D 1046 1715 D 1049 1714 D 1051 1713 D 1054 1712 D
  1057 1711 D 1060 1710 D 1063 1709 D 1065 1709 D 1068 1708 D 1071 1707 D
  1074 1706 D 1077 1705 D 1080 1704 D 1082 1703 D 1085 1702 D 1088 1701 D
  1091 1700 D 1094 1699 D 1096 1697 D 1099 1696 D 1102 1695 D 1105 1694 D
  1108 1693 D 1110 1692 D 1113 1691 D 1116 1690 D 1119 1689 D 1122 1688 D
  1125 1687 D 1127 1686 D 1130 1684 D 1133 1683 D 1136 1682 D 1139 1681 D
  1141 1680 D 1144 1679 D 1147 1678 D 1150 1676 D 1153 1675 D 1155 1674 D
  1158 1673 D 1161 1672 D 1164 1670 D 1167 1669 D 1169 1668 D 1172 1667 D
  1175 1665 D 1178 1664 D 1181 1663 D 1184 1662 D 1186 1660 D 1189 1659 D
  1192 1658 D 1195 1657 D 1198 1655 D 1200 1654 D 1203 1653 D 1206 1651 D
  1209 1650 D 1212 1649 D 1214 1647 D 1217 1646 D 1220 1645 D 1223 1643 D
  1226 1642 D 1229 1640 D 1231 1639 D 1234 1638 D 1237 1636 D 1240 1635 D
  1243 1633 D 1245 1632 D 1248 1630 D 1251 1629 D 1254 1627 D 1257 1626 D
  1259 1624 D 1262 1623 D 1265 1621 D 1268 1620 D T 1268 1620 M 1271 1618 D
  1274 1617 D 1276 1615 D 1279 1614 D 1282 1612 D 1285 1611 D 1288 1609 D
  1290 1608 D 1293 1606 D 1296 1604 D 1299 1603 D 1302 1601 D 1304 1599 D
  1307 1598 D 1310 1596 D 1313 1594 D 1316 1593 D 1319 1591 D 1321 1589 D
  1324 1588 D 1327 1586 D 1330 1584 D 1333 1582 D 1335 1581 D 1338 1579 D
  1341 1577 D 1344 1575 D 1347 1574 D 1349 1572 D 1352 1570 D 1355 1568 D
  1358 1566 D 1361 1565 D 1363 1563 D 1366 1561 D 1369 1559 D 1372 1557 D
  1375 1555 D 1378 1553 D 1380 1551 D 1383 1549 D 1386 1547 D 1389 1545 D
  1392 1543 D 1394 1541 D 1397 1539 D 1400 1537 D 1403 1535 D 1406 1533 D
  1408 1531 D 1411 1529 D 1414 1527 D 1417 1525 D 1420 1523 D 1423 1521 D
  1425 1519 D 1428 1517 D 1431 1515 D 1434 1512 D 1437 1510 D 1439 1508 D
  1442 1506 D 1445 1504 D 1448 1501 D 1451 1499 D 1453 1497 D 1456 1495 D
  1459 1492 D 1462 1490 D 1465 1488 D 1468 1485 D 1470 1483 D 1473 1481 D
  1476 1478 D 1479 1476 D 1482 1473 D 1484 1471 D 1487 1468 D 1490 1466 D
  1493 1464 D 1496 1461 D 1498 1459 D 1501 1456 D 1504 1453 D 1507 1451 D
  1510 1448 D 1512 1446 D 1515 1443 D 1518 1440 D 1521 1438 D 1524 1435 D
  1527 1432 D 1529 1430 D 1532 1427 D 1535 1424 D 1538 1421 D 1541 1419 D
  1543 1416 D 1546 1413 D 1549 1410 D 1552 1407 D 1555 1405 D 1557 1402 D
  1560 1399 D 1563 1396 D 1566 1393 D 1569 1390 D 1572 1387 D 1574 1384 D
  1577 1381 D 1580 1378 D 1583 1375 D 1586 1371 D 1588 1368 D 1591 1365 D
  1594 1362 D 1597 1359 D 1600 1355 D 1602 1352 D 1605 1349 D 1608 1346 D
  1611 1342 D 1614 1339 D 1617 1335 D 1619 1332 D 1622 1329 D 1625 1325 D
  1628 1322 D 1631 1318 D 1633 1314 D 1636 1311 D 1639 1307 D 1642 1304 D
  1645 1300 D 1647 1296 D 1650 1293 D 1653 1289 D 1656 1285 D 1659 1281 D
  1662 1277 D 1664 1273 D 1667 1269 D 1670 1265 D 1673 1261 D 1676 1257 D
  1678 1253 D 1681 1249 D 1684 1245 D 1687 1241 D 1690 1237 D 1692 1232 D
  1695 1228 D 1698 1224 D 1701 1219 D 1704 1215 D 1706 1210 D 1709 1206 D
  1712 1201 D 1715 1197 D 1718 1192 D 1721 1187 D 1723 1183 D 1726 1178 D
  1729 1173 D 1732 1168 D 1735 1163 D 1737 1158 D 1740 1153 D 1743 1148 D
  1746 1143 D 1749 1138 D 1751 1133 D 1754 1127 D 1757 1122 D 1760 1116 D
  1763 1111 D 1766 1105 D 1768 1100 D 1771 1094 D 1774 1088 D 1777 1083 D
  1780 1077 D 1782 1071 D 1785 1065 D 1788 1059 D 1791 1053 D 1794 1046 D
  1796 1040 D 1799 1034 D 1802 1027 D 1805 1021 D 1808 1014 D 1811 1007 D
  1813 1000 D 1816 993 D 1819 986 D 1822 979 D 1825 972 D 1827 965 D 1830 957 D
  1833 950 D 1836 942 D 1839 935 D 1841 927 D 1844 919 D 1847 911 D 1850 902 D
  1853 894 D 1856 886 D 1858 877 D 1861 868 D 1864 859 D 1867 850 D 1870 841 D
  1872 832 D 1875 822 D 1878 812 D 1881 802 D 1884 792 D 1886 782 D 1889 771 D
  1892 761 D 1895 750 D 1898 739 D 1900 727 D 1903 716 D 1906 704 D 1908 694 D
  632 2086 M 635 1912 D 638 1811 D 641 1742 D 644 1690 D 647 1649 D 649 1615 D
  652 1587 D 655 1564 D 658 1543 D 661 1525 D 663 1509 D 666 1495 D 669 1483 D
  672 1471 D 675 1461 D 677 1452 D 680 1443 D 683 1435 D 686 1428 D 689 1421 D
  692 1415 D 694 1409 D 697 1404 D 700 1399 D 703 1394 D 706 1390 D 708 1386 D
  711 1382 D 714 1378 D 717 1375 D 720 1371 D 722 1368 D 725 1365 D 728 1363 D
  731 1360 D 734 1357 D 737 1355 D 739 1353 D 742 1351 D 745 1349 D 748 1347 D
  751 1345 D 753 1343 D 756 1341 D 759 1339 D 762 1338 D 765 1336 D 767 1335 D
  770 1333 D 773 1332 D 776 1330 D 779 1329 D 781 1328 D 784 1327 D 787 1325 D
  790 1324 D 793 1323 D 796 1322 D 798 1321 D 801 1320 D 804 1319 D 807 1318 D
  810 1317 D 812 1316 D 815 1315 D 818 1314 D 821 1313 D 824 1313 D 826 1312 D
  829 1311 D 832 1310 D 835 1309 D 838 1308 D 841 1308 D 843 1307 D 846 1306 D
  849 1305 D 852 1305 D 855 1304 D 857 1303 D 860 1302 D 863 1302 D 866 1301 D
  869 1300 D 871 1300 D 874 1299 D 877 1298 D 880 1298 D 883 1297 D 886 1296 D
  888 1296 D 891 1295 D 894 1294 D 897 1294 D 900 1293 D 902 1292 D 905 1292 D
  908 1291 D 911 1290 D 914 1290 D 916 1289 D 919 1288 D 922 1288 D 925 1287 D
  928 1286 D 931 1286 D 933 1285 D 936 1284 D 939 1284 D 942 1283 D 945 1283 D
  947 1282 D 950 1281 D 953 1281 D 956 1280 D 959 1279 D 961 1279 D 964 1278 D
  967 1277 D 970 1277 D 973 1276 D 975 1275 D 978 1274 D 981 1274 D 984 1273 D
  987 1272 D 990 1272 D 992 1271 D 995 1270 D 998 1270 D 1001 1269 D 1004 1268
D
  1006 1267 D 1009 1267 D 1012 1266 D 1015 1265 D 1018 1265 D 1020 1264 D
  1023 1263 D 1026 1262 D 1029 1262 D 1032 1261 D 1035 1260 D 1037 1259 D
  1040 1259 D 1043 1258 D 1046 1257 D 1049 1256 D 1051 1255 D 1054 1255 D
  1057 1254 D 1060 1253 D 1063 1252 D 1065 1251 D 1068 1251 D 1071 1250 D
  1074 1249 D 1077 1248 D 1080 1247 D 1082 1246 D 1085 1246 D 1088 1245 D
  1091 1244 D 1094 1243 D 1096 1242 D 1099 1241 D 1102 1240 D 1105 1240 D
  1108 1239 D 1110 1238 D 1113 1237 D 1116 1236 D 1119 1235 D 1122 1234 D
  1125 1233 D 1127 1232 D 1130 1231 D 1133 1230 D 1136 1229 D 1139 1229 D
  1141 1228 D 1144 1227 D 1147 1226 D 1150 1225 D 1153 1224 D 1155 1223 D
  1158 1222 D 1161 1221 D 1164 1220 D 1167 1219 D 1169 1218 D 1172 1217 D
  1175 1216 D 1178 1215 D 1181 1214 D 1184 1213 D 1186 1212 D 1189 1211 D
  1192 1210 D 1195 1209 D 1198 1207 D 1200 1206 D 1203 1205 D 1206 1204 D
  1209 1203 D 1212 1202 D 1214 1201 D 1217 1200 D 1220 1199 D 1223 1198 D
  1226 1197 D 1229 1196 D 1231 1195 D 1234 1194 D 1237 1193 D 1240 1192 D
  1243 1191 D 1245 1190 D 1248 1189 D 1251 1189 D 1254 1188 D 1257 1188 D
  1259 1187 D 1262 1185 D 1265 1184 D 1268 1182 D 1271 1181 D 1274 1179 D
  1276 1178 D 1279 1176 D 1282 1175 D 1285 1173 D 1288 1172 D 1290 1170 D
  1293 1168 D 1296 1167 D 1299 1165 D 1302 1164 D 1304 1162 D 1307 1160 D
  1310 1159 D 1313 1157 D 1316 1155 D 1319 1154 D 1321 1152 D 1324 1150 D
  1327 1149 D 1330 1147 D 1333 1145 D 1335 1143 D 1338 1142 D 1341 1140 D
  1344 1138 D 1347 1136 D 1349 1135 D 1352 1133 D 1355 1131 D 1358 1129 D
  1361 1127 D 1363 1126 D 1366 1124 D 1369 1122 D 1372 1120 D 1375 1118 D
  1378 1116 D 1380 1114 D 1383 1112 D 1386 1111 D 1389 1109 D 1392 1107 D
  1394 1105 D T 1394 1105 M 1397 1103 D 1400 1101 D 1403 1099 D 1406 1097 D
  1408 1095 D 1411 1093 D 1414 1091 D 1417 1089 D 1420 1086 D 1423 1084 D
  1425 1082 D 1428 1080 D 1431 1078 D 1434 1076 D 1437 1074 D 1439 1072 D
  1442 1069 D 1445 1067 D 1448 1065 D 1451 1063 D 1453 1061 D 1456 1058 D
  1459 1056 D 1462 1054 D 1465 1051 D 1468 1049 D 1470 1047 D 1473 1044 D
  1476 1042 D 1479 1040 D 1482 1037 D 1484 1035 D 1487 1033 D 1490 1030 D
  1493 1028 D 1496 1025 D 1498 1023 D 1501 1020 D 1504 1018 D 1507 1015 D
  1510 1013 D 1512 1010 D 1515 1007 D 1518 1005 D 1521 1002 D 1524 1000 D
  1527 997 D 1529 994 D 1532 992 D 1535 989 D 1538 986 D 1541 983 D 1543 981 D
  1546 978 D 1549 975 D 1552 972 D 1555 969 D 1557 966 D 1560 964 D 1563 961 D
  1566 958 D 1569 955 D 1572 952 D 1574 949 D 1577 946 D 1580 943 D 1583 940 D
  1586 937 D 1588 933 D 1591 930 D 1594 927 D 1597 924 D 1600 921 D 1602 918 D
  1605 914 D 1608 911 D 1611 908 D 1614 904 D 1617 901 D 1619 898 D 1622 894 D
  1625 891 D 1628 887 D 1631 884 D 1633 880 D 1636 877 D 1639 873 D 1642 869 D
  1645 866 D 1647 862 D 1650 858 D 1653 855 D 1656 851 D 1659 847 D 1662 843 D
  1664 839 D 1667 835 D 1670 832 D 1673 828 D 1676 824 D 1678 820 D 1681 815 D
  1684 811 D 1687 807 D 1690 803 D 1692 799 D 1695 794 D 1698 790 D 1701 786 D
  1704 781 D 1706 777 D 1709 772 D 1712 768 D 1715 763 D 1718 759 D 1721 754 D
  1723 749 D 1726 745 D 1729 740 D 1732 735 D 1735 730 D 1737 725 D 1740 720 D
  1743 715 D 1746 710 D 1749 705 D 1751 700 D 1754 694 D
  911 1678 M 905 1670 D 901 1662 D 899 1656 D 897 1647 D 897 1637 D 899 1627 D
  901 1621 D 899 1625 D 897 1631 D 895 1640 D 895 1648 D 897 1658 D 901 1666 D
  905 1672 D 911 1678 D 956 1674 M 958 1672 D 961 1674 D 958 1676 D 956 1674 D
  951 1662 M 954 1660 D 954 1655 D 949 1643 D 949 1637 D 951 1635 D 958 1635 D
  963 1639 D 965 1643 D 954 1635 M 951 1637 D 951 1643 D 956 1655 D 956 1660 D
  954 1662 D 946 1662 D 941 1658 D 939 1655 D 993 1627 M 999 1635 D 1003 1643 D
  1006 1648 D 1007 1658 D 1007 1668 D 1006 1678 D 1003 1684 D 1006 1680 D
  1007 1674 D 1010 1664 D 1010 1656 D 1007 1647 D 1003 1639 D 999 1633 D
  993 1627 D 985 1621 D 1080 694 M
  982 2220 M 976 2213 D 972 2205 D 969 2198 D 968 2189 D 968 2179 D 969 2169 D
  972 2163 D 969 2167 D 968 2173 D 965 2183 D 965 2190 D 968 2200 D 972 2208 D
  976 2214 D 982 2220 D 1026 2216 M 1029 2214 D 1031 2216 D 1029 2218 D
  1026 2216 D 1021 2205 M 1024 2203 D 1024 2197 D 1019 2185 D 1019 2179 D
  1021 2177 D 1029 2177 D 1034 2181 D 1036 2185 D 1024 2177 M 1021 2179 D
  1021 2185 D 1026 2197 D 1026 2203 D 1024 2205 D 1017 2205 D 1012 2200 D
  1010 2197 D 1074 2216 M 1077 2214 D 1079 2216 D 1077 2218 D 1074 2216 D
  1069 2205 M 1072 2203 D 1072 2197 D 1067 2185 D 1067 2179 D 1069 2177 D
  1077 2177 D 1082 2181 D 1084 2185 D 1072 2177 M 1069 2179 D 1069 2185 D
  1074 2197 D 1074 2203 D 1072 2205 D 1065 2205 D 1060 2200 D 1058 2197 D
  1112 2169 M 1118 2177 D 1122 2185 D 1124 2190 D 1126 2200 D 1126 2210 D
  1124 2220 D 1122 2226 D 1124 2223 D 1126 2216 D 1128 2206 D 1128 2198 D
  1126 2189 D 1122 2181 D 1118 2175 D 1112 2169 D 1103 2163 D 1216 694 M
  846 1230 M 848 1230 D 849 1234 D 848 1222 D 848 1226 D 846 1230 D 844 1232 D
  839 1234 D 833 1234 D 828 1232 D 825 1228 D 825 1224 D 826 1220 D 828 1218 D
  839 1210 D 843 1206 D 825 1224 M 828 1220 D 839 1212 D 841 1210 D 843 1206 D
  843 1200 D 841 1197 D 839 1194 D 835 1192 D 828 1192 D 823 1194 D 822 1197 D
  820 1200 D 820 1204 D 818 1192 D 820 1197 D 822 1197 D 866 1192 M 873 1192 D
  882 1192 M 891 1192 D 887 1192 M 894 1234 D 897 1234 M 893 1234 D
  873 1234 M 878 1234 D 879 1197 D 870 1192 M 877 1234 D 878 1192 D 893 1234 D
  886 1192 D 943 694 M
  T X grestore
 %end(plot)
 %%BoundingBox:  0 0   495 586

