%Paper: hep-th/9311060
%From: boettche@aeg.phy.bnl.gov (Stefan Boettcher)
%Date: Wed, 10 Nov 93 09:25:05 EST


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\centerline{\bf DIMENSIONAL EXPANSION FOR THE ISING LIMIT}
\centerline{\bf OF QUANTUM FIELD THEORY}
\bigskip
\bigskip
\bigskip
\centerline{Carl M. Bender and Stefan Boettcher\footnote{$^*$}{Current
Address: Brookhaven National Laboratory; Upton, NY 11973}}
\medskip
\centerline{Department of Physics}
\medskip
\centerline{Washington University}
\medskip
\centerline{St. Louis, MO 63130}
\bigskip
\bigskip
\bigskip
\bigskip
\bigskip
\centerline{\bf ABSTRACT}
\bigskip

A recently-proposed technique, called the dimensional expansion, uses the
space-time dimension $D$ as an expansion parameter to extract nonperturbative
results in quantum field theory. Here we apply dimensional-expansion
methods to examine the Ising limit of a self-interacting scalar field theory.
We
compute the first few coefficients in the dimensional expansion for
$\gamma_{2n}$, the renormalized $2n$-point Green's function at zero momentum,
for $n\!=\!2$, 3, 4, and 5. Because the exact results for
$\gamma_{2n}$ are known at $D\!=\!1$ we can compare the predictions of the
dimensional expansion at this value of $D$. We find typical errors of less
than $5\%$. The radius of convergence of the dimensional expansion for
$\gamma_{2n}$ appears to be ${{2n}\over {n-1}}$. As a function of the
space-time dimension $D$, $\gamma_{2n}$ appears to rise monotonically with
increasing $D$ and we conjecture that it becomes infinite at
$D\!=\!{{2n}\over {n-1}}$.  We presume that for values of $D$ greater than this
critical value, $\gamma_{2n}$ vanishes identically because the corresponding
$\phi^{2n}$ scalar quantum field theory is free for $D\!>\!{{2n}\over{n-1}}$.
\footnote{}{PACS numbers: 11.10.-z, 11.90.+t, 02.90.+p}
\footnote{}{hep-th/9311060}
\vfill \eject

In a recent letter[1] we
proposed a new technique called the dimensional expansion, which can be used
to obtain nonperturbative results in quantum field theory. The dimensional
series uses the space-time dimension $D$ as an expansion parameter. The first
term in such an expansion is easy to obtain because a quantum field theory can
be solved in closed form in zero-dimensional space-time. An advantage of
dimensional expansions is that some of the nontrivial aspects of the
interaction
already
appear at $D\!=\!0$. (Traditional perturbative methods yield only
noninteractive
results in leading order.) The obvious question is how one can obtain the
coefficients of higher powers of $D$. A detailed explanation of how to do so
is given in a subsequent paper[2].

Here we use the dimensional expansion to compute the first four $\gamma_{2n}$,
the renormalized $2n$-point Green's functions at zero external momentum, for a
self-interacting scalar quantum field theory in the Ising limit. Specifically,
we calculate $\gamma_4$ to fourth order in powers of $D$, $\gamma_6$ to fifth
order in powers of $D$, $\gamma_8$ to sixth order in powers of $D$, and
$\gamma_{10}$ to seventh order in powers of $D$:
$$
\eqalign{
\gamma_4~&=~{1\over 12} \bigl[ 1+(1.180\pm0.001) D +(0.620\pm0.001) D^2 +
   (0.18\pm0.02) D^3 \cr
&~~~~~~~~~~~~+ (0.03\pm 0.02) D^4 +\ldots \bigr ]~~,\cr
\gamma_6~&=~{1\over 30} \bigl[ 1+(2.20\pm0.02) D + (2.30\pm0.03) D^2 +
   (1.50\pm0.03) D^3 + (0.55\pm0.04) D^4 \cr
&~~~~~~~~~~~~+ (0.12\pm0.04) D^5 +\ldots \bigr ] ~~,\cr
\gamma_8~&=~{1\over 56} \bigl[ 1+(3.0\pm0.1) D + (4.5\pm0.1) D^2 + (4.2\pm0.1)
D^3 + (2.6\pm0.1) D^4 \cr
&~~~~~~~~~~~+ (1.2\pm0.2) D^5 + (0.6\pm0.2) D^6 +\ldots \bigr ] ~~,\cr
\gamma_{10}~&=~{1\over 90} \bigl[ 1+(4.11\pm0.02)D + (8.0\pm0.1) D^2 +
(10.0\pm0.3) D^3 + (8.0\pm0.3) D^4 \cr
&~~~~~~~~~~~+ (4.5\pm0.3) D^5 + (1.8\pm0.3) D^6+(0.7\pm0.4)D^7 +\ldots  \bigr ]
{}~. \cr} \eqno(1)
$$

To obtain these dimensional
expansions we use the graphical methods described in Ref.~2. These graphical
methods rely on lattice strong-coupling techniques that were developed and
explained in an earlier series of papers[3,4,5,6]. For the Lagrangian
$$
{\cal L}~=~{1 \over 2} [{\partial \phi(x)}]^2
          ~+~{1 \over 2} m^2 {\phi(x)}^2 ~+~{1\over 4}g {\phi(x)}^4\eqno(2)
$$
the Ising limit[7,8,9] is defined as the limit in which the unrenormalized
coupling constant $g$ tends to infinity while the renormalized mass, $M$, is
held
fixed. The Ising limit is conveniently obtained by choosing $m^2\!\propto\!-g$.
In the limit $g\!\to\!\infty$ the theory asymptotically approaches a two-state
system. The Green's functions of this system are universal in the sense that
they are independent of the power of $\phi$ in the self-interaction term in
(2); $g\phi^{2k}$ gives the same results as $g\phi^4$ for all $k\!\geq\!2$.

Lattice strong-coupling methods are especially well suited to obtain the
dimensional expansion of Green's functions in quantum field theory because the
lattice integral for each graph is a {\it polynomial} in powers of $D$. This
property leads to an efficient organization of the graphs that contribute to
each order in the $D$-series. We achieve a high order in the graphical
expansion
by eliminating all graphs except those that contribute to the coefficients in
the dimensional expansion under consideration. We employ an intermediate
renormalization scheme to calculate the renormalized mass $M$ and dimensionless
renormalized $2n$-point scattering amplitudes $\gamma_{2n}$ at zero momentum.
We perform mass renormalization of the scattering amplitudes by eliminating the
bare mass $m$ in $\gamma_{2n}$ in favor of the renormalized mass $M$. We then
use Pad\'e extrapolation methods to derive a sequence of approximants for each
coefficient in the dimensional expansions in (1) for each of the scattering
amplitudes $\gamma_{2n}$ in the continuum limit. We believe that each
Pad\'e sequence gives an accurate approximation to the true
coefficient in the dimensional expansion for $\gamma_{2n}$ because these
dimensional series are in good numerical agreement with known exact results for
$\gamma_{2n}$[3]. The series in (1) are exact at $D\!=\!0$. At $D\!=\!1$ the
exact results are $\gamma_4\!=\!{1\over4}$, $\gamma_6\!=\!{1\over4}$,
$\gamma_8\!=\!{5\over{16}}$, $\gamma_{10}\!=\!{7\over{16}}$, and the results
for
(1) are $\gamma_4=0.250\pm0.003\,(1\%$ error),
$\gamma_6=0.26\pm0.01\,(4\%$ error), $\gamma_8=0.31\pm0.01\,(4\%$ error), and
$\gamma_{10}=0.42\pm0.02\,(5\%$ error).

We obtain the graphical rules for the lattice strong-coupling expansion by
observing that in the limit of large $g$ the kinetic term in the Lagrangian (2)
can be viewed as a small perturbation. Therefore, the generating function
$$
{\cal Z}[J]~=~{\cal N}~\int{\cal D} \phi(x)~exp \Bigl\{ ~-\textstyle \int\!
 d^{ D}x \bigl[ {1 \over 2} [{\partial \phi(x)}]^2
                    ~+~{1 \over 2} m^2 {\phi(x)}^2 ~+~{1\over 4}g {\phi(x)}^4
                    ~-~J(x) {\phi(x)} \bigr] \Bigr\} \eqno(3)
$$
for the quantum field theory associated with the Lagrangian (2)
can be rewritten as
$$
{\cal Z}[J]~=~exp \Bigl\{ {1 \over 2} \textstyle \int\!
d^{ D}xd^{ D}y {\textstyle {\delta \over {\delta J(x)}}}
                {\cal D}^{-1}(x-y)
{\textstyle {\delta \over {\delta J(y)}}} \Bigr\}~~
                {\cal Z}_0[J]~~, \eqno(4)
$$
where ${\cal D}^{-1}(x-y)={\partial}^2 {\delta}^{ D}(x-y)$ and
$$
{\cal Z}_0[J]~=~{\cal N} \int\! {\cal D} \phi~
                  exp \Bigl\{-\textstyle \int\! d^{ D}x \bigl[
                  {1 \over 2} m^2 \phi(x)^2
               ~+~{1\over 4}g\phi(x)^4~-~J(x)\phi(x)
                  \bigr] \Bigr\}~~.  \eqno(5)
$$

The factorization in (4) of the partition function leads to the
strong-coupling lattice expansion. By introducing a $D$-dimensional hypercubic
lattice with lattice spacing $a$ we  rewrite (5) as

$${\cal Z}_0[J]~=~{\cal N} \prod_i \int\limits_{-\infty}^{\infty}\! dt~
exp \Bigl\{ - {1 \over 2} a^{ D} m^2 t^2 - {1 \over 4} a^{ D} g t^4
+ a^{ D} J_i t \Bigr\}~~.   \eqno(6)
$$
Next, we expand in powers of $J_i$ and, to obtain the Ising limit, we set
$$
m^2~=~-\alpha g a^{2-D}~~, \eqno(7)
$$
where $\alpha$ is a dimensionless parameter considered to be small in the
strong-coupling limit:
$$
{\cal Z}_0[J]~=~{\cal N} \prod_i \sum_{n=0}^{\infty}
                  {1 \over {(2n)!}} (a^{ D} J_i)^{2n}
                  \int\limits_0^{\infty} dt~t^{n-1/2}~
exp \Bigl\{ - {1 \over 4} a^{ D} g \bigl[ t^2 - 2\,\alpha\,a^{2- D} t
 \bigr]     \Bigr\}~~.  \eqno(8)
$$
In the limit $g\!\to\!\infty$ the integral in (8) is asymptotic to $\alpha^n$
multiplied by a constant independent of $n$ which
we absorb into $\cal N$. Thus, we write ${\cal Z}_0[J]$ in (8) as
$$
{\cal Z}_0[J]~=~{\cal N} exp \Bigl\{ a^{ D} \sum_i \Bigl[
                  \sum\limits_{n=1}^{\infty} {1 \over {(2n)!}}
                  J_i^{2n} V_{2n} \Bigr] \Bigr\}~~,  \eqno(9)
$$
where the vertices are $V_2\!=\!a^2\alpha$, $V_4\!=\!-2a^{4+D}\alpha^2$,
$V_6\!=\!16a^{6+2D}\alpha^3$, $V_8\!=\!-272a^{8+3D}\alpha^4$,
$V_{10}\!=\!7936a^{10+4D}\alpha^5$ and so on. The propagator on the lattice can
be written in vector notation as ${\cal D}^{-1}\!=\!a^{-D-2}[(\underline
1)\!-\!2D(\underline 0)]$. This notation was introduced in Ref.~4 where this
discrete form of
the propagator was used to evaluate lattice integrals. The lattice
strong-coupling expansion is organized by the number of free propagators
${\cal D}^{-1}$ (in contrast to weak-coupling expansions where the number of
{\it vertices} and not the number of lines determines the order).

To compute $\gamma_{2n}$ it is necessary to calculate the
one-particle-irreducible $2n$-point functions
$\Lambda_{2n}$ for $n=1$, 2, 3, 4, and 5, in the strong-coupling
expansion and to find their Fourier transforms $\tilde {\Lambda}_{2n}$ in
momentum space at zero external momentum. We must also compute
${{\partial}\over{\partial (p^2)}} {\tilde \Lambda}_2(p^2)\vert_{p^2=0}$ to
obtain the wave-function renormalization constant defined by
$Z^{-1}\!\equiv\!1\!+\!{\partial \over {\partial (p^2)}}
{\tilde \Lambda}^{-1}_2 \vert_{p^2=0}$.
We define the scattering amplitudes $\gamma_{2n}$ as the {\it dimensionless}
renormalized one-particle-irreducible vertices at zero external momentum
$$
\gamma_{2n}~\equiv~{\tilde \Gamma}^R_{2n}(0,0,\ldots,0)
                M^{D(n-1)-2n} ~~,  \eqno(10)
$$
where $M$ is the renormalized mass defined as
$M^2\!\equiv\!{\tilde \Gamma}^R_2(0,0)$.
There are simple rules giving $\Gamma_{2n}$ in terms of $\Lambda_{2m}$,
$m\leq n$, which are explained in Ref.~4:
$$\eqalign{
  \Gamma_2~&=~\Lambda_2^{-1} ~~,\cr
  \Gamma_4~&=~-~\Lambda_4 \Lambda_2^{-4} ~~,\cr
  \Gamma_6~&=~-~\Lambda_6 \Lambda_2^{-6}
           ~+~{{6!} \over {2~(3!)^2}} \Lambda^2_4 \Lambda_2^{-7}
                                             ~~,\cr
  \Gamma_8~&=~-~\Lambda_8 \Lambda_2^{-8}
           ~+~ {{8!} \over {3!~5!}}
               \Lambda_4 \Lambda_6 \Lambda_2^{-9}
           ~-~{{8!} \over {(2!)^2~(3!)^2}}
               \Lambda^3_4 \Lambda_2^{-10} ~~,\cr
  \Gamma_{10}~&=~-~\Lambda_{10} \Lambda_2^{-10}
              ~+~ {{10!} \over {3!~7!}}
                  \Lambda_8 \Lambda_4 \Lambda_2^{-11}
              ~+~ {{10!} \over {2~(5!)^2}}
                  \Lambda^2_6 \Lambda_2^{-11}
              ~-~ {{10!} \over {2~3!~5!}}
                  \Lambda_6 \Lambda^2_4 \Lambda_2^{-12} \cr
              &~~~~-~ {{10!} \over {2~(3!)^2~4!}}
                  \Lambda_6 \Lambda^2_4 \Lambda_2^{-12}
              ~+~ {{10!} \over {2~(2!)^2~(3!)^2}}
                  \Lambda^4_4 \Lambda_2^{-13}
              ~+~ {{10!} \over {(3!)^4}}
                  \Lambda^4_4 \Lambda_2^{-13}~~. \cr
         } \eqno(11)
$$
We use the wave-function renormalization constant $Z$ to renormalize the
one-particle-irreducible vertices in an intermediate renormalization scheme
according to
${\tilde \Gamma}^R_{2n}(0,\ldots,0)\!=\!Z^n{\tilde \Gamma}_{2n}(0,\ldots,0)$.
In order to mass renormalize the scattering amplitudes $\gamma_{2n}$,
we eliminate the bare mass $m$, which is related to $\alpha$
through (7), in favor of the renormalized mass $M$. To that end, we simply
invert the relation obtained for the renormalized mass
$$\eqalign{
M^2a^2~&=~\alpha^{-1}-2\,D+ (2\,D-{2\over 3} )\alpha+ (4\,D
^{2}-{{26}\over{3}}\,D+{{194}\over{45}} )\alpha^{3} \cr
&~~~~~~~+  (32\,D^{3}-132\,D^{2}+{{2584}\over{15}}\,D-{{68164}\over{945}}
 )\alpha^{5} \cr
&~~~~~~~+ (-2048\,D^{5}-4096\,D^{4}-1024\,D^{3}-
800\,D^{2}+480\,D-64 )\alpha^{6} +\ldots\cr }   \eqno(12)
$$
to expand $\alpha$ in terms of $y\equiv a^{-2}M^{-2}$:
$$\eqalign{
\alpha~&=~y-2\,Dy^2+ (4\,D^2+2\,D-{2\over 3} )y^3
+ (-8\,D^3-12\,D^2+4\,D )y^4 + (16\,D^4+48\,D^3 \cr
&~~~~~~~-4\,D^2-14\,D+{{26}\over5} )y^5 + (-32\,D^5-160\,D^4
    -{{200}\over3}\,D^3+140\,D^2-52\,D ) y^6  \cr
&~~~~~+ (64\,D^6+480\,D^5+560\,D^4-720\,D^3+20\,D^2+272\,D-{{636}\over7}
 )y^7+\ldots~~. \cr}   \eqno(13)
$$
We then substitute (13) for $\alpha$ in every $\Gamma_{2n}^R$ to obtain
$$\eqalignno{
{\gamma_4}~&=~{y^{D/2} \over 12}~\bigr[1+4\,D\,y+ (4\,D^{2}-10\,D
 ) y^{2}+16\,D y^{3}+ (-80\,D^{2}+30\,D )y^{4} + (256\,D^{3} \cr
&~~~~~~~~~~~~~~+104\,D^{2}-192\,D ) y^{5}
+  (-704\,D^{4}-1736\,D^{3}+2508\,D^{2}-656\,D ) y^{6} \cr
&~~~~~~~~~~~~+ (1792\,D^{5}+10432\,D^{4}-11232\,D^{3}-3872\,D^{2}+4992\,D
 ) y^{7} + \ldots \bigr]~~, &(14)\cr
{\gamma_6}~&=~{y^D \over 30}~\bigl[ 1+6\,D\,y+ (12\,D^{2}-6\,D
 ) y^{2}+ (8\,D^{3}-12\,D^{2}-20\,D ) y^{3}\cr
&~~~~~~~~~~~~+ (48\,D^{2}+48\,D ) y^{4} + (-96\,D^{3}-816\,D ^{2}+528\,D
 ) y^{5} \cr
&~~~~~~~~~~~~+ (192\,D^{4} +4640\,D^{3}-2736\,D^ {2}-560\,D ) y^{6} \cr
&~~~~~~~~~~~~+ (-384\,D^{5}-18432\,D^{4}-10800\,D^{3}+46512\,D^{2}-23040\,D )
 y^{7} +\ldots \bigr],&(15) \cr
{\gamma_8}~&=~{y^{3D/2} \over 56}~\bigl[1+8\,D\,y+ (24\,D^{2}-8\,D
 ) y^{2}+ (32\,D^{3}-32\,D^{2} )y^{3} \cr
&~~~~~~~~~~~~+ (16\,D^{4}-32\,D^{3}-36\,D^{2}-18\,D ) y^{4}+ (
896\,D^{2}-448\,D ) y^{5}  \cr
&~~~~~~~~~~~~+ (-4192\,D^{3}-920\,D^{2}+ 2816\,D ) y^{6} \cr
&~~~~~~~~~~~~+ (13184\,D^{4}+52064\,D^{3}-92800\,D^ {2}+38400\,D ) y^{7}
+ \ldots \bigr] ~~, &(16)\cr
{\gamma_{10}}~&=~{y^{2D} \over 90}~\bigl[ 1+10\,D\,y+ (40\,D^{2}-10\,D
 ) y^{2}+ (80\,D^{3}-60\,D^{2} ) y^{3}  \cr
&~~~~~~~~~~~~+  (80\,D^{4}-120\,D^{3}+30\,D ) y^{4}+ (32\,D^{5}
-80\,D^{4}-300\,D^{2}+108\,D ) y^{5}  \cr
&~~~~~~~~~~~~+ (1280\,D^{3}+ 5040\,D^{2}-4240\,D ) y^{6} \cr
&~~~~~~~~~~~~+ (-2560\,D^{4}-64080\,D^{3}+76880\,D^{2}-23040\,D ) y^{7}
+\ldots  \bigr] ~~.&(17)\cr}
$$

The strong-coupling expansions in (14-17) were obtained by treating the
dimensionless parameter $\alpha\!=\!-a^{ D-2} m^2/g$ as small in the limit
where
the bare coupling $g$ tends to infinity. The relation in (12) explicitly
carries the assumption of smallness over to the parameter $y$.
This justifies the reversion of (12) into (13) and the subsequent
reexpansion of the scattering amplitudes $\gamma_{2n}$ in powers
of $y$. Up to this point we have taken the lattice spacing
$a$ to be held fixed. We expect that in the continuum limit $a\!\to\!0$ our
expressions for $\gamma_{2n}$ become the corresponding quantities of the
continuum theory. The continuum limit is subtle because as $a\to 0$ the
parameter $y$ that we have taken to be small actually becomes
infinite.
Hence, subsequent terms in this expansion for the scattering amplitudes
$\gamma_{2n}$ in (14-17) are increasingly singular as a series in powers of $y$
in the limit where $a \to 0$.

We use Pad\'e extrapolation techniques to extract information from
perturbation series like those in (14-17), where the perturbative parameter
tends to infinity. The Pad\'e extrapolation
method employed here uses as input a perturbation series of the form
$$
f(y)~=~y^r~(c_0~+~c_1 y~+~c_2 y^2~+~\ldots~)~~~~~~(r\not=0)~~, \eqno(18)
$$
where we assume that $f(\infty)$ is finite.  We first take the $r$th root of
both sides of (18) and divide by
$y$ to obtain
$$
{f(y)^{1/r} \over y}~=~(new~power~series~in~y)~. \eqno(19)
$$
Then we take the $N$th power of the right side of (19)
for $N=1,2,3,\ldots~$, reexpand and form the $(0,N)$-Pad\'e approximant.
By extracting the coefficient of $y^N$ in the denominator of the
$(0,N)$-Pad\'e and raising it to the power $-r/N$, we create a
sequence of Pad\'e extrapolants for $f(\infty)$[10].
We apply this method to the scattering amplitudes $\gamma_{2n}$ in (14-17) and
expand each $(0,N)$-Pad\'e approximant for all $\gamma_{2n}$ in
powers of $D$. For each $n$, we obtain a sequence in $N$ of
$(0,N)$-Pad\'e approximants for each coefficient in the $D$-series
of $\gamma_{2n}$. In Fig.~1 we plot the $(0,N)$-Pad\'e extrapolants for the
first four coefficients in the dimensional expansion of $\gamma_4$ as functions
of $1/N$.
We indicate the errors in the Pad\'e determination of the coefficients in (1).
We truncate the dimensional expansions for each $\gamma_{2n}$ after that
coefficient for which the estimated error of the following coefficient becomes
significant compared to its absolute size.

Observe that the series in (1) all have positive coefficients and therefore
each
$\gamma_{2n}$ is a monotonically rising function of $D$. Each of these
functions
is plotted in Fig.~2. For each $n$, $\gamma_{2n+2}$ is growing faster than
$\gamma_{2n}$ for increasing $D$. We believe that the radius of convergence of
the $D$-series for $\gamma_{2n}$ is likely to be $D\!=\!{{2n}\over{n-1}}$, the
space-time dimension for which the coupling constant $g$ of a $g\phi^{2n}$
theory becomes dimensionless and the theory becomes renormalizable. Since we
expect that for values of $D\!>{{2n}\over{n-1}}$, $\gamma_{2n}$
vanishes[11,12,13,14,15,16], we assume that there is a singularity (possibly a
natural boundary) in the complex-$D$ plane at $D\!=\!{{2n}\over{n-1}}$.
\vfill \eject

\centerline{\bf REFERENCES}
\bigskip
\item{[1]} C. M. Bender, S. Boettcher, and L. Lipatov, Phys. Rev. Lett.
{\bf 68}, 3674 (1992).
\medskip
\item{[2]} C. M. Bender, S. Boettcher, and L. Lipatov, Phys. Rev. D {\bf 46},
5557 (1992)
\medskip
\item{[3]} C. M. Bender, F. Cooper, G. S. Guralnik, H. Moreno, R. Roskies,
and D. H. Sharp, Phys. Rev. Lett. {\bf 45}, 501 (1980).
\medskip
\item{[4]} C. M. Bender, F. Cooper, G. S. Guralnik, R. Roskies, and D. H.
Sharp,
Phys. Rev. D {\bf 23}, 2976 (1981).
\medskip
\item{[5]} C. M. Bender, F. Cooper, G. S. Guralnik, R. Roskies, and D. H.
Sharp,
Phys. Rev. D {\bf 23}, 2999 (1981).
\medskip
\item{[6]} C. M. Bender, F. Cooper, G. S. Guralnik, R. Roskies, and D. H.
Sharp,
Phys. Rev. D {\bf 24}, 2683 (1981).
\medskip
\item{[7]} G. A. Baker Jr., Phys. Rev. D {\bf 15}, 1552 (1977).
\medskip
\item{[8]} G. A. Baker Jr., in ``Phase Transitions and Critical Phenomena,''
C. Domb and J. L. Lebowitz, eds., Vol. 9, p. 233 (Academic, London, 1984).
\medskip
\item{[9]} G. A. Baker Jr., Phys. Rev. Lett. {\bf 69}, 3264 (1992).
\medskip
\item{[10]} C. M. Bender, Los Alamos Science {\bf 2}, 76 (1981).
\medskip
\item{[11]} M. Aizenman, Phys. Rev. Lett. {\bf 47}, 1 (1981).
\medskip
\item{[12]} M. Aizenman, Commun. Math. Phys. {\bf 86}, 1 (1982).
\medskip
\item{[13]} J. Fr\"ohlich, Nucl. Phys. B {\bf 200} [FS4], 281 (1982).
\medskip
\item{[14]} J. Fr\"ohlich, in ``Progress in Gauge Field Theory,'' eds.
G. 't Hooft, A. Jaffe, H. Lehmann, P. K. Mitter, I. M. Singer and A. Stora,
p. 169 (Plenum, New York, 1984).
\medskip
\item{[15]} M. L\"uscher and P. Weisz, Nucl. Phys. B {\bf 290} [FS20], 25
(1987).
\medskip
\item{[16]} D. Callaway, Physics Reports {\bf 167}, 179 (1988) and
References therein.
\vfill \eject

\centerline{\bf FIGURE CAPTIONS}
\bigskip
\parindent=1in
\item{Figure 1.} Pad\'e extrapolation for the first four
coefficients, $b_i$ ($i=1,\ldots,4)$, in the dimensional expansion of
$\gamma_4\!=\!
{1\over{12}}(1\!+\!b_1D\!+\!b_2D^2\!+\!b_3D^3\!+\!b_4D^4\!+\!\ldots)$.
For each coefficient $b_i$ we plot the value of the $(0,N)$-Pad\'e (shown as
cross) as a function of $1/N$ for $N=1,\ldots,11$. The continuum value of each
$b_i$ is the extrapolation of the sequence to $N\!=\!\infty$. In Eq. (1) we
list
the results of this procedure for $\gamma_4$, $\gamma_6$, $\gamma_8$, and
$\gamma_{10}$.
\bigskip
\item{Figure 2.} Plot of $\gamma_4$, $\gamma_6$, $\gamma_8$, and $\gamma_{10}$
in Eq. (1) as functions of $D$. For each $n$, $\gamma_{2n}$ rises monotonically
and $\gamma_{2n+2}$ is growing faster than $\gamma_{2n}$ for increasing $D$.
We believe that the radius of convergence of the $D$-series for each
$\gamma_{2n}$ in Eq. (1) is $D\!=\!{{2n}\over{n-1}}$.
\bye




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/AryOfDxSpace [56 51 34 31] def
/AryOfDySpace [90 81 53 48] def
/AryOfDyBaseLine [ 0 0 0 0 ] def % filled in at init time
% input: character style index
/SetCharStyle
    {/sychar exch def typeface findfont
    AryOfDySpace sychar get scalefont setfont
    } def
/AryOfDashSolid [] def
/AryOfDashDotted [12 24] def
/AryOfDashDotDash [12 24 96 24] def
/AryOfDashShortDash [24 24] def
/AryOfDashLongDash [144 24] def
/AryOfAryOfDash
  [AryOfDashSolid AryOfDashDotted AryOfDashDotDash
   AryOfDashShortDash AryOfDashLongDash] def
% input: line style index
/SetLineStyle
    {AryOfAryOfDash exch get dup length setdash } def
/SetAryOfDyBaseLine
    {/mxT matrix def
    /fontT typeface findfont def
    0 1 AryOfDyBaseLine length 1 sub
        {/sycharT exch def
        % transform dyBaseline into user space at current font size
        AryOfDyBaseLine sycharT
            0 fontT /FontBBox get 1 get		% stack = (0 dyBaseline)
                AryOfDySpace sycharT get dup mxT scale
                fontT /FontMatrix get mxT concatmatrix
            dtransform
        exch pop put
        } for
    } def
/typeface /Courier def		% Should be fixed pitch
SetAryOfDyBaseLine
% end of 4014 prologue
90 rotate 0 -8.5 inch translate
0.38 inch 0.35 inch translate
/dxWidInch 10.24 def
/dyHtInch 7.8 def
ScaleCoords
0 SetCharStyle
0 SetLineStyle
xHome yHome moveto
1575 2512 m
0 -71 rl
s
1579 2512 m
0 -71 rl
s
1579 2478 m
6 7 rl
7 4 rl
7 0 rl
10 -4 rl
7 -7 rl
4 -10 rl
0 -7 rl
-4 -10 rl
-7 -6 rl
-10 -4 rl
-7 0 rl
-7 4 rl
-6 6 rl
s
1599 2489 m
7 -4 rl
7 -7 rl
3 -10 rl
0 -7 rl
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-7 -6 rl
-7 -4 rl
s
1565 2512 m
14 0 rl
s
1648 2450 m
4 2 rl
7 7 rl
0 -48 rl
s
1657 2457 m
0 -46 rl
s
1648 2411 m
20 0 rl
s
1575 1687 m
0 -71 rl
s
1579 1687 m
0 -71 rl
s
1579 1653 m
6 7 rl
7 4 rl
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4 -10 rl
0 -7 rl
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-7 -7 rl
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s
1599 1664 m
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3 -10 rl
0 -7 rl
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s
1565 1687 m
14 0 rl
s
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s
1662 1634 m
4 -2 rl
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s
1641 1591 m
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s
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s
1575 1027 m
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s
1579 1027 m
0 -72 rl
s
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-7 -7 rl
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s
1599 1003 m
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s
1565 1027 m
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s
1643 965 m
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6 3 rl
10 0 rl
6 -3 rl
3 -4 rl
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-7 0 rl
s
1662 974 m
4 -3 rl
2 -4 rl
0 -7 rl
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s
1662 953 m
4 -2 rl
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0 2 rl
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3 -2 rl
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s
1668 949 m
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s
1575 697 m
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s
1579 697 m
0 -71 rl
s
1579 663 m
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s
1599 674 m
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3 -10 rl
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s
1565 697 m
14 0 rl
s
1662 639 m
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s
1664 644 m
0 -48 rl
s
1664 644 m
-25 -34 rl
36 0 rl
s
1655 596 m
16 0 rl
s
3084 2211 m
-1237 131 rl
-413 42 rl
-206 16 rl
-124 9 rl
-82 6 rl
-60 5 rl
-44 6 rl
-34 3 rl
-28 3 rl
-22 1 rl
s
3084 1387 m
-1237 67 rl
-413 26 rl
-206 12 rl
-124 7 rl
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-44 3 rl
-34 2 rl
-28 0 rl
-22 0 rl
s
3084 804 m
-1237 -15 rl
-413 0 rl
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-124 2 rl
-82 1 rl
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-34 -1 rl
-28 -1 rl
-22 1 rl
s
3084 575 m
-1237 -26 rl
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-44 -2 rl
-34 0 rl
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s
3057 2183 m
55 55 rl
s
3057 2238 m
55 -55 rl
s
1819 2314 m
55 55 rl
s
1819 2369 m
55 -55 rl
s
1406 2356 m
55 55 rl
s
1406 2411 m
55 -55 rl
s
1201 2373 m
55 55 rl
s
1201 2428 m
55 -55 rl
s
1077 2381 m
54 55 rl
s
1077 2436 m
54 -55 rl
s
994 2387 m
55 55 rl
s
994 2442 m
55 -55 rl
s
935 2393 m
55 55 rl
s
935 2448 m
55 -55 rl
s
891 2398 m
55 55 rl
s
891 2453 m
55 -55 rl
s
856 2402 m
55 55 rl
s
856 2457 m
55 -55 rl
s
829 2404 m
55 55 rl
s
829 2459 m
55 -55 rl
s
806 2406 m
55 55 rl
s
806 2461 m
55 -55 rl
s
3057 1359 m
55 55 rl
s
3057 1414 m
55 -55 rl
s
1819 1426 m
55 55 rl
s
1819 1481 m
55 -55 rl
s
1406 1452 m
55 55 rl
s
1406 1507 m
55 -55 rl
s
1201 1464 m
55 55 rl
s
1201 1519 m
55 -55 rl
s
1077 1472 m
54 55 rl
s
1077 1527 m
54 -55 rl
s
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55 55 rl
s
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55 -55 rl
s
935 1483 m
55 54 rl
s
935 1537 m
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s
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55 54 rl
s
891 1540 m
55 -54 rl
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55 55 rl
s
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55 -55 rl
s
829 1488 m
55 55 rl
s
829 1543 m
55 -55 rl
s
806 1488 m
55 55 rl
s
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55 -55 rl
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3057 776 m
55 55 rl
s
3057 831 m
55 -55 rl
s
1819 762 m
55 55 rl
s
1819 817 m
55 -55 rl
s
1406 762 m
55 55 rl
s
1406 817 m
55 -55 rl
s
1201 764 m
55 55 rl
s
1201 819 m
55 -55 rl
s
1077 767 m
54 54 rl
s
1077 821 m
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s
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55 55 rl
s
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55 -55 rl
s
935 768 m
55 55 rl
s
935 823 m
55 -55 rl
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891 767 m
55 55 rl
s
891 822 m
55 -55 rl
s
856 767 m
55 54 rl
s
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55 -54 rl
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829 766 m
55 55 rl
s
829 821 m
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806 821 m
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3057 547 m
55 55 rl
s
3057 602 m
55 -55 rl
s
1819 521 m
55 55 rl
s
1819 576 m
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1406 515 m
55 55 rl
s
1406 570 m
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s
1201 514 m
55 55 rl
s
1201 569 m
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s
1077 514 m
54 55 rl
s
1077 569 m
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s
994 514 m
55 55 rl
s
994 569 m
55 -55 rl
s
935 511 m
55 55 rl
s
935 566 m
55 -55 rl
s
891 510 m
55 55 rl
s
891 565 m
55 -55 rl
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856 510 m
55 55 rl
s
856 565 m
55 -55 rl
s
829 510 m
55 55 rl
s
829 565 m
55 -55 rl
s
806 510 m
55 55 rl
s
806 565 m
55 -55 rl
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2475 0 rl
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s
708 495 m
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s
708 2932 m
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s
807 495 m
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s
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s
906 495 m
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s
906 2932 m
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s
1005 495 m
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s
1005 2932 m
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s
1104 495 m
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s
1104 2856 m
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s
1203 495 m
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s
1203 2932 m
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s
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s
1302 2932 m
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s
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s
1500 495 m
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s
1500 2932 m
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s
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s
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s
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1995 495 m
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2094 2856 m
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s
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s
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s
2391 2932 m
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s
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s
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s
2589 2856 m
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s
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s
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s
2886 2932 m
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s
2985 2932 m
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3046 660 m
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s
3046 1155 m
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s
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2970 1320 m
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s
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s
3046 1485 m
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s
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s
3046 1650 m
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s
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s
3046 1815 m
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3046 1980 m
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s
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114 0 rl
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2970 2145 m
114 0 rl
s
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s
3046 2310 m
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s
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s
3046 2475 m
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3046 2640 m
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s
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3046 2805 m
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554 395 m
-11 -3 rl
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11 -4 rl
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s
554 395 m
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s
561 323 m
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s
608 330 m
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s
656 395 m
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s
656 395 m
-7 -3 rl
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0 -10 rl
3 -17 rl
4 -7 rl
3 -3 rl
7 -4 rl
s
663 323 m
7 4 rl
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s
1049 395 m
-11 -3 rl
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s
1049 395 m
-7 -3 rl
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4 -3 rl
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s
1056 323 m
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s
1103 330 m
-3 -3 rl
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s
1134 381 m
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s
1162 395 m
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s
1131 330 m
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s
1141 334 m
17 -11 rl
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s
1543 395 m
-11 -3 rl
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s
1543 395 m
-7 -3 rl
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s
1550 323 m
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s
1597 330 m
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1659 395 m
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2039 395 m
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s
2039 395 m
-6 -3 rl
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s
2046 323 m
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s
2093 330 m
-3 -3 rl
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s
2162 385 m
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s
2145 395 m
-7 -3 rl
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0 -20 rl
3 -10 rl
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6 -4 rl
s
2148 323 m
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s
2535 395 m
-11 -3 rl
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s
2535 395 m
-7 -3 rl
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0 -10 rl
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3 -7 rl
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s
2541 323 m
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s
2589 330 m
-3 -3 rl
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3 4 rl
-3 3 rl
s
2634 395 m
-11 -3 rl
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0 -11 rl
3 -7 rl
11 -3 rl
13 0 rl
10 3 rl
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s
2634 395 m
-7 -3 rl
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4 -7 rl
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s
2647 364 m
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3 7 rl
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s
2634 364 m
-11 -3 rl
-3 -3 rl
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13 0 rl
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s
2634 364 m
-7 -3 rl
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s
2647 323 m
7 4 rl
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s
3014 381 m
6 4 rl
11 10 rl
0 -72 rl
s
3027 392 m
0 -69 rl
s
3014 323 m
30 0 rl
s
3078 330 m
-3 -3 rl
3 -4 rl
4 4 rl
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s
3126 395 m
-10 -3 rl
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0 -10 rl
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11 4 rl
7 10 rl
3 17 rl
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-11 3 rl
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s
3126 395 m
-6 -3 rl
-4 -4 rl
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-3 -17 rl
0 -10 rl
3 -17 rl
3 -7 rl
4 -3 rl
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3133 323 m
7 4 rl
4 3 rl
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0 10 rl
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s
421 532 m
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421 532 m
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0 -9 rl
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3 -7 rl
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428 460 m
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3 7 rl
4 18 rl
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s
476 467 m
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3 -4 rl
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524 532 m
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0 -9 rl
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524 532 m
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4 -7 rl
3 -3 rl
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530 460 m
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4 18 rl
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s
421 1357 m
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421 1357 m
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0 -9 rl
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3 -7 rl
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428 1285 m
6 4 rl
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0 9 rl
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s
476 1292 m
-3 -3 rl
3 -4 rl
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510 1357 m
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503 1323 m
7 6 rl
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530 1333 m
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0 -7 rl
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510 1357 m
34 0 rl
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510 1354 m
17 0 rl
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411 2169 m
7 4 rl
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s
425 2180 m
0 -69 rl
s
411 2111 m
30 0 rl
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476 2118 m
-3 -3 rl
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524 2183 m
-10 -3 rl
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524 2183 m
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3 -3 rl
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530 2111 m
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4 17 rl
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s
411 2994 m
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425 3004 m
0 -68 rl
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411 2936 m
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476 2943 m
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3 -4 rl
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510 3008 m
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503 2973 m
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530 2984 m
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510 3008 m
34 0 rl
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510 3004 m
17 0 rl
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1748 244 m
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1762 255 m
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1748 186 m
31 0 rl
s
1865 271 m
-62 -108 rl
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1889 258 m
0 -72 rl
s
1892 258 m
42 -65 rl
s
1892 251 m
42 -65 rl
s
1934 258 m
0 -72 rl
s
1879 258 m
13 0 rl
s
1923 258 m
20 0 rl
s
1879 186 m
20 0 rl
s
217 1666 m
72 0 rl
s
217 1669 m
72 0 rl
s
217 1655 m
0 42 rl
3 10 rl
4 3 rl
7 3 rl
10 0 rl
7 -3 rl
4 -3 rl
3 -10 rl
0 -28 rl
s
217 1697 m
3 6 rl
4 4 rl
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10 0 rl
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4 -4 rl
3 -6 rl
s
289 1655 m
0 25 rl
s
271 1740 m
48 0 rl
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271 1742 m
43 28 rl
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275 1742 m
44 28 rl
s
271 1770 m
48 0 rl
s
271 1733 m
0 9 rl
s
271 1763 m
0 14 rl
s
319 1733 m
0 14 rl
s
188 1749 m
3 -7 rl
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s
188 1749 m
3 -4 rl
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11 -2 rl
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s
236 1754 m
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showpage

%%Trailer
grestore
ps4014sav restore

save /ps4014sav exch def
/inch{72 mul}def
/xHome 0 def
/yHome 3071 def
/xLeftMarg 0 def
/m /moveto load def
/l /lineto load def
/rl /rlineto load def
/r /rmoveto load def
/s /stroke load def
/ml {/sychar exch def AryOfDxSpace sychar get neg 0 rmoveto} def
/mr {/sychar exch def AryOfDxSpace sychar get 0 rmoveto} def
/mu {/sychar exch def 0 AryOfDySpace sychar get rmoveto} def
/md {/sychar exch def 0 AryOfDySpace sychar get neg rmoveto} def
/cr1 {/sychar exch def
      xLeftMarg currentpoint exch pop AryOfDySpace sychar get sub moveto} def
/cr2 {xLeftMarg currentpoint exch pop moveto} def
/sh /show load def
% scale the coordinate space
% input: size of image area (x, y) in inches
/ScaleCoords {72 4096 dxWidInch div div 72 3120 dyHtInch div div scale} def
/AryOfDxSpace [56 51 34 31] def
/AryOfDySpace [90 81 53 48] def
/AryOfDyBaseLine [ 0 0 0 0 ] def % filled in at init time
% input: character style index
/SetCharStyle
    {/sychar exch def typeface findfont
    AryOfDySpace sychar get scalefont setfont
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/AryOfDashSolid [] def
/AryOfDashDotted [12 24] def
/AryOfDashDotDash [12 24 96 24] def
/AryOfDashShortDash [24 24] def
/AryOfDashLongDash [144 24] def
/AryOfAryOfDash
  [AryOfDashSolid AryOfDashDotted AryOfDashDotDash
   AryOfDashShortDash AryOfDashLongDash] def
% input: line style index
/SetLineStyle
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/SetAryOfDyBaseLine
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% end of 4014 prologue
90 rotate 0 -8.5 inch translate
0.38 inch 0.35 inch translate
/dxWidInch 10.24 def
/dyHtInch 7.8 def
ScaleCoords
0 SetCharStyle
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xHome yHome moveto
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2401 2331 m
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2403 2336 m
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2403 2336 m
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2394 2288 m
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1710 2355 m
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1713 2359 m
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1764 2365 m
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1802 2336 m
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1805 2288 m
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1463 2355 m
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1466 2359 m
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1521 2365 m
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1517 2365 m
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1550 2336 m
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1550 2336 m
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1560 2315 m
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1550 2315 m
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1550 2315 m
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1560 2288 m
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1116 2355 m
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1119 2359 m
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1174 2365 m
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1170 2365 m
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1199 2327 m
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1208 2333 m
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1199 2288 m
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1252 2336 m
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1252 2336 m
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1256 2288 m
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609 597 m
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609 536 m
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4 27 rl
s
609 517 m
4 0 rl
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609 508 m
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609 495 m
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708 495 m
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708 2932 m
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807 495 m
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906 495 m
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906 2932 m
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1005 495 m
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476 467 m
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524 532 m
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524 532 m
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530 460 m
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421 1151 m
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421 1151 m
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476 1086 m
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510 1151 m
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503 1117 m
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510 1151 m
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510 1148 m
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411 1756 m
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425 1767 m
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411 1698 m
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476 1705 m
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524 1770 m
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524 1770 m
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530 1698 m
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411 2375 m
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425 2386 m
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411 2317 m
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476 2324 m
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510 2389 m
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503 2355 m
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530 2365 m
7 -3 rl
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4 -10 rl
0 -7 rl
-4 -10 rl
-7 -7 rl
-7 -4 rl
s
510 2389 m
34 0 rl
s
510 2386 m
17 0 rl
17 3 rl
s
404 2994 m
4 -3 rl
-4 -4 rl
-3 4 rl
0 3 rl
3 7 rl
4 3 rl
10 4 rl
13 0 rl
10 -4 rl
4 -3 rl
3 -7 rl
0 -7 rl
-3 -7 rl
-11 -7 rl
-16 -6 rl
-7 -4 rl
-7 -7 rl
-3 -9 rl
0 -11 rl
s
431 3008 m
7 -4 rl
3 -3 rl
4 -7 rl
0 -7 rl
-4 -7 rl
-10 -7 rl
-13 -6 rl
s
401 2943 m
3 4 rl
7 0 rl
17 -7 rl
10 0 rl
7 3 rl
3 4 rl
s
411 2947 m
17 -11 rl
13 0 rl
4 4 rl
3 7 rl
0 6 rl
s
476 2943 m
-3 -3 rl
3 -4 rl
3 4 rl
-3 3 rl
s
524 3008 m
-10 -4 rl
-7 -10 rl
-4 -17 rl
0 -10 rl
4 -17 rl
7 -10 rl
10 -4 rl
6 0 rl
11 4 rl
7 10 rl
3 17 rl
0 10 rl
-3 17 rl
-7 10 rl
-11 4 rl
-6 0 rl
s
524 3008 m
-7 -4 rl
-3 -3 rl
-4 -7 rl
-3 -17 rl
0 -10 rl
3 -17 rl
4 -7 rl
3 -3 rl
7 -4 rl
s
530 2936 m
7 4 rl
4 3 rl
3 7 rl
4 17 rl
0 10 rl
-4 17 rl
-3 7 rl
-4 3 rl
-7 4 rl
s
1828 258 m
0 -72 rl
s
1831 258 m
0 -72 rl
s
1817 258 m
34 0 rl
10 -3 rl
7 -7 rl
4 -7 rl
3 -11 rl
0 -16 rl
-3 -10 rl
-4 -7 rl
-7 -7 rl
-10 -4 rl
-34 0 rl
s
1851 258 m
7 -3 rl
7 -7 rl
3 -7 rl
4 -11 rl
0 -16 rl
-4 -10 rl
-3 -7 rl
-7 -7 rl
-7 -4 rl
s
showpage

%%Trailer
grestore
ps4014sav restore

