%Paper: hep-th/9402078
%From: Juergen Baacke <baacke@x4u2.desy.de>
%Date: Mon, 14 Feb 1994 15:57:58 +0100 (MET)


%%One figure is appended as a postscript file to be stripped off

%format{latex}
\documentstyle[12pt]{article}
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\begin{document}
\begin{titlepage}
\begin{flushright}
DO-TH-93/19EA \\
February 1994 \\
\end{flushright}
\vspace{20mm}
\begin{center}
{\Large
Quantum Fluctuations of the electroweak
sphaleron: Erratum and Addendum}
\vspace{10mm}

{\large  J. Baacke \footnote{e-mail: baacke@het.physik.
uni-dortmund.de}
and S. Junker} \\
\vspace{15mm}

{\large Institut f\"ur Physik, Universit\"at Dortmund} \\
{\large D-44221 Dortmund 50} \\ {\large Germany}
\vspace{40mm}
\end{center}
\end{titlepage}
We have presented recently \cite{BaaJu} an exact computation of the
fluctuation determinant of the electroweak sphaleron.
 The results disagreed substantially from
those of an earlier evaluation of this quantity \cite{CarLi}.
Moreover none of the two exact computations agreed with analytical
estimates \cite{Sha,CarMcLe} that are expected to be good at small Higgs
masses, essentially approximations in which higher
gradient terms are neglected, a fact that has been repeatedly
criticized (see e.g. \cite{HeKrSch}).

While we thought that possibly the gradient type
expansions were to blame for this discrepancy, the failure
of compatibility between approximations and exact results
can be traced back to our treatment of the tadpole
contributions. Indeed we removed all tadpole contributions
to the Higgs field {\it completely}, in a misinterpretation
of the renormalization and rescaling prescription of
Refs.  \cite{CarLi,CarMcLe}; however, a finite piece
has to be restituted. In order to understand this point
which has considerable numerical consequences we review
shortly these contributions (see \cite{DoJa}, especially
Appendix C). This cannot be done consistently within the
three dimensional asymptotic theory since the approximation
of dismissing all but the lowest Matsubara frequency is not
justified for a these divergent contributions.

The tadpole contribution including all Matsubara frequencies reads
\begin{equation}
{\cal V}[\Phi]_{Tad} =
\frac{T}{2} \sum_i       \int\frac{d^3p}{(2\pi)^3}\sum_{n=-\infty}^\infty
\frac{1}{p^2+m_i^2+(2\pi n T)^2} c_i\int d^3x (H_0^2(\vec x)-1)
\end{equation}
Here $m_i$ are the masses circulating in the loop, $c_i$ are
their couplings to the Higgs field and $H_0(\vec x)$ is the
Higgs profile. The coefficients $c_i$ can be identified as
coefficients of the terms
proportional to $H_0^2-1$ in the diagonal elements of the potential
given in Appendix A of \cite{BaaJu}, they are given below.
The momentum integral including the factor $T/2$ can be rewritten
\cite{DoJa}
\begin{equation}
\int \frac{d^3p}{(2\pi)^3} \left( \frac{1}{4 \sqrt{p^2+m^2}}
+\frac{1}{2 \sqrt{p^2+m^2}} \frac{1}{\exp{\sqrt{p^2+m^2}/T}-1}
\right)
\end{equation}
The first term is the $T=0$ contribution which goes into the
Higgs mass renormalization. The second term can be expanded at
high temperature as
\begin{equation}
\frac{T}{  \pi}\int x^2dx \frac{1}{\sqrt{x^2+m^2/T^2}}
\frac{1}{\exp{\sqrt{x^2+m^2/T^2}}-1} \approx
\frac{T^2}{24} - \frac{mT}{8\pi}
\end{equation}
up to terms of order $\ln T$ or lower.
The term quadratic in $T$ can be absorbed \cite{CarLi,CarMcLe}
into the $T$ dependence of the
vacuum expectation value of the Higgs field. The linear term is
part of the well known $T\Phi^3$ term of the effective potential
and without this contribution the latter is incomplete (see
e.g. the discussion of this term in Appendix A of \cite{HeKrSch}).
This is
the reason why our data must fail to approach the estimate based
on the effective potential.

Specifying the contributions of the different fields by \cite{BaaJu}
$m_i=M_W , c_i=M_W^2$ for altogether six components of gauge boson
and subtracted ghost fields, $m_i=M_W , c_i= (M_H^2+2M_W^2)/2$
for the three Goldstone boson fields and by $m_i=M_H , c_i=3
M_H^2/2$ for the physical Higgs field we find a contribution to the
effective action proportional to $T^2$
\begin{equation}
\frac{T^2}{8}(3M_W^2+M_H^2)\int d^3x (H_0^2-1)
\end{equation}
in agreement with standard results. The term linear in $T$ yields
now
\begin{equation}
-\frac{T}{8\pi}\left(M_W \left(9M_W^2+\frac{3}{2}M_H^2\right)
+\frac{3}{2}M_H^3\right)\int d^3x (H_0^2-1).
\end{equation}
This term is positive. Since the fluctuation determinant is related to
\\$-{\cal V}_{1-loop}/T$ and the term is to be restituted in order
to parallel the treatment of the effective potential, we find
a large negative contribution to $\ln \kappa$.

We present in Table 1 our previous results with the
completely removed tadpoles as $\ln \tilde{\kappa}$,
in units of $(gv)^6$ as
used in \cite{CarLi} and the results with the restituted piece as
$\ln \kappa$. It is the latter one that is correct and that has
to be compared with the
$\Phi^3$ estimate derived from the effective potential which,
if only the gauge loops are taken into account, takes the form
\cite{Sha}
\begin{equation}
\frac{3M_W^3}{4\pi}\int d^3x (H_0^3-1)
\end{equation}
We give the corresponding estimates in the last row of Table 1,
labelled ``$\Phi^3$''. This estimate differs from the one given
in \cite{CarLi} by a factor $\sqrt{8}$ which is due to a
mistake there: the
term is originally \cite{Sha} given for a Higgs
field normalized to the vacuum
expectation value $v$, while the one used in \cite{CarLi} has
vacuum expectation value $v/\sqrt{2}$ \footnote{This statement is
made with the agreement of Larry McLerran}.
We observe that our corrected data and this estimate are now
well consistent. Actually, since $\kappa$ is a dimensionful
quantity $\ln \kappa$ depends on the scale used, therefore
 one can only expect
that the exact data and the estimate become {it parallel} as $M_H \to 0$;
the absolute agreement in this limit is somewhat fortitious.

Our corrected data, the analytic estimate and the data of Carson
et al. \cite{CarLi} are presented in Figure 1, showing now a reasonable
general agreement. Whether the remaining differences are within or
outside the error margins of the involved numerical computations is
a question that seems hard to answer. Unfortunately also the disagreement
with the estimate of Ref. \cite{CarMcLe} persists.


\noindent
{\bf Acknowledgments}

One of us (J.B.) thanks K. Goeke and A. Ringwald for making him aware,
at different times, of Ref. \cite{HeKrSch}, W. Buchmueller and
L. McLerran for most useful discussions, and the DESY
for hospitality.


\begin{thebibliography}{99}

\bibitem{BaaJu} J. Baacke and S. Junker, Phys. Rev. {\bf D49},2055
(1994)

\bibitem{CarLi} L. Carson,X. Li,L. McLerran and R.-T. Wang,
Phys. Rev. {\bf D42}, 2127 (1990)

\bibitem{Sha} M. Shaposhnikov, Nucl. Phys. {\bf B287}, 757 (1987)

\bibitem{CarMcLe}L. Carson and L. McLerran, Phys. Rev. {\bf D41},
647 (1990)

\bibitem{HeKrSch} M. Hellmund, J.  Kripfganz and M. G. Schmidt,
Heidelberg preprint HD-THEP-93-23, and private communications
by many others

\bibitem{DoJa} Dolan and R. Jackiw, Phys. Rev. {\bf D9}, 3320
(1974)

\end{thebibliography}

\vspace{20mm}
\begin{center}
\section*{Table 1}
\vspace{6mm}
\begin{tabular}{|c||c|c|c|c|c|c|c|}\hline
$\xi$ & 0.4 & 0.5 & 0.6 & 0.8 & 1.0 & 1.5 & 2.0 \\  \hline
$\lambda/g^2$ &.02&.031&.045&.08&.125&.281&.5\\ \hline
$\ln \tilde{\kappa}$ &2.02&1.73&1.52&1.34&1.32&1.46&1.55\\ \hline
$\ln \kappa$ &-46.80&-31.00&-22.30&-13.64&-9.64&-5.96&-5.14\\ \hline
$\Phi^3$ &-45.23&-29.43&-20.68&-11.87&-7.73&-3.57&-2.08\\ \hline
\end{tabular}
\end{center}
\section*{Figure Captions}

\noindent
Fig. 1 {\bf The Fluctuation Determinant}

We plot the logarithm of the fluctuation determinant $\kappa$
as a function of the ratio $\lambda/g^2$. Our corrected results
are given as triangles, those of Ref. \cite{CarLi} as squares.
The solid line is the estimate based on the effective potential.
\end{document}


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2131 2078 L
2131 2078 L
2131 2078 L
2131 2078 L
2132 2078 L
2132 2078 L
2132 2078 L
2132 2078 L
2132 2078 L
2132 2078 L
2132 2078 L
2133 2078 L
2133 2078 L
2133 2078 L
2133 2078 L
2133 2078 L
2133 2078 L
2133 2078 L
2133 2078 L
2134 2078 L
2134 2078 L
2134 2078 L
2134 2078 L
2134 2078 L
2134 2078 L
2134 2078 L
2135 2078 L
2135 2078 L
2135 2078 L
2135 2078 L
2135 2078 L
2135 2078 M B
460 359 M 443 387 L
427 359 L
460 359 L
CF M
B
579 937 M 563 965 L
547 937 L
579 937 L
CF M
B
680 1256 M 664 1284 L
648 1256 L
680 1256 L
CF M
B
837 1573 M 821 1601 L
805 1573 L
837 1573 L
CF M
B
958 1719 M 942 1747 L
926 1719 L
958 1719 L
CF M
B
1179 1854 M 1163 1882 L
1147 1854 L
1179 1854 L
CF M
B
1336 1884 M 1320 1912 L
1303 1884 L
1336 1884 L
CF M
1336 1884 M B
895 1779 M 895 1805 L
868 1805 L
868 1779 L
895 1779 L
CF M
B
1194 1987 M 1194 2014 L
1167 2014 L
1167 1987 L
1194 1987 L
CF M
B
1522 1936 M 1522 1963 L
1495 1963 L
1495 1936 L
1522 1936 L
CF M
B
2148 1610 M 2148 1637 L
2122 1637 L
2122 1610 L
2148 1610 L
CF M
1696 507 M CS [] 0 setdash M
1706 575 M 1712 575 L
1719 571 L
1722 568 L
1725 562 L
1745 517 L
1748 510 L
1751 507 L
1712 575 M 1719 568 L
1722 562 L
1741 517 L
1745 510 L
1751 507 L
1754 507 L
1728 552 M 1703 507 L
1728 552 M 1706 507 L
1825 588 M 1767 485 L
1857 552 M 1850 549 L
1847 546 L
1844 539 L
1844 533 L
1847 527 L
1850 523 L
1857 520 L
1863 520 L
1870 523 L
1873 527 L
1876 533 L
1876 539 L
1873 546 L
1870 549 L
1863 552 L
1857 552 L
1850 549 M 1847 543 L
1847 530 L
1850 523 L
1870 523 M 1873 530 L
1873 543 L
1870 549 L
1873 546 M 1876 549 L
1883 552 L
1883 549 L
1876 549 L
1847 527 M 1844 523 L
1841 517 L
1841 514 L
1844 507 L
1854 504 L
1870 504 L
1879 501 L
1883 498 L
1841 514 M 1844 510 L
1854 507 L
1870 507 L
1879 504 L
1883 498 L
1883 494 L
1879 488 L
1870 485 L
1850 485 L
1841 488 L
1838 494 L
1838 498 L
1841 504 L
1850 507 L
1900 571 M 1902 569 L
1900 567 L
1898 569 L
1898 571 L
1900 575 L
1902 577 L
1908 578 L
1915 578 L
1921 577 L
1923 575 L
1925 571 L
1925 567 L
1923 563 L
1917 559 L
1908 555 L
1904 553 L
1900 550 L
1898 544 L
1898 538 L
1915 578 M 1919 577 L
1921 575 L
1923 571 L
1923 567 L
1921 563 L
1915 559 L
1908 555 L
1898 542 M 1900 544 L
1904 544 L
1913 540 L
1919 540 L
1923 542 L
1925 544 L
1904 544 M 1913 538 L
1921 538 L
1923 540 L
1925 544 L
1925 548 L
1931 507 M CS [] 0 setdash M
443 1715 M CS [] 0 setdash M
375 1731 M 443 1731 L
375 1734 M 443 1734 L
375 1722 M 375 1734 L
443 1722 M 443 1744 L
398 1766 M 443 1766 L
398 1770 M 443 1770 L
408 1770 M 401 1776 L
398 1786 L
398 1792 L
401 1802 L
408 1805 L
443 1805 L
398 1792 M 401 1799 L
408 1802 L
443 1802 L
398 1757 M 398 1770 L
443 1757 M 443 1779 L
443 1792 M 443 1815 L
398 1892 M 443 1879 L
398 1895 M 443 1882 L
398 1924 M 401 1927 L
401 1930 L
398 1927 L
398 1921 L
401 1914 L
414 1901 L
417 1895 L
417 1888 L
417 1895 M 420 1901 L
440 1908 L
443 1911 L
417 1895 M 420 1898 L
440 1904 L
443 1908 L
443 1914 L
440 1921 L
430 1927 L
443 1937 M CS [] 0 setdash M
CS [] 0 setdash M
255 255 M 2261 255 L
255 255 M 255 323 L
443 255 M 443 289 L
554 255 M 554 289 L
632 255 M 632 289 L
693 255 M 693 289 L
743 255 M 743 289 L
785 255 M 785 289 L
821 255 M 821 289 L
853 255 M 853 289 L
882 255 M 882 323 L
1070 255 M 1070 289 L
1181 255 M 1181 289 L
1259 255 M 1259 289 L
1320 255 M 1320 289 L
1369 255 M 1369 289 L
1411 255 M 1411 289 L
1448 255 M 1448 289 L
1480 255 M 1480 289 L
1508 255 M 1508 323 L
1697 255 M 1697 289 L
1807 255 M 1807 289 L
1886 255 M 1886 289 L
1946 255 M 1946 289 L
1996 255 M 1996 289 L
2038 255 M 2038 289 L
2074 255 M 2074 289 L
2107 255 M 2107 289 L
2135 255 M 2135 323 L
142 129 M 171 196 M 162 193 L
155 184 L
152 168 L
152 158 L
155 142 L
162 132 L
171 129 L
178 129 L
187 132 L
194 142 L
197 158 L
197 168 L
194 184 L
187 193 L
178 196 L
171 196 L
165 193 L
162 190 L
159 184 L
155 168 L
155 158 L
159 142 L
162 135 L
165 132 L
171 129 L
178 129 M 184 132 L
187 135 L
191 142 L
194 158 L
194 168 L
191 184 L
187 190 L
184 193 L
178 196 L
223 135 M 219 132 L
223 129 L
226 132 L
223 135 L
268 196 M 258 193 L
252 184 L
248 168 L
248 158 L
252 142 L
258 132 L
268 129 L
274 129 L
284 132 L
290 142 L
293 158 L
293 168 L
290 184 L
284 193 L
274 196 L
268 196 L
261 193 L
258 190 L
255 184 L
252 168 L
252 158 L
255 142 L
258 135 L
261 132 L
268 129 L
274 129 M 280 132 L
284 135 L
287 142 L
290 158 L
290 168 L
287 184 L
284 190 L
280 193 L
274 196 L
322 184 M 329 187 L
338 196 L
338 129 L
335 193 M 335 129 L
322 129 M 351 129 L
801 129 M 830 196 M 821 193 L
814 184 L
811 168 L
811 158 L
814 142 L
821 132 L
830 129 L
837 129 L
846 132 L
853 142 L
856 158 L
856 168 L
853 184 L
846 193 L
837 196 L
830 196 L
824 193 L
821 190 L
817 184 L
814 168 L
814 158 L
817 142 L
821 135 L
824 132 L
830 129 L
837 129 M 843 132 L
846 135 L
850 142 L
853 158 L
853 168 L
850 184 L
846 190 L
843 193 L
837 196 L
882 135 M 878 132 L
882 129 L
885 132 L
882 135 L
917 184 M 923 187 L
933 196 L
933 129 L
930 193 M 930 129 L
917 129 M 946 129 L
1476 129 M 1496 184 M 1502 187 L
1512 196 L
1512 129 L
1508 193 M 1508 129 L
1496 129 M 1524 129 L
2071 129 M 2090 184 M 2097 187 L
2106 196 L
2106 129 L
2103 193 M 2103 129 L
2090 129 M 2119 129 L
2164 196 M 2154 193 L
2148 184 L
2145 168 L
CS M
2145 158 L
2148 142 L
2154 132 L
2164 129 L
2170 129 L
2180 132 L
2187 142 L
2190 158 L
2190 168 L
2187 184 L
2180 193 L
2170 196 L
2164 196 L
2158 193 L
2154 190 L
2151 184 L
2148 168 L
2148 158 L
2151 142 L
2154 135 L
2158 132 L
2164 129 L
2170 129 M 2177 132 L
2180 135 L
2183 142 L
2187 158 L
2187 168 L
2183 184 L
2180 190 L
2177 193 L
2170 196 L
255 2261 M 2261 2261 L
255 2261 M 255 2192 L
443 2261 M 443 2226 L
554 2261 M 554 2226 L
632 2261 M 632 2226 L
693 2261 M 693 2226 L
743 2261 M 743 2226 L
785 2261 M 785 2226 L
821 2261 M 821 2226 L
853 2261 M 853 2226 L
882 2261 M 882 2192 L
1070 2261 M 1070 2226 L
1181 2261 M 1181 2226 L
1259 2261 M 1259 2226 L
1320 2261 M 1320 2226 L
1369 2261 M 1369 2226 L
1411 2261 M 1411 2226 L
1448 2261 M 1448 2226 L
1480 2261 M 1480 2226 L
1508 2261 M 1508 2192 L
1697 2261 M 1697 2226 L
1807 2261 M 1807 2226 L
1886 2261 M 1886 2226 L
1946 2261 M 1946 2226 L
1996 2261 M 1996 2226 L
2038 2261 M 2038 2226 L
2074 2261 M 2074 2226 L
2107 2261 M 2107 2226 L
2135 2261 M 2135 2192 L
255 255 M 255 2261 L
255 288 M 289 288 L
255 324 M 289 324 L
255 361 M 289 361 L
255 398 M 289 398 L
255 434 M 289 434 L
255 471 M 289 471 L
255 507 M 289 507 L
255 544 M 289 544 L
255 581 M 289 581 L
255 617 M 323 617 L
255 654 M 289 654 L
255 690 M 289 690 L
255 727 M 289 727 L
255 764 M 289 764 L
255 800 M 289 800 L
255 837 M 289 837 L
255 873 M 289 873 L
255 910 M 289 910 L
255 947 M 289 947 L
255 983 M 323 983 L
255 1020 M 289 1020 L
255 1056 M 289 1056 L
255 1093 M 289 1093 L
255 1130 M 289 1130 L
255 1166 M 289 1166 L
255 1203 M 289 1203 L
255 1239 M 289 1239 L
255 1276 M 289 1276 L
255 1313 M 289 1313 L
255 1349 M 323 1349 L
255 1386 M 289 1386 L
255 1422 M 289 1422 L
255 1459 M 289 1459 L
255 1496 M 289 1496 L
255 1532 M 289 1532 L
255 1569 M 289 1569 L
255 1605 M 289 1605 L
255 1642 M 289 1642 L
255 1679 M 289 1679 L
255 1715 M 323 1715 L
255 1752 M 289 1752 L
255 1788 M 289 1788 L
255 1825 M 289 1825 L
255 1862 M 289 1862 L
255 1898 M 289 1898 L
255 1935 M 289 1935 L
255 1971 M 289 1971 L
255 2008 M 289 2008 L
255 2045 M 289 2045 L
255 2081 M 323 2081 L
255 2118 M 289 2118 L
255 2154 M 289 2154 L
255 2191 M 289 2191 L
255 2228 M 289 2228 L
-14 583 M 0 612 M 55 612 L
107 644 M 107 583 L
110 651 M 110 583 L
110 651 M 75 603 L
126 603 L
97 583 M 119 583 L
161 651 M 152 648 L
145 638 L
142 622 L
142 612 L
145 596 L
152 587 L
161 583 L
168 583 L
177 587 L
184 596 L
187 612 L
187 622 L
184 638 L
177 648 L
168 651 L
161 651 L
155 648 L
152 644 L
148 638 L
145 622 L
145 612 L
148 596 L
152 590 L
155 587 L
161 583 L
168 583 M 174 587 L
177 590 L
180 596 L
184 612 L
184 622 L
180 638 L
177 644 L
174 648 L
168 651 L
-14 949 M 0 978 M 55 978 L
81 1004 M 84 1001 L
81 998 L
78 1001 L
78 1004 L
81 1010 L
84 1014 L
94 1017 L
107 1017 L
116 1014 L
119 1007 L
119 998 L
116 991 L
107 988 L
97 988 L
107 1017 M 113 1014 L
116 1007 L
116 998 L
113 991 L
107 988 L
113 985 L
119 978 L
123 972 L
123 962 L
119 956 L
116 953 L
107 949 L
94 949 L
84 953 L
81 956 L
78 962 L
78 965 L
81 969 L
84 965 L
81 962 L
116 982 M 119 972 L
119 962 L
116 956 L
113 953 L
107 949 L
161 1017 M 152 1014 L
145 1004 L
142 988 L
142 978 L
145 962 L
152 953 L
161 949 L
168 949 L
177 953 L
184 962 L
187 978 L
187 988 L
184 1004 L
177 1014 L
168 1017 L
161 1017 L
155 1014 L
152 1010 L
148 1004 L
145 988 L
145 978 L
148 962 L
152 956 L
155 953 L
161 949 L
168 949 M 174 953 L
177 956 L
180 962 L
184 978 L
184 988 L
180 1004 L
177 1010 L
174 1014 L
168 1017 L
-14 1316 M 0 1344 M 55 1344 L
81 1370 M 84 1367 L
81 1364 L
78 1367 L
78 1370 L
81 1376 L
84 1380 L
94 1383 L
107 1383 L
116 1380 L
119 1376 L
123 1370 L
123 1364 L
119 1357 L
110 1351 L
94 1344 L
87 1341 L
81 1335 L
78 1325 L
78 1316 L
107 1383 M 113 1380 L
116 1376 L
119 1370 L
119 1364 L
116 1357 L
107 1351 L
94 1344 L
78 1322 M 81 1325 L
87 1325 L
103 1319 L
113 1319 L
119 1322 L
123 1325 L
87 1325 M 103 1316 L
116 1316 L
119 1319 L
123 1325 L
123 1332 L
161 1383 M 152 1380 L
145 1370 L
142 1354 L
142 1344 L
145 1328 L
152 1319 L
161 1316 L
168 1316 L
177 1319 L
184 1328 L
187 1344 L
187 1354 L
184 1370 L
177 1380 L
168 1383 L
161 1383 L
155 1380 L
152 1376 L
148 1370 L
145 1354 L
145 1344 L
148 1328 L
152 1322 L
155 1319 L
161 1316 L
168 1316 M 174 1319 L
177 1322 L
180 1328 L
184 1344 L
184 1354 L
180 1370 L
177 1376 L
174 1380 L
CS M
168 1383 L
-14 1682 M 0 1710 M 55 1710 L
87 1736 M 94 1739 L
103 1749 L
103 1682 L
100 1746 M 100 1682 L
87 1682 M 116 1682 L
161 1749 M 152 1746 L
145 1736 L
142 1720 L
142 1710 L
145 1694 L
152 1685 L
161 1682 L
168 1682 L
177 1685 L
184 1694 L
187 1710 L
187 1720 L
184 1736 L
177 1746 L
168 1749 L
161 1749 L
155 1746 L
152 1742 L
148 1736 L
145 1720 L
145 1710 L
148 1694 L
152 1688 L
155 1685 L
161 1682 L
168 1682 M 174 1685 L
177 1688 L
180 1694 L
184 1710 L
184 1720 L
180 1736 L
177 1742 L
174 1746 L
168 1749 L
132 2047 M 161 2115 M 152 2112 L
145 2102 L
142 2086 L
142 2076 L
145 2060 L
152 2051 L
161 2047 L
168 2047 L
177 2051 L
184 2060 L
187 2076 L
187 2086 L
184 2102 L
177 2112 L
168 2115 L
161 2115 L
155 2112 L
152 2108 L
148 2102 L
145 2086 L
145 2076 L
148 2060 L
152 2054 L
155 2051 L
161 2047 L
168 2047 M 174 2051 L
177 2054 L
180 2060 L
184 2076 L
184 2086 L
180 2102 L
177 2108 L
174 2112 L
168 2115 L
2261 255 M 2261 2261 L
2261 288 M 2226 288 L
2261 324 M 2226 324 L
2261 361 M 2226 361 L
2261 398 M 2226 398 L
2261 434 M 2226 434 L
2261 471 M 2226 471 L
2261 507 M 2226 507 L
2261 544 M 2226 544 L
2261 581 M 2226 581 L
2261 617 M 2192 617 L
2261 654 M 2226 654 L
2261 690 M 2226 690 L
2261 727 M 2226 727 L
2261 764 M 2226 764 L
2261 800 M 2226 800 L
2261 837 M 2226 837 L
2261 873 M 2226 873 L
2261 910 M 2226 910 L
2261 947 M 2226 947 L
2261 983 M 2192 983 L
2261 1020 M 2226 1020 L
2261 1056 M 2226 1056 L
2261 1093 M 2226 1093 L
2261 1130 M 2226 1130 L
2261 1166 M 2226 1166 L
2261 1203 M 2226 1203 L
2261 1239 M 2226 1239 L
2261 1276 M 2226 1276 L
2261 1313 M 2226 1313 L
2261 1349 M 2192 1349 L
2261 1386 M 2226 1386 L
2261 1422 M 2226 1422 L
2261 1459 M 2226 1459 L
2261 1496 M 2226 1496 L
2261 1532 M 2226 1532 L
2261 1569 M 2226 1569 L
2261 1605 M 2226 1605 L
2261 1642 M 2226 1642 L
2261 1679 M 2226 1679 L
2261 1715 M 2192 1715 L
2261 1752 M 2226 1752 L
2261 1788 M 2226 1788 L
2261 1825 M 2226 1825 L
2261 1862 M 2226 1862 L
2261 1898 M 2226 1898 L
2261 1935 M 2226 1935 L
2261 1971 M 2226 1971 L
2261 2008 M 2226 2008 L
2261 2045 M 2226 2045 L
2261 2081 M 2192 2081 L
2261 2118 M 2226 2118 L
2261 2154 M 2226 2154 L
2261 2191 M 2226 2191 L
2261 2228 M 2226 2228 L
CS [] 0 setdash M
stroke
grestore
showpage
end


