%Paper: 9203221
%From: OSCARH@hep.physics.mcgill.ca (OSCAR HERNANDEZ (514)398-6492)
%Date: Thu, 26 Mar 1992 18:43:12 -0500 (EST)


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%%%%%%%%%%%%%%%%%%%%%%%%%the paper%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\preprint{hep-ph@\rightline{UCLA/92/TEP/9}\rightline{MCGILL/92--11}
\vskip 0.3in plus 0.1in
\title{\titlefont Improved Heavy Quark Effective Theory Currents}
\vskip 0.85in plus 0.4in
\centerline{\bf Oscar F. Hern\'andez}
\vskip 0.12in plus 0.02in
\addressline{Department of Physics}
\addressline{McGill University}
\addressline{Ernest Rutherford Physics Building}
\addressline{Montr\'eal, Qu\'e., Canada H3A 2T8}
\vskip 0.2in plus 0.02in
\centerline{\bf Brian R. Hill}
\vskip 0.12in plus 0.02in
\addressline{Department of Physics}
\addressline{University of California}
\addressline{Los Angeles, CA~~90024}
\vskip 0.1in plus 0.02in
\abstract
It is hoped that the accuracy of a variety of lattice calculations
will be improved by perturbatively eliminating effects proportional to the
lattice spacing.  In this paper, we apply this improvement program to the heavy
quark effective theory currents which cause a heavy quark to decay to a light
quark, and renormalize the resulting operators to order $\alphaS$.
We find a small decrease in the amount that the operator needs to be
renormalized, relative to the unimproved case.
\vskip 0.1in plus 0.02in
\date{3/92}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newsec{Introduction}%
It is hoped that the accuracy of a variety of lattice
calculations will be improved by the elimination of effects proportional to
powers of the lattice spacing, $a$~\ref\program{K. Symanzik, Nucl. Phys. B
{\bf 226} (1983) 187.}\ref\LandW{
M. L\"uscher and P. Weisz, Commun. Math. Phys. {\bf 97} (1985) 59.}.
This improvement program can be implemented
order by order in $a$ and the strong coupling $g^2$.  Calculationally, it is
relatively easy to perform the leading order in either of these expansions. For
matrix elements of vector currents, both types of leading order corrections are
thought to be in
the~20 to~30\% range~\ref\Tallahassee{E. Gabrielli {\it et al},
in {\sl Lattice 90,} ed. by U. M. Heller {\it
et al,} Nucl. Phys. B (Proc. Suppl.) {\bf 20} (1991) 448\semi G.
Heatlie {\it et al,} \npb352,91,266 .},
so it is at the same time important to
include them, and reasonable to stop at this order.

Much of the application of the improvement program to hadronic matrix elements
has been to the order $a$
improvement of the Wilson fermion action and operators
\ref\NNN{H. W. Hamber and C. M. Wu, \plb133,83,351 ; \plb136,84,255 .}--%
\nref\gab{E. Gabrielli {\it et al,} \npb362,91,475 .}%
\ref\SandW{B. Sheikholeslami and R. Wohlert, \npb259,85,572 .}.
Alternative methods for measuring matrix elements involving
quarks that are heavy relative to the QCD scale
have been proposed~\ref\Etalk{E. Eichten, in {\sl Field
Theory on the Lattice,} edited by A.~Billoire {\it et al.}, Nucl.
Phys. B (Proc. Suppl.) {\bf 4} (1988) 170.}\ref\LandTtalk{G. P.
Lepage and B. A. Thacker, {\it ibid}, p. 199.}.
The action and operators in this method can be thought of
as discretizations of the heavy quark effective field theory action and
operators~\LandTtalk
\ref\eft{W. E. Caswell and G. P. Lepage, \plb167,86,437\semi
H. D. Politzer and M. B. Wise, \plb208,88,504 \semi
E. Eichten and B. Hill, \plb234,90,511 \semi
B. Grinstein, \npb339,90,253 \semi
H.~Georgi, \plb240,90,447 \semi
M. J. Dugan, M. Golden, and B. Grinstein, HUTP/91--A045, to appear in Phys.
Lett. B.}.
In this paper, we study the application of the improvement program to this
action and the currents which cause a heavy quark to decay to a light quark.

The outline of the paper is as follows.  In the next section, we discuss
improvement of the heavy quark action, and in section three, we discuss
improvement of the aforementioned currents.  As is generally the case, the
improved lattice currents need to be related  to their continuum counterparts.
This is done perturbatively in $g^2$ where the scale is set by the lattice
spacing.  Although quite different in motivation, this perturbative calculation
is similar to the renormalization of other discretizations of the heavy quark
effective theory currents performed in
reference~\ref\fbsplit{O. F. Hern\'andez and B. R. Hill, UCLA/91/TEP/51, to
appear in Phys. Lett.~B.}. That calculation and the present one rely on the
framework and results of reference~\ref\fb{E. Eichten and B. Hill,
\plb240,90,193.}, which are summarized in section~four before the new
perturbative results are presented. In section five we give our conclusions.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newsec{Improvement of the Heavy Quark Action}%
In this section, we argue that
under two criteria of improvement, there is no need to modify the heavy quark
action.  The arguments are tree level, but this is adequate for the leading
order improvement discussed in the introduction.

Roughly, the on-shell improvement condition formulated by L\"uscher and
Weisz~\LandW\ is
that spectral quantities such as the location of single-particle poles
should not be corrected by terms proportional to powers of the lattice spacing.
Examining the momentum space propagator for the heavy quark action in
the continuum~\eft,
%
\eqn\contprop{{1\over p_0+i\epsilon},}
%
and comparing it with its lattice counterpart~\fb,
%
\eqn\latticeprop{{1\over{-i}(e^{ip_0a}-1)/a+i\epsilon},}
%
we see that although the propagators differ at order $a$, the
pole is located at $p_0=0$ to all orders in $a$.  We immediately conclude that
the action has no need of improvement under the on-shell improvement condition,
to all orders in $a$ at tree level.

A second argument yielding this conclusion is to compare
the heavy quark propagator in position space in an external gauge field
on the lattice to the same thing in the
continuum.\myfoot{$^\dagger$}{We thank Estia
Eichten for discussions leading to this argument.}
In the continuum, the propagator from $x$ to $y$
is~\ref\olderpapers{J. M. Cornwall and G. Tiktopoulos, \prd15,77,2937 \semi
E. Eichten and F. Feinberg, \prd23,81,2724 .},
%
\eqn\poscontprop{-i\delta^3({\bf y}-{\bf x})\theta(y_0-x_0)
P\exp{-ig\int_{x_0}^{y_0}dt\thinspace A_0(t,{\bf x})}.}
The discretization of the action with a nearest neighbor,
one-sided time derivative follows from this propagator~\fb.
On the lattice, the heavy quark propagator from~$m$ to~$n$ in a
background field is~\Etalk\LandTtalk,
%
\eqn\poslattprop{{-i\over a^3}\delta_{\bf nm}\theta(n_0-m_0)
U_0(n-\zerohat)^\dagger
U_0(n-2\zerohat)^\dagger
\cdots
U_0(m)^\dagger.}
%
In this expression, $\delta_{\bf nm}$ is the three-dimensional
Kronecker $\delta$--function,
$\theta(n_0-m_0)$ is $1$ if $n_0\ge m_0$ and $0$ otherwise, and $U_0(m)$ is
the lattice gauge link in the time direction at site $m$.

If the correspondence between continuum and lattice gauge fields is~\LandW,
%
\eqn\correspondence{U_0(n)=
P\exp{iga\int_0^1dt\thinspace A_0((n_0+1-t)a,{\bf n}a)},}
%
then the lattice propagator perfectly reproduces its continuum counterpart.
If one modifies the heavy quark action (by choosing any other
discretization of the time derivative)
the position space lattice propagator~\poslattprop\ is multiplied by a
$c$-number factor depending only on $n_0{-}m_0$ but is otherwise unchanged
(as long as the discretization does not include spatial links).  With the
correspondence~\correspondence, there is clearly no closer approximation
to the continuum propagator~\poscontprop\ than the propagator~\poslattprop\
obtained from the action currently in use.
% (for which this $c$-number factor is identically one).
%A number of these arguments are modified if a mass term is present in
%the heavy quark propagator.  The location of the pole is at
%i_l_0={1\over a}\ln(1+ma). We note that one solution to this difficulty is to
%identify the combination 1\over a\ln(1+ma) as the mass.  This also works with
%the light quark propagator example given above (but better not have worked if
%I hadn't suppressed Dirac structure and the other three dimensions, else
%everyone has missed the simplest solution to the improvement problem). We also
%note that such a term doesn't appear until order $g^2$.
%Unfortunately,the term that does appear is order $1/a$, which is dangerous for
%all arguments which rely on power counting in $a$.  To dispose of this concern
%completely, one should base the argument on the renormalized propagator which
%has as a condition that the location of the pole is at zero.  The
%mass counterterms are introduced to
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newsec{Improved Heavy-Light Currents}%
Since the lattice heavy quark action is not in need of improvement, we review
the situation for the light quark action.  Expressions for improved
lattice heavy-light currents quickly follow from this discussion.

The improved Wilson fermion action proposed by Sheikholeslami and
Wohlert~\SandW\ is
obtained from an improved action with next-to-nearest neighbor couplings~\NNN\
by a change of variables.  The resulting action is the Wilson action
plus an additional piece,
%
\eqn\DSI{\Delta S_I=-ia^4{ar\over4}\sum_{n\mu\nu}
\bar q(n)\sigma_{\mu\nu}gP_{\mu\nu}(n)q(n).}
%
$P_{\mu\nu}(n)$ is the sum of plaquettes defined in reference~\gab\
and goes to $F_{\mu\nu}(n)$
in the continuum limit, while $\sigma_{\mu\nu}=[\gamma_\mu,\gamma_\nu]/2$.
The action has the advantage that it only contains nearest neighbor couplings.
One can most easily obtain improved operators for use with this action by
starting with local bilinears and making the replacement~\Tallahassee\SandW,
%
\eqn\transformation{q(n)\rightarrow q(n)-{r\over2}\sum_\mu\gamma_\mu
\left[U_\mu(x)q(n+\mu)-U_\mu(n-\muhat)^\dagger q(n-\muhat)\right].}
%
A similar replacement is made for $\bar q(n).$

Actually, a two-parameter family of transformations, all of
which yield operators improved to order $a$, can be obtained by
using the equations of motion for the light and heavy quark.  The
effect of applying the equation of motion to the light quark field has
been discussed in reference \gab\ and the effect of applying the
equation of motion to the heavy quark field has been studied under
the guise of temporally split operators~\fbsplit.  In either case,
the operator renormalization is changed by an amount which comes
from the self-energy.  We will not consider this generalization further.

{\baselineskip=18pt plus 1pt
The most general heavy-light bilinear in the full continuum theory is,
%
\eqn\J{J(x)=\bar b(x) {\bf \Gamma} q(x).}
%
Here ${\bf \Gamma}$ is any Dirac matrix, and $q$ is the light quark field.
%
In the heavy quark effective theory, the corresponding operator~\J\ is~\eft,
%
\eqn\cont{\bdagger(x)\onezero{\bf \Gamma} q(x).}
%
In a Dirac basis, the two-by-four matrix preceding ${\bf \Gamma}$
takes the form $\onezero$ and projects onto the upper two rows of
${\bf \Gamma}.$  Applying the above recipe to the local operator
$b^\dagger(n)\onezero{\bf \Gamma q(n)}$ we are led to consider,
%
\eqn\improvedcurrent{b^\dagger(n)\onezero{\bf \Gamma} q(n)
-{r\over2}b^\dagger(n)\onezero{\bf \Gamma}\sum_\mu\gamma_\mu
\left[U_\mu(x)q(n+\mu)-U_\mu(n-\muhat)^\dagger q(n-\muhat)\right].}
%
The first term in this expression is the usual local current used to determine
hadronic matrix elements.  The calculations necessary to renormalize this part
of the current were mostly done in previous papers~\fb\ref\blp{Ph. Boucaud,
C. L. Lin, and O. P\`ene, \prd40,89,1529 ; \prd41,90,3541(E) .},
however it receives
additional contributions coming from the new term in the action \DSI.  The
remainder of the current,  which is naively order $a,$ will also affect the
renormalization of the current when it appears in loop diagrams with loop
momenta of order $a^{-1}$. We now turn to the perturbative renormalization of
the improved heavy-light current with the improved Wilson action of
Sheikholeslami and Wohlert.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newsec{Renormalization of the Improved Current}%
The lattice renormalization of
the heavy-light current \improvedcurrent\ gives the ratio of the lattice
operator to its counterpart in the continuum theory.  We can divide the
diagrams that contribute to this ratio into two parts.  Those that give
heavy and light quark wave function renormalization, and the 1PI
vertex correction diagrams.

The difference between wave function renormalization of the heavy quark on the
lattice and in the continuum is $g^2/(12\pi^2)$ times a constant $e=4.53$
computed in references~\fb\ and~\blp.\myfoot{$^\dagger$}{Here
we are using the reduced value of $e$ which is appropriate
if one fits correlation functions containing the propagator
\poslattprop\ to $A\e{-B(n_0{-}m_0)a}$~\fbsplit\fb\blp.}
The corresponding light quark wave function renormalization constant was
calculated in \gab.
The results for the~self~energy-graphs~which\eject}
we will need are given in terms of $\Delta_{\Sigma_1}$, defined in
reference \gab\ (when consulting their expressions for $\Delta_{\Sigma_1},$
note that we have taken $F_{0001}=1.31$~\ref\fzzzo{A. Gonzales-Arroyo and C. P.
Korthals-Altes, \npb205,83,46. }).

We now turn to the vertex correction diagrams of figures
\fig\veri{The vertex correction diagrams resulting from improving the
light quark operator} and \fig\verii{The vertex correction diagrams
resulting from improving the action}.  The techniques we use to
evaluate these diagrams have been discussed in detail in a number of
references~\fbsplit\fb, and will not be reviewed here.

At zero external momentum, \veri (a) vanishes.  As noted below
equation~\improvedcurrent, the operator has two pieces: one which is the
local unimproved operator, and an additional piece coming from improvement.
Thus the contribution of \veri (b) can be split up into two parts
which we will call the ``unimproved'' and ``improved'' contributions here
and in the following paragraph.  The ``unimproved'' contribution is~\fb\blp,
\eqn\verib{
{g^2\over 12 \pi^2} \big( d_1 + G d_2  \big)
.}
We have defined the
$c$-number $G$ by $G{\bf \Gamma}=\gamma_0{\bf \Gamma}\gamma_0$.
The analytical expressions for $d_1$ and $d_2$ can be found in reference~\fb.
We tabulate the constant
$\Delta_{\Sigma_1}$ and the constants $d_1$ and $d_2$ in Table~1
for several values of the Wilson mass parameter $r$.
Errors in Table~1 and~2 are at most \curlyO(1)\ in the last decimal place.
\topinsert
\vbox{
\def\tablerule{\noalign{\hrule}}
$$\vbox{\offinterlineskip\tabskip=0em
	\halign to 4in{\vrule #\tabskip=1em plus5em&
		\strut\hfil$#$\hfil&\vrule #&
      \hfil$#$\hfil&\vrule #&
      \hfil$#$\hfil&\vrule #&
      \hfil$#$\hfil&\tabskip=0em\vrule #\cr
		\tablerule
%		height 2pt&\omit&&&&&&&\cr
		&r&&d_1&&d_2&&\Delta_{\Sigma_1}&\cr
%		height 2pt&\omit&&&&&&&\cr
		\tablerule
		&1.00&&5.46&&-7.22&&-9.21&\cr
		\tablerule
		&0.75&&5.76&&-7.23&&-8.62&\cr
		\tablerule
		&0.50&&6.30&&-7.00&&-7.80&\cr
		\tablerule
		&0.25&&7.37&&-5.72&&-6.73&\cr
		\tablerule
		&0.00&&8.79&&\phantom{-}0.00&&-6.04&\cr
		\tablerule}}
$$
\centerline{Table 1. Previously Computed $r$-dependent
Quantities \gab\fb\blp.}
}
\endinsert

The ``improved'' contribution from diagram \veri (b) can be combined
with the contribution from the other three diagrams in figure \veri.
In order to compactly quote the analytic expressions for this result, we
use the notation of references
\ref\MandZ{G. Martinelli and Y.-C. Zhang, \plb123,83,433 .}
and~\ref\fhh{J. M. Flynn, O. F.  Hern\'andez, and B. R. Hill,
\prd43,91,3709 .}\
for the following commonly occurring combinations:
\def\summu{\sum\nolimits_\mu}
\eqn\Deltas{\eqalign{\Delta_1=\summu\sin^2{l_\mu\over2},\quad &
\Delta_4=\summu\sin^2{l_\mu}, \cr
\Delta_2=\Delta_4+4r^2\Delta_1^2 , \quad &
\Delta_5=\summu\sin^2{l_\mu\over2}\sin^2{l_\mu}.\cr
}}
Additional combinations, $\Delta_1^{(3)},$ $\Delta_2^{(3)},$ and
$\Delta_4^{(3)},$ are the same as above except the sums on $\mu$ run only
from~$1$ to~$3$.
Given these definitions the total ``improved'' contribution from
figure~\veri\ to the vertex renormalization is,
\eqn\totalveri{
 {g^2\over 12 \pi^2} ( \Delta d_1 + G \Delta d_2),
}
where,
\eqn\ij{
\Delta d_1 ={-3r^2 \over 16}\int {d^4 l \over \pi^2} {4-\Delta_1 \over
\Delta_2},
\quad \Delta d_2={-r^3 \over 2}\int {d^3 l \over \pi} {\Delta^{(3)}_1 \over
\Delta^{(3)}_2}.
}

We now turn to the contributions depicted in figure~\verii\ which are
due to the additional term in the
action~\DSI.  In these figures, the insertion of this term
is denoted with a cross.  A tadpole
diagram which vanishes has not been depicted.  The result
for the two diagrams depicted is,
\eqn\totalverii{
{g^2\over 12 \pi^2} ( \Delta d'_1 + G \Delta d'_2 ),
}
where,
\eqn\ijkl{
\Delta d'_1 ={r^2 \over 4}\int {d^4 l \over \pi^2}
{\Delta_4(3-\Delta_1)+\Delta_5 \over
4\Delta_1\Delta_2},
\quad \Delta d'_2 ={r \over 2}\int {d^3 l \over \pi}
{\Delta^{(3)}_4(1+r^2\Delta^{(3)}_1) \over
4\Delta^{(3)}_1\Delta^{(3)}_2}.
}

\topinsert
\def\tablerule{\noalign{\hrule}}
\vbox{
$$\vbox{\offinterlineskip\tabskip=0em
	\halign to 6in{\vrule #\tabskip=1em plus5em&
		\strut\hfil$#$\hfil&\vrule #&
      \hfil$#$\hfil&\vrule #&
      \hfil$#$\hfil&\vrule #&
      \hfil$#$\hfil&\vrule #&
      \hfil$#$\hfil&\tabskip=0em\vrule #\cr
		\tablerule
%		height 2pt&\omit&&&&&\cr
&r&&\Delta d_1&&\Delta d_2&&\Delta d_1'&&\Delta d_2'&\cr
%		height 2pt&\omit&&&&&&&\cr
		\tablerule
		&1.00&&-6.64&&-5.84&&3.43&&6.16&\cr
		\tablerule
		&0.75&&-5.30&&-3.75&&2.53&&4.94&\cr
		\tablerule
		&0.50&&-3.61&&-1.87&&1.55&&3.66&\cr
		\tablerule
		&0.25&&-1.52&&-0.45&&0.55&&2.15&\cr
		\tablerule
		&0.00&&\phantom{-}0.00&&\phantom{-}0.00&&0.00&&0.00&\cr
		\tablerule}}
$$
\centerline{Table 2. Changes to $d_1$ and $d_2$ as a function of $r$.}
}
\endinsert

Combining the various results in this section, we find that the
ratio of the lattice to continuum operators is,
\eqn\ratiolec{
1+{g^2\over12\pi^2}\left[
(d_1+\Delta d_1 + \Delta d'_1)
+(d_2+\Delta d_2 + \Delta d'_2)G
+{1\over2}e-{1\over2}\Delta_{\Sigma_1}-1 \right]\!.}
The dependance on $\mu a$ has been eliminated by setting $\mu=1/a$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newsec{Conclusions}%
We illustrate the use of the results of the previous section for the case of
most
interest, the current which determines the $B$ meson decay constant, $f_B$.  In
that case, for reasonable values of the input parameters, the ratio of the
continuum effective theory to full theory bilinears is numerically
$0.98$~\eft\blp.
% More accurately 0.980
We need to multiply this by the ratio given in eq.~\ratiolec.
For ${\bf \Gamma}=\gamma_0\gamma_5,$ the constant $G$
(which appears in \ratiolec) is~$-1$. We take $g^2=1.8$ which is appropriate
for effects arising from the scale $\pi/a$ with $1/a=2\gev$ as argued
in~\ref\LandM{G. P.
Lepage and P. B. Mackenzie, in {\sl Lattice 90,} ed. by U. M. Heller
{\it et al,} Nucl. Phys. B (Proc. Suppl.) {\bf 20} (1991) 173.}.
Taking values from tables~1 and~2 with $r=1.00$,
equation~\ratiolec\ gives % more accurately 1.228
1.23 and the product of the two ratios is 1.20 (as compared to 1.28 in the
unimproved case).  To obtain the physical value of $f_B,$ one divides
the lattice results for the improved current by this number.

The operator we have renormalized is corrected both to order $g^2$ and to
order~$a$.   As noted in the introduction both of these corrections are thought
to be in the 20 to 30\% range.  Analytically, the next perturbative
corrections are proportional to $g^2 a,$ $g^4,$ or $a^2$ times
powers of $g^2\ln a$~\Tallahassee, and there is numerical evidence
that these further corrections are at the few per cent level for
currents made from two Wilson fermion fields~\Tallahassee.  Thus it is hoped
that the improved currents renormalized here will lead to a considerably
more precise lattice determination of $f_B$ and other heavy quark
matrix elements.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip 0.2in
{\it Note Added.} While preparing this manuscript we learned of unpublished
work on this  subject~\ref\asquared{G. Martinelli and G. C. Rossi,
unpublished.}\ref\BandP{A. Borrelli and C. Pittori, unpublished.} referenced in
a paper on lattice measurements of several quantities using an improved
action~\ref\results{G. Martinelli, C. T. Sachrajda, G. Salina, and A. Vladikas,
SHEP 91/92--6.}.   Martinelli and Rossi~\asquared\ argue that
it is unnecessary to change the heavy quark action up to order $a^2.$
This is consistent with our conclusion in section~two that the heavy quark
action is not in need of improvement at any order in $a$.  The result of
Borrelli and Pittori~\BandP\ cited in reference~\results\ is consistent with
our factor above if one takes $g^2=1.0$ rather than~$1.8$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgements
OFH was supported in part by the National Science and Engineering Research
Council of Canada, and les Fonds FCAR du Qu\'ebec.
BRH was supported in part by the Department of Energy under Contract No.
DE--AT03--88ER 40383 Mod~A006--Task~C.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\baselineskip=16pt\listrefs\listfigs\bye


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showpage

%! lwplot file
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showpage

%! lwplot file
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72 300 div dup scale
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showpage

%! lwplot file
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72 300 div dup scale
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1196 1771 m 1198 1767 c
1196 1769 m 1200 1767 c
1236 1814 m 1236 1809 c
1241 1809 c
1241 1814 c
1236 1814 c
1238 1814 m 1238 1809 c
1236 1812 m 1241 1812 c
1236 1798 m 1236 1767 c
1238 1796 m 1238 1769 c
1229 1798 m 1241 1798 c
1241 1767 c
1229 1767 m 1247 1767 c
1232 1798 m 1236 1796 c
1234 1798 m 1236 1794 c
1236 1769 m 1232 1767 c
1236 1771 m 1234 1767 c
1241 1771 m 1243 1767 c
1241 1769 m 1245 1767 c
1286 1796 m 1288 1794 c
1290 1796 c
1288 1798 c
1286 1798 c
1281 1796 c
1279 1794 c
1270 1798 m 1265 1796 c
1263 1794 c
1261 1789 c
1261 1785 c
1263 1780 c
1265 1778 c
1270 1776 c
1274 1776 c
1279 1778 c
1281 1780 c
1283 1785 c
1283 1789 c
1281 1794 c
1279 1796 c
1274 1798 c
1270 1798 c
1265 1794 m 1263 1789 c
1263 1785 c
1265 1780 c
1279 1780 m 1281 1785 c
1281 1789 c
1279 1794 c
1270 1798 m 1268 1796 c
1265 1791 c
1265 1782 c
1268 1778 c
1270 1776 c
1274 1776 m 1277 1778 c
1279 1782 c
1279 1791 c
1277 1796 c
1274 1798 c
1263 1780 m 1261 1778 c
1259 1773 c
1259 1771 c
1261 1767 c
1263 1764 c
1270 1762 c
1279 1762 c
1286 1760 c
1288 1758 c
1263 1767 m 1270 1764 c
1279 1764 c
1286 1762 c
1259 1771 m 1261 1769 c
1268 1767 c
1279 1767 c
1286 1764 c
1288 1760 c
1288 1758 c
1286 1753 c
1279 1751 c
1265 1751 c
1259 1753 c
1256 1758 c
1256 1760 c
1259 1764 c
1265 1767 c
1265 1751 m 1261 1753 c
1259 1758 c
1259 1760 c
1261 1764 c
1265 1767 c
1306 1798 m 1306 1778 c
1308 1771 c
1311 1769 c
1315 1767 c
1322 1767 c
1326 1769 c
1329 1771 c
1331 1776 c
1308 1796 m 1308 1776 c
1311 1771 c
1299 1798 m 1311 1798 c
1311 1776 c
1313 1769 c
1315 1767 c
1331 1798 m 1331 1767 c
1342 1767 c
1333 1796 m 1333 1769 c
1324 1798 m 1335 1798 c
1335 1767 c
1302 1798 m 1306 1796 c
1304 1798 m 1306 1794 c
1335 1771 m 1338 1767 c
1335 1769 m 1340 1767 c
1358 1798 m 1358 1767 c
1360 1796 m 1360 1769 c
1351 1798 m 1363 1798 c
1363 1767 c
1378 1794 m 1378 1796 c
1376 1796 c
1376 1791 c
1381 1791 c
1381 1796 c
1378 1798 c
1374 1798 c
1369 1796 c
1365 1791 c
1363 1785 c
1351 1767 m 1369 1767 c
1354 1798 m 1358 1796 c
1356 1798 m 1358 1794 c
1358 1769 m 1354 1767 c
1358 1771 m 1356 1767 c
1363 1771 m 1365 1767 c
1363 1769 m 1367 1767 c
1396 1785 m 1421 1785 c
1421 1789 c
1419 1794 c
1416 1796 c
1410 1798 c
1405 1798 c
1398 1796 c
1394 1791 c
1392 1785 c
1392 1780 c
1394 1773 c
1398 1769 c
1405 1767 c
1410 1767 c
1416 1769 c
1421 1773 c
1419 1787 m 1419 1789 c
1416 1794 c
1396 1791 m 1394 1787 c
1394 1778 c
1396 1773 c
1416 1785 m 1416 1791 c
1414 1796 c
1410 1798 c
1405 1798 m 1401 1796 c
1398 1794 c
1396 1787 c
1396 1778 c
1398 1771 c
1401 1769 c
1405 1767 c
1484 1805 m 1484 1803 c
1486 1803 c
1486 1805 c
1484 1805 c
1484 1807 m 1486 1807 c
1488 1805 c
1488 1803 c
1486 1800 c
1484 1800 c
1482 1803 c
1482 1805 c
1484 1809 c
1486 1812 c
1493 1814 c
1502 1814 c
1509 1812 c
1511 1809 c
1513 1805 c
1513 1800 c
1511 1796 c
1504 1791 c
1493 1787 c
1488 1785 c
1484 1780 c
1482 1773 c
1482 1767 c
1509 1809 m 1511 1805 c
1511 1800 c
1509 1796 c
1502 1814 m 1506 1812 c
1509 1805 c
1509 1800 c
1506 1796 c
1502 1791 c
1493 1787 c
1482 1771 m 1484 1773 c
1488 1773 c
1500 1771 c
1509 1771 c
1513 1773 c
1488 1773 m 1500 1769 c
1509 1769 c
1511 1771 c
1488 1773 m 1500 1767 c
1509 1767 c
1511 1769 c
1513 1773 c
1513 1778 c
1589 1823 m 1585 1818 c
1580 1812 c
1576 1803 c
1574 1791 c
1574 1782 c
1576 1771 c
1580 1762 c
1585 1755 c
1589 1751 c
1580 1809 m 1578 1803 c
1576 1794 c
1576 1780 c
1578 1771 c
1580 1764 c
1585 1818 m 1583 1814 c
1580 1807 c
1578 1794 c
1578 1780 c
1580 1767 c
1583 1760 c
1585 1755 c
1609 1814 m 1609 1767 c
1611 1769 c
1616 1769 c
1611 1812 m 1611 1771 c
1602 1814 m 1614 1814 c
1614 1769 c
1614 1791 m 1616 1796 c
1620 1798 c
1625 1798 c
1632 1796 c
1636 1791 c
1638 1785 c
1638 1780 c
1636 1773 c
1632 1769 c
1625 1767 c
1620 1767 c
1616 1769 c
1614 1773 c
1634 1791 m 1636 1787 c
1636 1778 c
1634 1773 c
1625 1798 m 1629 1796 c
1632 1794 c
1634 1787 c
1634 1778 c
1632 1771 c
1629 1769 c
1625 1767 c
1605 1814 m 1609 1812 c
1607 1814 m 1609 1809 c
1654 1823 m 1658 1818 c
1663 1812 c
1667 1803 c
1670 1791 c
1670 1782 c
1667 1771 c
1663 1762 c
1658 1755 c
1654 1751 c
1663 1809 m 1665 1803 c
1667 1794 c
1667 1780 c
1665 1771 c
1663 1764 c
1658 1818 m 1661 1814 c
1663 1807 c
1665 1794 c
1665 1780 c
1663 1767 c
1661 1760 c
1658 1755 c
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