%Paper: 
%From: PALLANTE@roma2.infn.it
%Date: Thu, 4 Aug 1994 18:15:39 +0200 (WET-DST)

%
%   begin of Latex file
%
\documentstyle[12pt]{article}

\renewcommand{\baselinestretch}{1.2}
\renewcommand{\topfraction}{1}
\renewcommand{\bottomfraction}{1}
\renewcommand{\textfraction}{0}
\def\thefootnote{\fnsymbol{footnote}}
\topmargin 2.5cm
\sloppy
\parindent 0.5cm
\textwidth 15.0cm
\textheight 21.5cm
\hoffset=-0.8cm
\voffset=-3cm
%
%\let\Huge=\large
%\let\huge=\large
%\let\Large=\normalsize
%\let\large=\normalsize
%************************
% \footline={\hfill}
% \headline={\hss\tenrm\folio\hss}
% \voffset=1\baselineskip
%***********************
%\newcommand{p1k2}{p_1\cdot k_2}
%\newcommand{p2k1}{p_2\cdot k_1}
%\newcommand{l12}{l_1^2}
%\newcommand{l22}{l_2^2}
%\newcommand{\q12}{q_1^2}
%\newcommand{\q22}{q_2^2}
%\newcommand{\q14}{q_1^4}
%\newcommand{\q24}{q_2^4}
%\newcommand{\l14}{l_1^4}
%\newcommand{\l24}{l_2^4}
\newcommand{\eps}{\epsilon}
\newcommand{\epsmnab}{\epsilon_{\mu\nu\alpha\beta}}
\newcommand{\Udag}{U^{\dagger}}
\newcommand{\udag}{u^{\dagger}}
\newcommand{\s}{\Sigma}
\newcommand{\fmunu}{f_{\mu\nu}}
\newcommand{\Lmunu}{L_{\mu\nu}}
\newcommand{\Rmunu}{R_{\mu\nu}}
\newcommand{\Lab}{L_{\alpha\beta}}
\newcommand{\Rab}{R_{\alpha\beta}}
\newcommand{\Rtab}{\tilde{R}_{\alpha\beta}}
\newcommand{\Rtmunu}{\tilde{R}_{\mu\nu}}
\newcommand{\lmu}{{\cal L}_{\mu}}
\newcommand{\lnu}{{\cal L}_{\nu}}
\newcommand{\lalfa}{{\cal L}_{\alpha}}
\newcommand{\lbeta}{{\cal L}_{\beta}}
\newcommand{\llambda}{{\cal L}_{\lambda}}
\newcommand{\DUdag}{DU^{\dagger}}
\newcommand{\Dslash}{\mbox{ $\not \!\! D$}}
\newcommand{\half}{\mbox{\small $\frac{1}{2}$}}
\newcommand{\quarter}{\mbox{\small $\frac{1}{4}$}}
\newcommand{\piT}{\tilde{\pi}}
\newcommand{\rhoT}{\tilde{\rho}}
\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\lab}{\protect\label}
\newcommand{\cutoff}{\Lambda_\chi}
\newcommand{\capt}{\protect\caption}
\newcommand{\si}{\stackrel{\leftharpoonup}{d}}
\newcommand{\sict}{\stackrel{\leftharpoonup}{d}^{C^T}}
\newcommand{\sictt}{\stackrel{\leftharpoonup}{d}^{C^T2}}
\newcommand{\sic}{\stackrel{\leftharpoonup}{d}^{C}}
\newcommand{\de}{\stackrel{\rightharpoonup}{d}}
\newcommand{\dect}{\stackrel{\rightharpoonup}{d}^{C^T}}
\newcommand{\dectt}{\stackrel{\rightharpoonup}{d}^{C^T2}}
\newcommand{\dec}{\stackrel{\rightharpoonup}{d}^{C}}

\begin{document}

\begin{titlepage}
\begin{flushright}
LNF 94/031 (P)\\
ROM2F 94/12\\
\\
\end{flushright}
\vspace{0.5cm}
\begin{center}

{
%\Large
\bf
The hadronic vacuum polarization contribution to the \\
muon g-2 in the quark-resonance model}

\vspace*{1.0cm}

\vspace*{1.2cm}
         {\bf E. Pallante\footnotemark
      \footnotetext{ email: pallante@vaxtov.roma2.infn.it \\
                     \hspace*{6.5mm}fax: +39-6-9403-427 }} \\
{\footnotesize{ I.N.F.N., Laboratori Nazionali di Frascati,
Via E. Fermi, 00044 Frascati ITALY.}}

\end{center}

\vspace*{1.5cm}

\begin{abstract}

The hadronic vacuum polarization contribution to the anomalous magnetic
moment of the muon is parametrized by using the quark-resonance model
formulated in \cite{QR}.  In this context a recent prediction obtained
within the ENJL model \cite{RAF} can be affected by two additional
contributions: the next to leading corrections in the inverse cutoff
expansion and the gluonic corrections. Motivated by the necessity of
reaching a highly accurate theoretical prediction of the hadronic
contribution to the muon g-2, we study in detail both the effects.

\end{abstract}
\vspace*{1.0cm}
\hspace*{1.0cm} Short Title: The muon g-2 in the Quark-Resonance model.
\vspace*{2.0cm}
\begin{flushleft}
LNF 94/031 (P)\\
ROM2F 94/12\\
\end{flushleft}
%\vspace*{1.cm}
\footnotetext{
Work partially supported by the EEC Human Capital and Mobility program.}
\end{titlepage}
\newpage

Effective chiral Lagrangians {\em \`a la} Nambu-Jona Lasinio are a good
theoretical framework to understand the hadronic vacuum polarization
contribution to the anomalous magnetic moment of the muon, given by the
diagram in Figure 1.

Phenomenological estimates of $a^h_\mu =(g^h-2)/2$ are obtained from the
best fit of the $e^+e^-\to hadrons$ total cross section $\sigma^h(t)$
through the usual dispersion relation

\beq
a_\mu^h = {1\over 4\pi^3 }\int_{4m_\pi^2}^\infty~dt~K(t)~\sigma^h(t),
\eeq

whith the QED function $K(t)$ given by:

\beq
K(t) = \int_0^1~dx~{x^2(1-x)\over x^2+(1-x){t/ m_\mu^2}} .
\eeq

The most recent numerical estimates give the following values:

\begin{eqnarray} && \bullet ~~7.07(.066)(.17)\cdot 10^{-8}~~~~~~
\cite{Kinoshita} \nonumber\\ &&\bullet ~~6.84(.11)\cdot 10^{-8}
{}~~~~~~~~~~~~~~\cite{Casas} \nonumber\\ &&\bullet ~~7.100(.105)(.49)\cdot
10^{-8}~~~~ \cite{Kurdadze} , \lab{PHEN} \end{eqnarray}

where the first error is statistical and the second one is systematic.
In what follows we give a theoretical picture to understand the above
numerical values.

$a^h_\mu$ is related to the renormalized hadronic photon self-energy
$\Pi^h_R(Q^2)$ through the following integral \cite{RAF,RAFOLD}

\beq
a^h_\mu = {\alpha\over \pi}\int_0^1~dx (1-x) \biggl [ -e^2~\Pi^h_R\biggl (
{x^2\over 1-x} m_\mu^2\biggr )\biggr ] .
\lab{INTE}
\eeq

$\Pi^h_R(Q^2)$ is given in terms of the vector two point function
$\Pi_V^1(Q^2)$ which has been extensively analyzed in refs.
\cite{QR,2point}:

\beq \Pi^h_R(Q^2) = \sum_{i=u,d,s}Q_i^2~ (\Pi_V^1(Q^2)-\Pi_V^1(0))=
{2\over 3} (\Pi_V^1(Q^2)-\Pi_V^1(0)) , \eeq

where $Q_i=(2/3,-1/3,-1/3)$ are the charges of the SU(3) flavour quarks
u,d,s and the renormalized photon self-energy satisfies the constraint
$\Pi^h_R(0)=0$.

Because the typical momenta of the off-shell photons are of the order of
the squared muon mass $(Q^2\sim .01 GeV^2)$ the integral is dominated by
the low energy contribution to $\Pi^h_R(Q^2)$.  By doing the Taylor
expansion of $\Pi^h_R(Q^2)$ at $Q^2=0$ and by imposing $\Pi^h_R(0)=0$
one has

\beq \Pi^h_R(Q^2) = Q^2\Pi_R^{h\prime} (Q^2)=Q^2 \biggl [ {d\Pi^h_R\over
dQ^2}(0) + {1\over 2}Q^2{d^2\Pi^h_R\over (dQ^2)^2}(0) +~ ....~\biggr ] .
\eeq

Notice that $\Pi_R^{h \prime} (Q^2)$ coincides with the first derivative
only at $Q^2=0$. The term with the second derivative contains one
additive power of the ratios $Q^2/\cutoff^2$ and $Q^2/M_Q^2$, where
$\cutoff\sim 1$ GeV is the ultraviolet low-energy cutoff and $M_Q\sim
300$ MeV is the infrared low-energy cutoff and are the natural
dimensionful parameters of the long-distance expansion.

By first approximation $\Pi^h_R(Q^2)\sim Q^2 {d\Pi^h_R\over dQ^2}(0)$
and the integral of eq. (\ref{INTE}) gives the value for $a^h_\mu$

\beq
a^h_\mu = \biggl ( {\alpha\over \pi}\biggr )^2 m_\mu^2 ~{4\pi^2\over 3}
\biggl [ -{2\over 3}{d\Pi_V^1\over dQ^2}(0)\biggr ] .
\lab{FIRSTA}
\eeq

The calculation of $a_\mu^h$ reduces then to the calculation of the
first derivative of the vector two-point function at $Q^2=0$, as was
already pointed out in ref. \cite{RAF}.  In ref. \cite{2point} the
long-distance behaviour of the vector function was derived in the ENJL
framework.  In \cite{QR} we have shown that the inclusion in the
bosonized NJL Lagrangian of higher dimensional quark-resonance vertices,
which are suppressed respect to the leading lowest dimensional
four-quark operator by powers of the inverse cutoff $\cutoff$,
 generates next-to-leading power corrections to the leading logarithms
(NPLL) of the parameters of the effective resonance Lagrangian which are
responsible of their $Q^2$ dependence.

As a consequence, higher dimensional quark-resonance operators modify
the long distance behaviour of the vector Green's function predicted by
the ENJL model and enter in the determination of $a_\mu^h$ through eq.
(\ref{INTE}).  Already in the first approximation of eq. (\ref{FIRSTA})
the NPLL corrections proportional to $Q^2$, i.e. of the type
$Q^2/\cutoff^2\ln (\cutoff^2/M_Q^2 )$ give contribution to the
derivative at $Q^2=0$; the derivative at $Q^2=0$ is sensitive to the
behaviour at shorter distances.

We proceed as follows. Using the first approximation of eq.
(\ref{FIRSTA}) we review the ENJL prediction already derived in
\cite{RAF}. Then we study the relevance of two sources of corrections:
the NPLL contributions and the gluon contributions.  Beyond the first
approximation we analyze the sensitivity to both of them via the
evaluation of the dispersive integral of eq. (\ref{INTE}) over the
long-distance part of $\Pi^h_R(Q^2)$.

The vector two-point function in the ENJL model and in the chiral limit
can be parametrized in terms of the running photon-vector coupling
$f_V(Q^2)$ and the squared mass of the vector resonance $M_V^2(Q^2)$ as
follows:

\beq
\Pi_V^1(Q^2) = {2f_V^2(Q^2)M_V^2(Q^2)\over M_V^2(Q^2)+ Q^2},
\lab{VEC}
\eeq

or equivalently in terms of the vector two-point function in the mean
field approximation $\overline{\Pi}_V^1(Q^2)$ \cite{2point}

\beq
\Pi_V^1(Q^2) ={ \overline{\Pi}_V^1(Q^2)\over 1+ Q^2
{8\pi^2 G_V\over N_c\cutoff^2} \overline{\Pi}_V^1(Q^2) } ,
\lab{MF}
\eeq

where $G_V(\cutoff )$ is the coefficient of the four-quark vector-like
interaction in the ENJL Lagrangian.  The $Q^2$ dependent parameters are
given by:

\begin{eqnarray} \overline{\Pi}_V^1(Q^2)&=& {N_c\over 16\pi^2} 8
\int_0^1~d\alpha~\alpha (1-\alpha )~ \Gamma (0, \alpha_Q) \nonumber\\
f^2_V(Q^2)&=&{1\over 2}\overline{\Pi}_V^1(Q^2) \nonumber\\ M_V^2(Q^2)&=&
{N_c\cutoff^2\over 8\pi^2 G_V} \biggl (\overline{\Pi}_V^1(Q^2)\biggr
)^{-1} , \lab{PARA} \end{eqnarray}

with the incomplete Gamma function $\Gamma (0,\alpha_Q) = -\ln\alpha_Q
-\gamma_E + {\cal{O}}(\alpha_Q)$, $\alpha_Q = (M_Q^2 + \alpha (1-\alpha
)Q^2)/ \cutoff^2$ and $\gamma_E = 0.5772...$ is the Euler's constant.
Expressions (\ref{VEC}) and (\ref{MF}) with the parameters (\ref{PARA})
correspond to the diagram of Figure 2a, which is the infinite
resummation of linear chains of constituent quark bubbles with the
insertion of the leading four-quark vector operator of the ENJL
Lagrangian.  Linear chains of quark bubbles are of order $N_c$, while
loops of chains of quark bubbles are of order 1 in the $1/N_c$
expansion.

The derivative at $Q^2=0$ is given by

\beq {d\Pi_V^1\over dQ^2}(0)={d\overline{\Pi}_V^1\over dQ^2}(0)-
{\overline{\Pi}_V^1}(0)^2 {8\pi^2 G_V\over N_c\cutoff^2 }
 = 2{df_V^2\over dQ^2}(0) - 2 {f_V^2(0)\over M_V^2(0)} .
\lab{DEFI}
\eeq

The ENJL model gives the following prediction:

\beq {d\Pi_V^1\over dQ^2}(0) = -{N_c\over 16\pi^2}{1\over M_Q^2} {4\over
15} \biggl [ e^{-{M_Q^2\over \cutoff^2}} + {5\over 6} {1-g_A\over g_A}
\Gamma (0, {M_Q^2\over \cutoff^2})\biggr ] , \lab{NJLP} \eeq

where $g_A$ is the mixing parameter between the axial-vector and the
pseudoscalar mesons in the bosonized ENJL action and it is related to
the vector meson mass and the vector coupling $G_V$ by the following
relations \cite{2point}:

\beq {1-g_A\over g_A}= 4M_Q^2{G_V\over\cutoff^2}\Gamma
(0,M_Q^2/\cutoff^2 ) ={6M_Q^2\over M_V^2}.  \lab{RELA} \eeq

With the values of the best fit 1 of ref. \cite{ENJL} $M_Q=265 MeV$,
$\cutoff = 1.165 GeV$ and $g_A = 0.61$ we obtain for $a_\mu^h$ the value
$a^h_\mu = 8.66\cdot 10^{-8}$, which corresponds to the value
${d\Pi_V^1\over dQ^2}(0)=-0.164$ of the first derivative.  In formula
(\ref{NJLP}) the incomplete Gamma function has been approximated to its
leading logarithmically divergent part.  This corresponds to the
substitutions $\Gamma (0,\alpha_Q) = -\ln\alpha_Q -\gamma_E$ and
$exp(-M_Q^2/ \cutoff^2)=1$, where $exp(-M_Q^2/ \cutoff^2)$ comes from
the derivative of $\Gamma (0,\alpha_Q)$ at $Q^2=0$.

Chiral loop corrections due to $\pi , K$ exchanges, are not included in
the above value of $a_\mu^h$. They are given by the diagram in Figure 2b
and are next-to-leading (i.e. O(1)) in the $1/N_c$ expansion; in the
ENJL framework they are generated by diagrams with loops of chains of
quark bubbles. This contribution has been derived in \cite{RAF} using
{\em ChPt} with a value of $(0.71\pm 0.07)\cdot 10^{-8}$.  The two
summed contributions give

\beq a_\mu(had) = a_\mu ({\hbox{Fig. 2a}}) + a_\mu(\chi {\hbox{loops}})
= 9.37\cdot 10^{-8}, \lab{NUM1} \eeq

which is a rather high value compared to the phenomenological estimates
in (\ref{PHEN}). Beyond the fact that the first derivative approximation
can be not sufficiently accurate, we first analyze the effects of the
two ``extra'' contributions we have mentioned.

The first source of ``extra'' corrections can be derived in the
framework of the Quark-Resonance model formulated in \cite{QR}.

The vector two-point function calculated with the inclusion of NTL
vertices corresponds to the diagram of Figure 2c.: the vector resonance
exchange diagram plus the ``local'' diagram. The infinite resummation of
linear chains of quark bubbles of Figure 2a is a part of the
contributions to the renormalized vector resonance propagator.  %, i.e.
the insertion %of the q-q-R vertex generated by the bosonization of the
vector four-quark %leading vertex proportional to $G_V$.

The complete set of NTL corrections (i.e. $1/\cutoff^2$), includes two
types of contributions \cite{QR}:

\begin{description} \item[I.] NTL genuine power corrections (NTLP) of
the type: ${Q^2\over \cutoff^2 },{M_Q^2\over \cutoff^2 }$


\item[II.] NTL power corrections to the leading logs (NPLL) of the type:
 ${Q^2\over \cutoff^2 }\ln \alpha_Q , {M_Q^2\over \cutoff^2 }\ln
\alpha_Q$.  \end{description}

Class I is generated by an infinite number of higher dimensional
quark-resonance vertices, while class II is generated by a finite number
of $1/\cutoff^2$ vertices \cite{QR}.
 A best fit to the experimental $e^+e^-\to hadrons$ cross section in the
$I=1,J=1$ channel and in the intermediate $Q^2$ region (500 MeV - 900
MeV) has shown the numerical relevance of the new NPLL counterterms
proportional to $Q^2$ \cite{QR}, while NPLL corrections proportional to
the IR cutoff $M_Q$ are negligible in this regime. They are not
negligible when treating the very low energy behaviour $(Q^2\to 0)$ of
the Green's functions, as it happens in the present case.

In table (\ref{TERMS}) the type of quark-resonance vertices we need are
listed.  We distinguish four sectors: the derivative one, the vector
one, the scalar one and the scalar-vector one.  NTL corrections
proportional to $Q^2$ get contributions from the derivative and the
vector sets, while NTL corrections proportional to $M_Q^2$ get
contributions also from the scalar and scalar-vector sets when the
scalar field assumes its VEV, $<H>=M_Q$, which plays the role of the IR
cutoff of the effective theory.

The full running of the vector resonance parameters ($Z_V =$ vector wave
function renormalization constant, $f_V$ = vector-photon coupling and
$M_V$ = vector mass) is parametrized as follows:

\begin{eqnarray} Z_V(Q^2)&=& {N_c\over 16\pi^2} {1\over 3}
\int_0^1~d\alpha~ \Gamma (0,\alpha_Q ) \biggl [ 6\alpha (1-\alpha ) +
12\alpha (1-\alpha )\beta_V^1 {Q^2\over \cutoff^2} + \beta_M^1
{M_Q^2\over \cutoff^2} P_1(\alpha )\biggr ] \nonumber\\
M_V^2(Q^2)&=&{N_c\over 16\pi^2}{\cutoff^2\over 2{\tilde{G}_V}}{1\over
Z_V} \biggl [ 1 + \beta_M^2 {M_Q^2\over \cutoff^2} \int_0^1~d\alpha~
\Gamma (0,\alpha_Q ) P_2(\alpha )\biggr ] \nonumber\\ f_V(Q^2)&=&
{1\over \sqrt{Z_V}} {N_c\over 16\pi^2}{\sqrt{2}\over 3}
\int_0^1~d\alpha~ \Gamma (0,\alpha_Q ) \biggl [ 6\alpha (1-\alpha ) +
6\alpha (1-\alpha )\beta_\Gamma^1
 {Q^2\over \cutoff^2} +\nonumber\\
&& 6\alpha (1-\alpha )\beta_V^1
{Q^2\over \cutoff^2} +\beta_M^3 {M_Q^2\over \cutoff^2} P_3 (\alpha
)\biggr ] .  \lab{QRPAR} \end{eqnarray}

In the calculation of $f_V$ and $Z_V$ we have not included NTLP
corrections.  They give a correction which is almost $1\%$ of the
leading logarithmic term.  Genuine power corrections can be generated by
the quadratically or more divergent part of diagrams that contain higher
dimensional vertices which are suppressed by inverse powers of the UV
cutoff $\cutoff$.

NTLP corrections to the squared vector mass and proportional to $M_Q$
can be reabsorbed in the renormalization of the vector coupling $G_V$ as
follows:

\beq {\cutoff^2\over \tilde{G}_V} = {\cutoff^2\over G_V} \biggl [ 1+
\delta^\prime {M_Q^2\over \cutoff^2}\biggr ] .
\eeq

In eq. (\ref{QRPAR}) the dependence upon the Feynman parameter $\alpha$
is always of the form $\alpha (1-\alpha )$ for the $Q^2/\cutoff^2$
corrections.  The polynomials $P_i(\alpha)$ are explicitely calculable
for each NPLL counterterm.  We will put them equal to the leading
polynomial $\alpha (1-\alpha)$ which is still a good approximation. With
this assumption the three vector parameters $Z_V$, $M_V^2$ and $f_V^2$
at $Q^2=0$ are given by:

\begin{eqnarray} Z_V(0)&=& {N_c\over 16\pi^2} {1\over 3} \Gamma
(0,M_Q^2/\cutoff^2 ) \biggl [ 1 +{\beta_M^1\over 6} {M_Q^2\over
\cutoff^2} \biggr ] \nonumber\\
M_V^2(0)&=&{N_c\over 16\pi^2}
{\cutoff^2\over 2{\tilde{G}_V}} {1\over Z_V(0)} \biggl [ 1+
{\beta_M^2\over 6} {M_Q^2\over \cutoff^2} \Gamma (0,M_Q^2/\cutoff^2 )
\biggr ]
   \nonumber\\
f_V^2(0)&=&{2\over 9}{1\over Z_V(0)} \biggl ({N_c\over
16\pi^2}\biggr )^2 \Gamma^2 (0, M_Q^2/\cutoff^2 )\biggl [ 1+{
\beta_M^3\over 6} {M_Q^2\over \cutoff^2} \biggr ]^2 .  \end{eqnarray}

The NPLL corrections proportional to $M_Q^2$ do modify the ENJL leading
prediction of the parameters of the effective meson Lagrangian at zero
energy.  One way to estimate the new coefficients $\beta_M^i$, is to do
the best fit of the whole set of the leading $(M_Q, \cutoff ,g_A)$ and
NPLL parameters (with and without gluonic corrections) at $Q^2=0$ using
as inputs the experimental values of the low energy meson parameters;
although a sufficiently accurate determination of the $\beta_M^i$ is
still not accessible with the present uncertainty on the very low energy
experimental data.

The ratio $f_V^2/M_V^2$ and the derivative of the squared coupling at
$Q^2=0$, which enter in eq. (\ref{DEFI}), are given by the following
expressions and retaining up to $1/\cutoff^2$ terms:

\begin{eqnarray}
{f_V^2(0)\over M_V^2(0)}&=&{2\over 3} {N_c\over 16\pi^2} {1\over
{M_V^2(0)}^{ENJL}} \Gamma (0, M_Q^2/\cutoff^2 )\biggl [ 1+
 { \beta_M^3\over 3}{M_Q^2\over \cutoff^2} -{\beta_M^2\over
6}{M_Q^2\over \cutoff^2}\Gamma (0, M_Q^2/\cutoff^2 ) \biggr ]
\nonumber\\ f_V^{2\prime}(0)&=&{2\over 3}{N_c\over 16\pi^2}\biggl \{
\Gamma (0, M_Q^2/\cutoff^2 ){2\beta_\Gamma^1\over \cutoff^2} -{1\over
5M_Q^2} e^{-{M_Q^2/\cutoff^2}}
 \biggl [ 1+ {2\beta_M^3 -\beta_M^1\over 6}{M_Q^2\over \cutoff^2} \biggr
]\biggr \} .
\lab{23}
\end{eqnarray}

Again the approximation $\Gamma (0,\alpha_Q) = -\ln\alpha_Q -\gamma_E$
and $exp(-M_Q^2/ \cutoff^2)=1$ is understood and we have used the
relation of eq. (\ref{RELA}) which defines ${M_V^2(0)}^{ENJL}$. We
assume its numerical value given by the fit 1 of ref.  \cite{ENJL} which
is $(0.811)^2~GeV^2$.

The size of the $Q^2$ corrections has been determined by the best fit of
ref. \cite{QR}:

\beq
\beta_\Gamma^1 = -0.75\pm 0.01~~~~~~~~~~~\beta_V^1=-0.79\pm 0.01.
\eeq

The sign of $\beta_\Gamma^1, \beta_V^1$ is negative and increases the
value of $a^h_\mu$. By including only $Q^2$ type corrections the value
of $a^h_\mu$ increases to $1.22\cdot 10^{-7}$.  Comparable values of the
$\beta_M^i$ coefficients give as a maximum range of variation of
$a_\mu^h$ $1.20\cdot 10^{-7} \div 1.24\cdot 10^{-7}$. This proves that
$Q^2$ dependent contributions give the bulk of the NPLL corrections to
$a_\mu^h$ calculated within the first derivative approximation.

Gluonic corrections can be parametrized following ref. \cite{2point}.
The leading contribution in the $1/N_c$ expansion involves only one
unknown parameter $g$ which is related to the lowest dimensional gluon
vacuum condensate:

\beq
g = {\pi^2\over 6N_c M_Q^4} <{\alpha_s\over \pi} GG> .
\lab{GLU}
\eeq

Because $<{\alpha_s\over \pi} GG>$ is $O(N_c)$ $g$ is $O(1)$ and the
leading gluonic correction to the two-point vector function is still
$O(N_c)$.  The role of gluonic corrections in effective low energy
fermion models is still an unsolved theoretical problem. What is clear
is that the two gluon condensate of eq. (\ref{GLU}) is only the low
energy ($<\cutoff$) `residue' of the standard two gluon condensate which
is phenomenologically estimated through QCD sum rules. The latter
suggest an $O(1)$ $g$ parameter, while best fits of the ENJL parameters,
using as inputs the experimental values of $f_\pi , L_i$ and the meson
resonances' parameters \cite{ENJL}, strongly favour a value of $g\le
0.5$.  The first phenomenological estimation, usually referred to as the
``standard value'', has been obtained by Shifman et al. (SVZ) \cite{SVZ}
by studying the charmonium channel and using Operator Product Expansion:
they obtain $<{\alpha_s\over \pi} GG> = 0.012~GeV^4$ (i.e. g=1.3).  A
compatible value $<{\alpha_s} GG>= (3.9\pm 1.0) 10^{-2}~GeV^4$ has been
obtained \cite{Launer} from the $e^+e^-\to I=1$ hadron cross section and
using moment sum rules ratio.  The most recent estimation via FESR
(Finite Energy sum rules) \cite{FESR} gives a significative higher value
$<{\alpha_s\over \pi} GG> = 0.044^{+ 0.015}_{-0.021}~GeV^4$ (i.e. $g\sim
4.8$), although the error in all cases has to be conservatively taken
around $40\%$.

In what follows we will use three values of g which are acceptable in
the ENJL model. The IR cutoff $M_Q$, the UV cutoff $\cutoff$ and the
axial-pseudoscalar mixing parameter $g_A$, extracted from the best fits
of experimental data, vary as functions of g as it is summarized in
table (\ref{GDIP}).  $M_Q$ decreases sensitively by increasing g.

Leading gluonic corrections can be expressed as an additive contribution
to the function $\overline{\Pi}_V^1(Q^2)$ which is written for $N_c=3$
as \cite{Narison}

\beq
\overline{\Pi}_V^{1g}(Q^2)= {3gM_Q^4\over 2\pi^2 Q^4} (-1 + 3 {\cal
I}_2 -2{\cal I}_3 ),
\eeq

\noindent in terms of the $Q^2$ dependent functions

\beq
{\cal I}_N = \int_0^1~d\alpha~
{1\over [ 1+ {Q^2\over M_Q^2} \alpha (1-\alpha )]^N} .
\eeq

\noindent Expanding at small $Q^2$ we obtain their values at $Q^2=0$:

\beq
\overline{\Pi}_V^{1g}(0)= -{3g\over 20\pi^2}
{}~~~~~~~~~\overline{\Pi}_V^{\prime 1g}(0)= {6g\over 70\pi^2M_Q^2} ,
\eeq

\noindent which give the following expression for the derivative:

\begin{eqnarray}
{d\Pi_V^1\over dQ^2}(0) &=& {d{\Pi_V^1}^ {g=0}\over
dQ^2}(0) +\Delta^g \nonumber\\
\Delta^g&=& {6g\over 70\pi^2M_Q^2} -
\biggl ({3g\over 20\pi^2}\biggr )^2 {8\pi^2 G_V\over N_c\cutoff^2}
\biggl ( 1- {40\pi^2\over 3g} {\Pi_V^1}^ {g=0}(0)\biggr ) .
\lab{DGLUON}
\end{eqnarray}

Notice that ${\Pi_V^1}^{g=0}(0) = {\overline{\Pi}_V^1}^{ g=0}(0)$; all
the quantities with superscript $g=0$ are those at g=0 with the
parameters $M_Q$, $\cutoff$ and $g_A$ rescaled according to table
(\ref{GDIP}) for a given g in formula (\ref{DGLUON}).  In table
(\ref{GDERT}) we give the gluonic corrections to the derivative at
$Q^2=0$ with g = 0.25, 0.5.

Gluonic contributions decrease $a_\mu^h$ towards a better agreement with
the phenomenological estimates (\ref{PHEN}).

The evaluation of the full dispersive integral (\ref{INTE}) requires the
knowledge of the long distance (ld) plus the short distance (sd)
behaviour of $\Pi_R^h(Q^2)$. We can do our best performing the matching
between the ld prediction coming from the effective theory and the sd
prediction coming from perturbative QCD.  In ref. \cite{RAF} a value of
$a_\mu^h(Fig. 2a)=6.7\cdot 10^{-8}$ has been obtained in the ENJL
framework with a best fitted matching point $\hat{x}\simeq 0.91$ which
corresponds to an euclidean
$\hat{Q}^2=\hat{x}^2/(1-\hat{x})m_\mu^2\simeq (320)^2~MeV^2$.  The value
obtained for $a_\mu^h$ is quite better than the first approximation
value $8.66\cdot 10^{-8}$.

To see how ``extra'' corrections modify the ENJL prediction is
sufficient to study the integral (\ref{INTE}) over the ld part in the
range $\int_0^{\hat{x}}$ for different values of $\hat{x}$ corresponding
to an equivalent value of $\hat{Q}^2 = (0.3)^2,(0.5)^2, (0.8)^2~GeV^2$.

NPLL corrections proportional to $Q^2$ in the QR model lead to a
 vector two-point function which decreases faster in $Q^2$ then the ENJL
prediction.  The dispersive integral gives for $a_\mu^h$ the values of
table (\ref{QRTOT}).

The correction induced by higher order terms proportional to $Q^2$ and
which are relevant in the intermediate $Q^2$ region is about one
percent.  This proves that $a_\mu^h$ is practically only sensitive to
perturbative corrections which modify the very low $Q^2$ region, i.e.
$Q^2\leq (500MeV)^2$.

Gluonic corrections modify the ENJL vector two-point function for all
$Q^2$ as shown in figure 3. They modify the ENJL prediction of $a_\mu^h$
as summarized in table (\ref{GTOT}).  The corrections are $10\%$ for
g=0.5 and $15\%$ for g=0.25. They decrease $a_\mu^h$. A better
determination of non-gluonic contributions to $a_\mu^h$ can be used to
constrain the value of the g parameter in low energy effective fermion
models.

We conclude that the ENJL prediction is reliable within $30\%$.  Both
gluonic corrections and next-to-leading higher dimensional
quark-resonance interactions have to be taken into account if one wants
to reach an accuracy better than $30\%$.  The first derivative
approximation is not sufficiently accurate to estimate the hadronic
vacuum polarization contribution to $a_\mu^h$, although the quantity is
sensitive only to ``extra'' corrections in the low $Q^2$ region ($Q <
500$ MeV).  NPLL corrections proportional to $Q^2$ do not affect
$a_\mu^h$.  NPLL corrections proportional to $M_Q^2$ and gluonic
corrections are relevant in the low $Q^2$ region.  Their inclusion can
explain the full agreement of the ENJL prediction with the
phenomenological estimates (\ref{PHEN}) of $a_\mu^h$.

\vspace{3cm}
{\bf Acknowledgements}. I am grateful to Eduardo de Rafael for having
called my attention to this problem and for useful discussions.
\vfill\eject

\centerline {{\bf FIGURE CAPTIONS}}
%\vskip2.truecm

\begin{description}

\item[1)] Hadronic vacuum polarization diagram to the anomalous
magnetic moment of the muon.

\item[2a)] The vector two-point function in the ENJL model and leading
in the $1/N_c$ expansion. It is given by the infinite resummation of
linear chains of constituent quark bubbles; each four-quark vertex is
the leading vertex of the ENJL model with coupling $G_V$.

\item[2b)] Chiral loops corrections to the vector two-point function.
They are of O(1) (i.e. next-to-leading) in the $1/N_c$ expansion and in
the ENJL model they are generated by the infinite resummation of loops
of chains of constituent quark bubbles.

\item[2c)] The vector two-point function in the QR model and leading in
the $1/N_c$ expansion. It is given by the ``local'' diagram and the
vector-exchange diagram with the renormalized vector meson propagator.
 The diagram of figure 2a is a part of the contribution to the renormalized
vector meson propagator.

\item[3)] The vector two-point function $\Pi_V^1(Q^2)$ in the ENJL model
without gluonic corrections (solid line) and with gluonic corrections
for g=0.25, 0.5 (dashed lines).

\end{description}

\vskip 2.truecm

\centerline {{\bf TABLE CAPTIONS}}
%\vskip2.truecm

\begin{description}

\item[1)] $1/\cutoff^2$ quark-resonance vertices of the sectors: I)
derivative, II) vector, III) scalar and IV) scalar-vector which give
contribution to the NPLL corrections.

\item[2)] Values of $M_Q$, $\cutoff$ and $g_A$ of the ENJL model
obtained from the best fits of ref. \cite{ENJL} to the experimental data
of low energy parameters and with fixed values of the gluonic paramater
g=0, 0.25, 0.5.

\item[3)] Gluonic corrections to $a_\mu^h$ in the first derivative
approximation for two values of the gluonic parameter $g$ favoured by
the ENJL model.  $\Delta^g$ is the gluonic correction to the first
derivative as defined in eq. (\ref{DGLUON}), $d\Pi_V^1/dQ^2(0)$ is the
first derivative including gluonic corrections. In the last two columns
the numerical value of $a_\mu^h$ and the variation in percentage are
shown.

\item[4)] Numerical values of $a_\mu^h$ obtained through the dispersion
relation (\ref{INTE}), where the integral is performed on the
long-distance part of $\Pi_R^h$ predicted by the QR model in the range
$0<Q^2<\hat{Q}^2$ and compared to the ENJL prediction.

\item[5)] Numerical values of $a_\mu^h$ obtained through the dispersion
relation (\ref{INTE}), where the integral is performed on the
long-distance part of $\Pi_R^h$ predicted by the ENJL model in the range
$0<Q^2<\hat{Q}^2$ without the inclusion of gluonic corrections (g=0) and
including gluonic corrections with g=0.25, 0.5.


\end{description}
\vfill\eject
\newpage

\begin{thebibliography}{99}

\bibitem{QR} E. Pallante and R. Petronzio, Preprint ROM2F 37/93,
submitted to Nucl. Phys.  B

\bibitem{RAF}
E. de Rafael, {\em Phys. Lett. } {\bf B322} (1994) 239.

\bibitem{Kinoshita} T. Kinoshita, B. Ni$\check{z}$i\'c and Y. Okamoto,
{\em Phys. Rev.} {\bf D31}
 (1985) 2108.

\bibitem{Casas} J.A. Casas, C. Lopez and F.J. Yndur\'ain, {\em Phys.
Rev.} {\bf D32} (1985) 736.

\bibitem{Kurdadze}
L.M. Kurdadze et al., {\em Sov. J. Nucl. Phys.} {\bf 40} (1984) 286.

\bibitem{RAFOLD} B.E. Lautrup, A. Peterman and E. de Rafael, {\em Phys.
Rep.} {\bf 3C} (1972) 193.

\bibitem{2point} J. Bijnens, E. de Rafael and H. Zheng, Preprint CERN-TH
6924/93, CTP-93/P2917, NORDITA-93/43 N,P.

\bibitem{ENJL} J. Bijnens, C. Bruno and E. de Rafael, {\em Nucl. Phys.}
{\bf B390} (1993) 501.

\bibitem{SVZ}

M.A. Shifman, A.I. Vainshtein and V.I. Zakharov,
{\em Nucl. Phys.} {\bf B147} (1979) 385, 448.

\bibitem{Launer}

G. Launer, S. Narison and R. Tarrach, {\em Z. Phys.} {\bf C 26} (1984)
433.

\bibitem{FESR}

R.A. Bertlmann et al., {\em Z. Phys.} {\bf C 39} (1988) 231.


\bibitem{Narison}

S. Narison, ``QCD Spectral Sum Rules'', World Scientific Lecture Notes
in Physics, Vol. 26.

\end{thebibliography}
\vfill
\newpage


\begin{table}
%\centering
\begin{tabular}{l}
DERIVATIVE \\
\\
$\beta_\Gamma^1~\bar{Q}\gamma_\nu d^\mu \Gamma_{\mu\nu} Q
+\beta_\Gamma^2~\bar{Q}\gamma_\mu \{\de^\mu ,\de^2 \} Q $  \\
\\
VECTOR \\
\\
$\beta_V^1 \bar{Q}\gamma_\mu d^2 W_{\mu}^+ Q+
\beta_V^2\bar{Q} \gamma_\mu \{ W_\mu^+ ,\de^2\} Q+
\beta_V^3 \bar{Q}\gamma_\mu \{\{\de_\mu ,\de_\nu\} ,W_{\nu}^+\} Q+
\beta_V^4 \bar{Q}\gamma_\mu  [ \Gamma_{\mu\nu}, W_\nu^+] Q +$ \\
$\beta_V^5 \bar{Q}\gamma_\mu\{ W^{+2}, \de^\mu \} Q +
\beta_V^6 \bar{Q}\gamma_\mu [ d^\mu W^{+}_\nu, W_\nu^+ ]Q+
\beta_V^7 \bar{Q}\gamma_\mu (W^{+}_\mu W^{+}_\nu \de_\nu + \de_\nu
W^{+}_\nu W^{+}_\mu )Q +   $          \\
$\beta_V^8 \bar{Q}\gamma_\mu (W^{+}_\nu W^{+}_\mu \de_\nu + \de_\nu
W^{+}_\mu W^{+}_\nu )Q +
\beta_V^9 \bar{Q}\gamma_\mu [ d^\nu W^{+}_\mu, W_\nu^+ ]Q $\\
            \\
 \\
SCALAR        \\
\\
$\beta_S^1~ \bar{Q}H^3 Q+ \beta_S^2~\bar{Q}\gamma_\mu
\{ H^2, \de^\mu\} Q +
\beta_S^3  ~\bar{Q}\{ H, \de^2 \} Q$ \\
\\
SCALAR-VECTOR            \\
\\
$\beta_{SV}^1 \bar{Q}\{W^{+2},H\} Q
+\beta_{SV}^2 \bar{Q}W_\mu^+ H W_\mu^+ Q
+ \beta_{SV}^3\bar{Q} \gamma_\mu\{ H^2, W_\mu^+\} Q+
\beta_{SV}^4\bar{Q} \gamma_\mu HW_\mu^+ H Q $ \\
$+\beta_{SV}^5 \bar{Q} (W_\mu^+ H \de_\mu + \de_\mu H W_\mu^+)Q
+\beta_{SV}^6 \bar{Q} (HW_\mu^+  \de_\mu + \de_\mu  W_\mu^+H)Q$.
             \\
\end{tabular}
\capt[1]{\lab{TERMS}}
\end{table}
\vskip 3cm

The covariant derivative of the vector field $W_\mu^+$ and the scalar
field $H$ is defined as $d_\mu {\cal O} = \partial_\mu {\cal O}
 + [\Gamma_\mu ,{\cal O}]$, while the covariant derivative on the {\em
constituent} quark field $Q$ is defined as $d_\mu Q = \partial_\mu Q+
\Gamma_\mu Q$.  $\Gamma_\mu = 1/2 (\xi^\dagger \partial_\mu\xi +
\xi\partial_\mu\xi^\dagger )$
 is the vector current constructed with the square root of the
pseudoscalar meson field $U=\xi^2=exp(i\phi/f_\pi )$.  The identity
$\Gamma_{\mu\nu}= -{i\over 2}f_{\mu\nu}^+ +{1\over 4} [\xi_\mu , \xi_\nu
]$ introduces the field strenght of the electromagnetic field
$f_{\mu\nu}^+= \xi F_{\mu\nu}\xi^\dagger +\xi^\dagger F_{\mu\nu}\xi$
which generates the photon-vector interaction associated with the
coupling $f_V$.

\vfill \newpage

\begin{table}
\centering
\begin{tabular}{|c|c|c|c|}\hline
  & & & \\
$g$ & $M_Q$ (MeV)& $\cutoff$ (GeV) & $g_A$\\ \hline
    & & &\\
$g=0$    & 265& 1.165 & 0.61\\
    & && \\
$g=0.25$    & 246& 1.062 & 0.62\\
    & & &\\
$g=0.5$    & 204& 1.090 &0.66\\
    & & \\ \hline
\end{tabular}
\capt[1]{\lab{GDIP}}
\end{table}
\vspace{2cm}

\begin{table}
\centering
\begin{tabular}{|c|c|c|c|c|}\hline
  & & & & \\
$g$ & $\Delta^g$  & ${d\Pi_V^1/dQ^2}(0)$   & $a_\mu^h$  &
$\Delta a_\mu/a_\mu (\% )$ \\ \hline
    & & & & \\
$g=0.25$    & 0.048   & -0.136  &$7.2\cdot 10^{-8}$    & $-17\%$
\\
    & & & & \\
$g=0.5$    & 0.134   &-0.133   & $7.0\cdot 10^{-8}$   &  $-19\%$
\\
    & & & & \\
\hline
\end{tabular}
\capt[1]{\lab{GDERT}}
\end{table}
%\vfill
%\newpage
\vspace{2cm}

\begin{table}
\centering
\begin{tabular}{|c|c|c|}\hline
  & &  \\
$\hat{Q}^2$ $GeV^2$&
$a_\mu^{QR}\times 10^{8}$ & $a_\mu^{ENJL}\times 10^{8}$ \\
\hline
     & &\\
$(0.5)^2$ & 6.9 & 6.8\\
    & &\\
$(0.8)^2$ & 7.3 & 7.1 \\
&  & \\ \hline
\end{tabular}
\capt[1]{\lab{QRTOT}}
\end{table}
\vspace{2cm}

\begin{table}
\centering
\begin{tabular}{|c|c|c|c|}\hline
  & & & \\ $\hat{Q}^2$ $GeV^2$& $a_\mu^h$ g=0 & $a_\mu^h$ g=0.25
&$a_\mu^h$ g=0.5 \\
\hline & & & \\
$(0.3)^2$ & $5.9\cdot 10^{-8}$ &
$5.0\cdot 10^{-8}$ & $5.3\cdot 10^{-8}$ \\ & & & \\ $(0.5)^2$ &
$6.8\cdot 10^{-8}$ & $5.9\cdot 10^{-8}$ & $6.3\cdot 10^{-8}$
\\ & & & \\
$(0.8)^2$ & $7.1\cdot 10^{-8}$ & $6.2\cdot 10^{-8}$ & $6.6\cdot 10^{-8}$
\\ & & & \\ \hline
\end{tabular}
\capt[1]{\lab{GTOT}}
\end{table}

\vfill\eject
\newpage
\end{document}

%%% figure 3 - postscript file
%%%
 initgraphics
 0.072 0.072 scale  %units are mils
     0 rotate
   250  1500 translate
 /vsml /Helvetica findfont  97 scalefont def
 /smal /Helvetica findfont 139 scalefont def
 /larg /Helvetica findfont 194 scalefont def
 /vlrg /Helvetica findfont 250 scalefont def
1
     9 setlinewidth
  4020  2763 moveto
  4004  2758 lineto
  3994  2748 lineto
  3989  2738 lineto
  3984  2717 lineto
  3984  2702 lineto
  3989  2681 lineto
  3994  2671 lineto
  4004  2661 lineto
  4020  2656 lineto
  4030  2656 lineto
  4045  2661 lineto
  4056  2671 lineto
  4061  2681 lineto
  4066  2702 lineto
  4066  2717 lineto
  4061  2738 lineto
  4056  2748 lineto
  4045  2758 lineto
  4030  2763 lineto
  4020  2763 lineto
  4020  2763 moveto
  4009  2758 lineto
  3999  2748 lineto
  3994  2738 lineto
  3989  2717 lineto
  3989  2702 lineto
  3994  2681 lineto
  3999  2671 lineto
  4009  2661 lineto
  4020  2656 lineto
  4030  2656 moveto
  4040  2661 lineto
  4051  2671 lineto
  4056  2681 lineto
  4061  2702 lineto
  4061  2717 lineto
  4056  2738 lineto
  4051  2748 lineto
  4040  2758 lineto
  4030  2763 lineto
  4004  2666 moveto
  4004  2671 lineto
  4009  2681 lineto
  4020  2686 lineto
  4025  2686 lineto
  4035  2681 lineto
  4040  2671 lineto
  4045  2635 lineto
  4051  2630 lineto
  4061  2630 lineto
  4066  2640 lineto
  4066  2645 lineto
  4040  2671 moveto
  4045  2651 lineto
  4051  2640 lineto
  4056  2635 lineto
  4061  2635 lineto
  4066  2640 lineto
  4220  2784 moveto
  4210  2774 lineto
  4199  2758 lineto
  4189  2738 lineto
  4184  2712 lineto
  4184  2692 lineto
  4189  2666 lineto
  4199  2645 lineto
  4210  2630 lineto
  4220  2620 lineto
  4210  2774 moveto
  4199  2753 lineto
  4194  2738 lineto
  4189  2712 lineto
  4189  2692 lineto
  4194  2666 lineto
  4199  2651 lineto
  4210  2630 lineto
  4323  2748 moveto
  4328  2733 lineto
  4328  2763 lineto
  4323  2748 lineto
  4312  2758 lineto
  4297  2763 lineto
  4287  2763 lineto
  4271  2758 lineto
  4261  2748 lineto
  4256  2738 lineto
  4251  2722 lineto
  4251  2697 lineto
  4256  2681 lineto
  4261  2671 lineto
  4271  2661 lineto
  4287  2656 lineto
  4297  2656 lineto
  4312  2661 lineto
  4323  2671 lineto
  4287  2763 moveto
  4276  2758 lineto
  4266  2748 lineto
  4261  2738 lineto
  4256  2722 lineto
  4256  2697 lineto
  4261  2681 lineto
  4266  2671 lineto
  4276  2661 lineto
  4287  2656 lineto
  4323  2697 moveto
  4323  2656 lineto
  4328  2697 moveto
  4328  2656 lineto
  4307  2697 moveto
  4343  2697 lineto
  4374  2697 moveto
  4435  2697 lineto
  4435  2707 lineto
  4430  2717 lineto
  4425  2722 lineto
  4415  2728 lineto
  4400  2728 lineto
  4384  2722 lineto
  4374  2712 lineto
  4369  2697 lineto
  4369  2686 lineto
  4374  2671 lineto
  4384  2661 lineto
  4400  2656 lineto
  4410  2656 lineto
  4425  2661 lineto
  4435  2671 lineto
  4430  2697 moveto
  4430  2712 lineto
  4425  2722 lineto
  4400  2728 moveto
  4389  2722 lineto
  4379  2712 lineto
  4374  2697 lineto
  4374  2686 lineto
  4379  2671 lineto
  4389  2661 lineto
  4400  2656 lineto
  4466  2763 moveto
  4502  2656 lineto
  4471  2763 moveto
  4502  2671 lineto
  4538  2763 moveto
  4502  2656 lineto
  4456  2763 moveto
  4487  2763 lineto
  4518  2763 moveto
  4548  2763 lineto
 stroke
     9 setlinewidth
  4569  2784 moveto
  4579  2774 lineto
  4589  2758 lineto
  4600  2738 lineto
  4605  2712 lineto
  4605  2692 lineto
  4600  2666 lineto
  4589  2645 lineto
  4579  2630 lineto
  4569  2620 lineto
  4579  2774 moveto
  4589  2753 lineto
  4595  2738 lineto
  4600  2712 lineto
  4600  2692 lineto
  4595  2666 lineto
  4589  2651 lineto
  4579  2630 lineto
   498  4825 moveto
   605  4825 lineto
   498  4830 moveto
   605  4830 lineto
   498  4892 moveto
   605  4892 lineto
   498  4897 moveto
   605  4897 lineto
   498  4810 moveto
   498  4912 lineto
   605  4810 moveto
   605  4846 lineto
   605  4876 moveto
   605  4912 lineto
   477  4979 moveto
   487  4969 lineto
   503  4959 lineto
   523  4948 lineto
   549  4943 lineto
   569  4943 lineto
   595  4948 lineto
   616  4959 lineto
   631  4969 lineto
   641  4979 lineto
   487  4969 moveto
   508  4959 lineto
   523  4953 lineto
   549  4948 lineto
   569  4948 lineto
   595  4953 lineto
   610  4959 lineto
   631  4969 lineto
   498  5046 moveto
   503  5030 lineto
   513  5020 lineto
   523  5015 lineto
   544  5010 lineto
   559  5010 lineto
   580  5015 lineto
   590  5020 lineto
   600  5030 lineto
   605  5046 lineto
   605  5056 lineto
   600  5071 lineto
   590  5082 lineto
   580  5087 lineto
   559  5092 lineto
   544  5092 lineto
   523  5087 lineto
   513  5082 lineto
   503  5071 lineto
   498  5056 lineto
   498  5046 lineto
   498  5046 moveto
   503  5036 lineto
   513  5025 lineto
   523  5020 lineto
   544  5015 lineto
   559  5015 lineto
   580  5020 lineto
   590  5025 lineto
   600  5036 lineto
   605  5046 lineto
   605  5056 moveto
   600  5066 lineto
   590  5077 lineto
   580  5082 lineto
   559  5087 lineto
   544  5087 lineto
   523  5082 lineto
   513  5077 lineto
   503  5066 lineto
   498  5056 lineto
   595  5030 moveto
   590  5030 lineto
   580  5036 lineto
   575  5046 lineto
   575  5051 lineto
   580  5061 lineto
   590  5066 lineto
   626  5071 lineto
   631  5077 lineto
   631  5087 lineto
   621  5092 lineto
   616  5092 lineto
   590  5066 moveto
   610  5071 lineto
   621  5077 lineto
   626  5082 lineto
   626  5087 lineto
   621  5092 lineto
   468  5128 moveto
   472  5131 lineto
   475  5128 lineto
   472  5124 lineto
   468  5124 lineto
   462  5128 lineto
   458  5131 lineto
   455  5142 lineto
   455  5155 lineto
   458  5166 lineto
   462  5169 lineto
   468  5172 lineto
   475  5172 lineto
   482  5169 lineto
   489  5159 lineto
   496  5142 lineto
   499  5135 lineto
   506  5128 lineto
   516  5124 lineto
   527  5124 lineto
   455  5155 moveto
   458  5162 lineto
   462  5166 lineto
   468  5169 lineto
   475  5169 lineto
   482  5166 lineto
   489  5155 lineto
   496  5142 lineto
   520  5124 moveto
   516  5128 lineto
   516  5135 lineto
   523  5152 lineto
   523  5162 lineto
   520  5169 lineto
   516  5172 lineto
   516  5135 moveto
   527  5152 lineto
   527  5166 lineto
   523  5169 lineto
   516  5172 lineto
   510  5172 lineto
 stroke
     9 setlinewidth
   477  5198 moveto
   487  5208 lineto
   503  5219 lineto
   523  5229 lineto
   549  5234 lineto
   569  5234 lineto
   595  5229 lineto
   616  5219 lineto
   631  5208 lineto
   641  5198 lineto
   487  5208 moveto
   508  5219 lineto
   523  5224 lineto
   549  5229 lineto
   569  5229 lineto
   595  5224 lineto
   610  5219 lineto
   631  5208 lineto
  1600  7384 moveto
  1600  3077 lineto
  1600  3077 moveto
  7199  3077 lineto
  7199  3077 moveto
  7199  7384 lineto
  7199  7384 moveto
  1600  7384 lineto
  1600  3077 moveto
  1600  3077 moveto
  1600  3261 lineto
  1803  3077 moveto
  1803  3138 lineto
  2007  3077 moveto
  2007  3138 lineto
  2211  3077 moveto
  2211  3138 lineto
  2414  3077 moveto
  2414  3138 lineto
  2618  3077 moveto
  2618  3261 lineto
  2821  3077 moveto
  2821  3138 lineto
  3025  3077 moveto
  3025  3138 lineto
  3229  3077 moveto
  3229  3138 lineto
  3432  3077 moveto
  3432  3138 lineto
  3636  3077 moveto
  3636  3261 lineto
  3840  3077 moveto
  3840  3138 lineto
  4043  3077 moveto
  4043  3138 lineto
  4247  3077 moveto
  4247  3138 lineto
  4450  3077 moveto
  4450  3138 lineto
  4654  3077 moveto
  4654  3261 lineto
  4858  3077 moveto
  4858  3138 lineto
  5061  3077 moveto
  5061  3138 lineto
  5265  3077 moveto
  5265  3138 lineto
  5468  3077 moveto
  5468  3138 lineto
  5672  3077 moveto
  5672  3261 lineto
  5876  3077 moveto
  5876  3138 lineto
  6079  3077 moveto
  6079  3138 lineto
  6283  3077 moveto
  6283  3138 lineto
  6486  3077 moveto
  6486  3138 lineto
  6690  3077 moveto
  6690  3261 lineto
  6894  3077 moveto
  6894  3138 lineto
  7097  3077 moveto
  7097  3138 lineto
  1585  2985 moveto
  1569  2980 lineto
  1559  2965 lineto
  1554  2939 lineto
  1554  2924 lineto
  1559  2898 lineto
  1569  2883 lineto
  1585  2878 lineto
  1595  2878 lineto
  1610  2883 lineto
  1621  2898 lineto
  1626  2924 lineto
  1626  2939 lineto
  1621  2965 lineto
  1610  2980 lineto
  1595  2985 lineto
  1585  2985 lineto
  1585  2985 moveto
  1574  2980 lineto
  1569  2975 lineto
  1564  2965 lineto
  1559  2939 lineto
  1559  2924 lineto
  1564  2898 lineto
  1569  2888 lineto
  1574  2883 lineto
  1585  2878 lineto
  1595  2878 moveto
  1605  2883 lineto
  1610  2888 lineto
  1615  2898 lineto
  1621  2924 lineto
  1621  2939 lineto
  1615  2965 lineto
  1610  2975 lineto
  1605  2980 lineto
  1595  2985 lineto
  2492  2985 moveto
  2476  2980 lineto
  2466  2965 lineto
  2461  2939 lineto
  2461  2924 lineto
  2466  2898 lineto
  2476  2883 lineto
  2492  2878 lineto
  2502  2878 lineto
  2517  2883 lineto
  2528  2898 lineto
  2533  2924 lineto
  2533  2939 lineto
  2528  2965 lineto
  2517  2980 lineto
  2502  2985 lineto
  2492  2985 lineto
  2492  2985 moveto
  2481  2980 lineto
  2476  2975 lineto
  2471  2965 lineto
  2466  2939 lineto
  2466  2924 lineto
  2471  2898 lineto
  2476  2888 lineto
  2481  2883 lineto
  2492  2878 lineto
  2502  2878 moveto
  2512  2883 lineto
  2517  2888 lineto
 stroke
     9 setlinewidth
  2517  2888 moveto
  2522  2898 lineto
  2528  2924 lineto
  2528  2939 lineto
  2522  2965 lineto
  2517  2975 lineto
  2512  2980 lineto
  2502  2985 lineto
  2574  2888 moveto
  2569  2883 lineto
  2574  2878 lineto
  2579  2883 lineto
  2574  2888 lineto
  2620  2965 moveto
  2625  2960 lineto
  2620  2955 lineto
  2615  2960 lineto
  2615  2965 lineto
  2620  2975 lineto
  2625  2980 lineto
  2640  2985 lineto
  2661  2985 lineto
  2676  2980 lineto
  2682  2975 lineto
  2687  2965 lineto
  2687  2955 lineto
  2682  2944 lineto
  2666  2934 lineto
  2640  2924 lineto
  2630  2919 lineto
  2620  2908 lineto
  2615  2893 lineto
  2615  2878 lineto
  2661  2985 moveto
  2671  2980 lineto
  2676  2975 lineto
  2682  2965 lineto
  2682  2955 lineto
  2676  2944 lineto
  2661  2934 lineto
  2640  2924 lineto
  2615  2888 moveto
  2620  2893 lineto
  2630  2893 lineto
  2656  2883 lineto
  2671  2883 lineto
  2682  2888 lineto
  2687  2893 lineto
  2630  2893 moveto
  2656  2878 lineto
  2676  2878 lineto
  2682  2883 lineto
  2687  2893 lineto
  2687  2903 lineto
  3510  2985 moveto
  3494  2980 lineto
  3484  2965 lineto
  3479  2939 lineto
  3479  2924 lineto
  3484  2898 lineto
  3494  2883 lineto
  3510  2878 lineto
  3520  2878 lineto
  3535  2883 lineto
  3546  2898 lineto
  3551  2924 lineto
  3551  2939 lineto
  3546  2965 lineto
  3535  2980 lineto
  3520  2985 lineto
  3510  2985 lineto
  3510  2985 moveto
  3499  2980 lineto
  3494  2975 lineto
  3489  2965 lineto
  3484  2939 lineto
  3484  2924 lineto
  3489  2898 lineto
  3494  2888 lineto
  3499  2883 lineto
  3510  2878 lineto
  3520  2878 moveto
  3530  2883 lineto
  3535  2888 lineto
  3540  2898 lineto
  3546  2924 lineto
  3546  2939 lineto
  3540  2965 lineto
  3535  2975 lineto
  3530  2980 lineto
  3520  2985 lineto
  3592  2888 moveto
  3587  2883 lineto
  3592  2878 lineto
  3597  2883 lineto
  3592  2888 lineto
  3679  2975 moveto
  3679  2878 lineto
  3684  2985 moveto
  3684  2878 lineto
  3684  2985 moveto
  3628  2908 lineto
  3710  2908 lineto
  3664  2878 moveto
  3700  2878 lineto
  4528  2985 moveto
  4512  2980 lineto
  4502  2965 lineto
  4497  2939 lineto
  4497  2924 lineto
  4502  2898 lineto
  4512  2883 lineto
  4528  2878 lineto
  4538  2878 lineto
  4553  2883 lineto
  4564  2898 lineto
  4569  2924 lineto
  4569  2939 lineto
  4564  2965 lineto
  4553  2980 lineto
  4538  2985 lineto
  4528  2985 lineto
  4528  2985 moveto
  4517  2980 lineto
  4512  2975 lineto
  4507  2965 lineto
  4502  2939 lineto
  4502  2924 lineto
  4507  2898 lineto
  4512  2888 lineto
  4517  2883 lineto
  4528  2878 lineto
  4538  2878 moveto
  4548  2883 lineto
  4553  2888 lineto
  4558  2898 lineto
  4564  2924 lineto
  4564  2939 lineto
  4558  2965 lineto
  4553  2975 lineto
  4548  2980 lineto
  4538  2985 lineto
  4610  2888 moveto
  4605  2883 lineto
  4610  2878 lineto
  4615  2883 lineto
  4610  2888 lineto
  4712  2970 moveto
  4707  2965 lineto
  4712  2960 lineto
 stroke
     9 setlinewidth
  4712  2960 moveto
  4718  2965 lineto
  4718  2970 lineto
  4712  2980 lineto
  4702  2985 lineto
  4687  2985 lineto
  4671  2980 lineto
  4661  2970 lineto
  4656  2960 lineto
  4651  2939 lineto
  4651  2908 lineto
  4656  2893 lineto
  4666  2883 lineto
  4682  2878 lineto
  4692  2878 lineto
  4707  2883 lineto
  4718  2893 lineto
  4723  2908 lineto
  4723  2914 lineto
  4718  2929 lineto
  4707  2939 lineto
  4692  2944 lineto
  4687  2944 lineto
  4671  2939 lineto
  4661  2929 lineto
  4656  2914 lineto
  4687  2985 moveto
  4676  2980 lineto
  4666  2970 lineto
  4661  2960 lineto
  4656  2939 lineto
  4656  2908 lineto
  4661  2893 lineto
  4671  2883 lineto
  4682  2878 lineto
  4692  2878 moveto
  4702  2883 lineto
  4712  2893 lineto
  4718  2908 lineto
  4718  2914 lineto
  4712  2929 lineto
  4702  2939 lineto
  4692  2944 lineto
  5546  2985 moveto
  5530  2980 lineto
  5520  2965 lineto
  5515  2939 lineto
  5515  2924 lineto
  5520  2898 lineto
  5530  2883 lineto
  5546  2878 lineto
  5556  2878 lineto
  5571  2883 lineto
  5582  2898 lineto
  5587  2924 lineto
  5587  2939 lineto
  5582  2965 lineto
  5571  2980 lineto
  5556  2985 lineto
  5546  2985 lineto
  5546  2985 moveto
  5535  2980 lineto
  5530  2975 lineto
  5525  2965 lineto
  5520  2939 lineto
  5520  2924 lineto
  5525  2898 lineto
  5530  2888 lineto
  5535  2883 lineto
  5546  2878 lineto
  5556  2878 moveto
  5566  2883 lineto
  5571  2888 lineto
  5576  2898 lineto
  5582  2924 lineto
  5582  2939 lineto
  5576  2965 lineto
  5571  2975 lineto
  5566  2980 lineto
  5556  2985 lineto
  5628  2888 moveto
  5623  2883 lineto
  5628  2878 lineto
  5633  2883 lineto
  5628  2888 lineto
  5694  2985 moveto
  5679  2980 lineto
  5674  2970 lineto
  5674  2955 lineto
  5679  2944 lineto
  5694  2939 lineto
  5715  2939 lineto
  5730  2944 lineto
  5736  2955 lineto
  5736  2970 lineto
  5730  2980 lineto
  5715  2985 lineto
  5694  2985 lineto
  5694  2985 moveto
  5684  2980 lineto
  5679  2970 lineto
  5679  2955 lineto
  5684  2944 lineto
  5694  2939 lineto
  5715  2939 moveto
  5725  2944 lineto
  5730  2955 lineto
  5730  2970 lineto
  5725  2980 lineto
  5715  2985 lineto
  5694  2939 moveto
  5679  2934 lineto
  5674  2929 lineto
  5669  2919 lineto
  5669  2898 lineto
  5674  2888 lineto
  5679  2883 lineto
  5694  2878 lineto
  5715  2878 lineto
  5730  2883 lineto
  5736  2888 lineto
  5741  2898 lineto
  5741  2919 lineto
  5736  2929 lineto
  5730  2934 lineto
  5715  2939 lineto
  5694  2939 moveto
  5684  2934 lineto
  5679  2929 lineto
  5674  2919 lineto
  5674  2898 lineto
  5679  2888 lineto
  5684  2883 lineto
  5694  2878 lineto
  5715  2878 moveto
  5725  2883 lineto
  5730  2888 lineto
  5736  2898 lineto
  5736  2919 lineto
  5730  2929 lineto
  5725  2934 lineto
  5715  2939 lineto
  6659  2965 moveto
  6669  2970 lineto
  6685  2985 lineto
  6685  2878 lineto
  6680  2980 moveto
  6680  2878 lineto
  6659  2878 moveto
  6705  2878 lineto
 stroke
     9 setlinewidth
  1600  7199 moveto
  1600  7384 lineto
  1803  7322 moveto
  1803  7384 lineto
  2007  7322 moveto
  2007  7384 lineto
  2211  7322 moveto
  2211  7384 lineto
  2414  7322 moveto
  2414  7384 lineto
  2618  7199 moveto
  2618  7384 lineto
  2821  7322 moveto
  2821  7384 lineto
  3025  7322 moveto
  3025  7384 lineto
  3229  7322 moveto
  3229  7384 lineto
  3432  7322 moveto
  3432  7384 lineto
  3636  7199 moveto
  3636  7384 lineto
  3840  7322 moveto
  3840  7384 lineto
  4043  7322 moveto
  4043  7384 lineto
  4247  7322 moveto
  4247  7384 lineto
  4450  7322 moveto
  4450  7384 lineto
  4654  7199 moveto
  4654  7384 lineto
  4858  7322 moveto
  4858  7384 lineto
  5061  7322 moveto
  5061  7384 lineto
  5265  7322 moveto
  5265  7384 lineto
  5468  7322 moveto
  5468  7384 lineto
  5672  7199 moveto
  5672  7384 lineto
  5876  7322 moveto
  5876  7384 lineto
  6079  7322 moveto
  6079  7384 lineto
  6283  7322 moveto
  6283  7384 lineto
  6486  7322 moveto
  6486  7384 lineto
  6690  7199 moveto
  6690  7384 lineto
  6894  7322 moveto
  6894  7384 lineto
  7097  7322 moveto
  7097  7384 lineto
  1600  3220 moveto
  1661  3220 lineto
  1600  3507 moveto
  1661  3507 lineto
  1600  3794 moveto
  1784  3794 lineto
  1600  4082 moveto
  1661  4082 lineto
  1600  4369 moveto
  1661  4369 lineto
  1600  4656 moveto
  1661  4656 lineto
  1600  4943 moveto
  1661  4943 lineto
  1600  5230 moveto
  1784  5230 lineto
  1600  5517 moveto
  1661  5517 lineto
  1600  5804 moveto
  1661  5804 lineto
  1600  6092 moveto
  1661  6092 lineto
  1600  6379 moveto
  1661  6379 lineto
  1600  6666 moveto
  1784  6666 lineto
  1600  6953 moveto
  1661  6953 lineto
  1600  7240 moveto
  1661  7240 lineto
  1142  3850 moveto
  1126  3845 lineto
  1116  3830 lineto
  1111  3804 lineto
  1111  3789 lineto
  1116  3763 lineto
  1126  3748 lineto
  1142  3743 lineto
  1152  3743 lineto
  1167  3748 lineto
  1178  3763 lineto
  1183  3789 lineto
  1183  3804 lineto
  1178  3830 lineto
  1167  3845 lineto
  1152  3850 lineto
  1142  3850 lineto
  1142  3850 moveto
  1131  3845 lineto
  1126  3840 lineto
  1121  3830 lineto
  1116  3804 lineto
  1116  3789 lineto
  1121  3763 lineto
  1126  3753 lineto
  1131  3748 lineto
  1142  3743 lineto
  1152  3743 moveto
  1162  3748 lineto
  1167  3753 lineto
  1172  3763 lineto
  1178  3789 lineto
  1178  3804 lineto
  1172  3830 lineto
  1167  3840 lineto
  1162  3845 lineto
  1152  3850 lineto
  1224  3753 moveto
  1219  3748 lineto
  1224  3743 lineto
  1229  3748 lineto
  1224  3753 lineto
  1296  3850 moveto
  1280  3845 lineto
  1270  3830 lineto
  1265  3804 lineto
  1265  3789 lineto
  1270  3763 lineto
  1280  3748 lineto
  1296  3743 lineto
  1306  3743 lineto
  1321  3748 lineto
  1332  3763 lineto
  1337  3789 lineto
  1337  3804 lineto
  1332  3830 lineto
  1321  3845 lineto
  1306  3850 lineto
  1296  3850 lineto
  1296  3850 moveto
  1285  3845 lineto
  1280  3840 lineto
  1275  3830 lineto
  1270  3804 lineto
 stroke
     9 setlinewidth
  1270  3804 moveto
  1270  3789 lineto
  1275  3763 lineto
  1280  3753 lineto
  1285  3748 lineto
  1296  3743 lineto
  1306  3743 moveto
  1316  3748 lineto
  1321  3753 lineto
  1326  3763 lineto
  1332  3789 lineto
  1332  3804 lineto
  1326  3830 lineto
  1321  3840 lineto
  1316  3845 lineto
  1306  3850 lineto
  1373  3830 moveto
  1378  3825 lineto
  1373  3820 lineto
  1367  3825 lineto
  1367  3830 lineto
  1373  3840 lineto
  1378  3845 lineto
  1393  3850 lineto
  1414  3850 lineto
  1429  3845 lineto
  1434  3840 lineto
  1439  3830 lineto
  1439  3820 lineto
  1434  3809 lineto
  1419  3799 lineto
  1393  3789 lineto
  1383  3784 lineto
  1373  3773 lineto
  1367  3758 lineto
  1367  3743 lineto
  1414  3850 moveto
  1424  3845 lineto
  1429  3840 lineto
  1434  3830 lineto
  1434  3820 lineto
  1429  3809 lineto
  1414  3799 lineto
  1393  3789 lineto
  1367  3753 moveto
  1373  3758 lineto
  1383  3758 lineto
  1409  3748 lineto
  1424  3748 lineto
  1434  3753 lineto
  1439  3758 lineto
  1383  3758 moveto
  1409  3743 lineto
  1429  3743 lineto
  1434  3748 lineto
  1439  3758 lineto
  1439  3768 lineto
  1142  5286 moveto
  1126  5281 lineto
  1116  5266 lineto
  1111  5240 lineto
  1111  5225 lineto
  1116  5199 lineto
  1126  5184 lineto
  1142  5179 lineto
  1152  5179 lineto
  1167  5184 lineto
  1178  5199 lineto
  1183  5225 lineto
  1183  5240 lineto
  1178  5266 lineto
  1167  5281 lineto
  1152  5286 lineto
  1142  5286 lineto
  1142  5286 moveto
  1131  5281 lineto
  1126  5276 lineto
  1121  5266 lineto
  1116  5240 lineto
  1116  5225 lineto
  1121  5199 lineto
  1126  5189 lineto
  1131  5184 lineto
  1142  5179 lineto
  1152  5179 moveto
  1162  5184 lineto
  1167  5189 lineto
  1172  5199 lineto
  1178  5225 lineto
  1178  5240 lineto
  1172  5266 lineto
  1167  5276 lineto
  1162  5281 lineto
  1152  5286 lineto
  1224  5189 moveto
  1219  5184 lineto
  1224  5179 lineto
  1229  5184 lineto
  1224  5189 lineto
  1296  5286 moveto
  1280  5281 lineto
  1270  5266 lineto
  1265  5240 lineto
  1265  5225 lineto
  1270  5199 lineto
  1280  5184 lineto
  1296  5179 lineto
  1306  5179 lineto
  1321  5184 lineto
  1332  5199 lineto
  1337  5225 lineto
  1337  5240 lineto
  1332  5266 lineto
  1321  5281 lineto
  1306  5286 lineto
  1296  5286 lineto
  1296  5286 moveto
  1285  5281 lineto
  1280  5276 lineto
  1275  5266 lineto
  1270  5240 lineto
  1270  5225 lineto
  1275  5199 lineto
  1280  5189 lineto
  1285  5184 lineto
  1296  5179 lineto
  1306  5179 moveto
  1316  5184 lineto
  1321  5189 lineto
  1326  5199 lineto
  1332  5225 lineto
  1332  5240 lineto
  1326  5266 lineto
  1321  5276 lineto
  1316  5281 lineto
  1306  5286 lineto
  1414  5276 moveto
  1414  5179 lineto
  1419  5286 moveto
  1419  5179 lineto
  1419  5286 moveto
  1362  5209 lineto
  1444  5209 lineto
  1398  5179 moveto
  1434  5179 lineto
  1142  6722 moveto
  1126  6717 lineto
  1116  6702 lineto
  1111  6676 lineto
  1111  6661 lineto
 stroke
     9 setlinewidth
  1111  6661 moveto
  1116  6635 lineto
  1126  6620 lineto
  1142  6615 lineto
  1152  6615 lineto
  1167  6620 lineto
  1178  6635 lineto
  1183  6661 lineto
  1183  6676 lineto
  1178  6702 lineto
  1167  6717 lineto
  1152  6722 lineto
  1142  6722 lineto
  1142  6722 moveto
  1131  6717 lineto
  1126  6712 lineto
  1121  6702 lineto
  1116  6676 lineto
  1116  6661 lineto
  1121  6635 lineto
  1126  6625 lineto
  1131  6620 lineto
  1142  6615 lineto
  1152  6615 moveto
  1162  6620 lineto
  1167  6625 lineto
  1172  6635 lineto
  1178  6661 lineto
  1178  6676 lineto
  1172  6702 lineto
  1167  6712 lineto
  1162  6717 lineto
  1152  6722 lineto
  1224  6625 moveto
  1219  6620 lineto
  1224  6615 lineto
  1229  6620 lineto
  1224  6625 lineto
  1296  6722 moveto
  1280  6717 lineto
  1270  6702 lineto
  1265  6676 lineto
  1265  6661 lineto
  1270  6635 lineto
  1280  6620 lineto
  1296  6615 lineto
  1306  6615 lineto
  1321  6620 lineto
  1332  6635 lineto
  1337  6661 lineto
  1337  6676 lineto
  1332  6702 lineto
  1321  6717 lineto
  1306  6722 lineto
  1296  6722 lineto
  1296  6722 moveto
  1285  6717 lineto
  1280  6712 lineto
  1275  6702 lineto
  1270  6676 lineto
  1270  6661 lineto
  1275  6635 lineto
  1280  6625 lineto
  1285  6620 lineto
  1296  6615 lineto
  1306  6615 moveto
  1316  6620 lineto
  1321  6625 lineto
  1326  6635 lineto
  1332  6661 lineto
  1332  6676 lineto
  1326  6702 lineto
  1321  6712 lineto
  1316  6717 lineto
  1306  6722 lineto
  1429  6707 moveto
  1424  6702 lineto
  1429  6697 lineto
  1434  6702 lineto
  1434  6707 lineto
  1429  6717 lineto
  1419  6722 lineto
  1403  6722 lineto
  1388  6717 lineto
  1378  6707 lineto
  1373  6697 lineto
  1367  6676 lineto
  1367  6645 lineto
  1373  6630 lineto
  1383  6620 lineto
  1398  6615 lineto
  1409  6615 lineto
  1424  6620 lineto
  1434  6630 lineto
  1439  6645 lineto
  1439  6651 lineto
  1434  6666 lineto
  1424  6676 lineto
  1409  6681 lineto
  1403  6681 lineto
  1388  6676 lineto
  1378  6666 lineto
  1373  6651 lineto
  1403  6722 moveto
  1393  6717 lineto
  1383  6707 lineto
  1378  6697 lineto
  1373  6676 lineto
  1373  6645 lineto
  1378  6630 lineto
  1388  6620 lineto
  1398  6615 lineto
  1409  6615 moveto
  1419  6620 lineto
  1429  6630 lineto
  1434  6645 lineto
  1434  6651 lineto
  1429  6666 lineto
  1419  6676 lineto
  1409  6681 lineto
  7138  3220 moveto
  7199  3220 lineto
  7138  3507 moveto
  7199  3507 lineto
  7015  3794 moveto
  7199  3794 lineto
  7138  4082 moveto
  7199  4082 lineto
  7138  4369 moveto
  7199  4369 lineto
  7138  4656 moveto
  7199  4656 lineto
  7138  4943 moveto
  7199  4943 lineto
  7015  5230 moveto
  7199  5230 lineto
  7138  5517 moveto
  7199  5517 lineto
  7138  5804 moveto
  7199  5804 lineto
  7138  6092 moveto
  7199  6092 lineto
  7138  6379 moveto
  7199  6379 lineto
  7015  6666 moveto
  7199  6666 lineto
  7138  6953 moveto
  7199  6953 lineto
  7138  7240 moveto
  7199  7240 lineto
 stroke
     9 setlinewidth
  2597  6902 moveto
  2587  6896 lineto
  2582  6891 lineto
  2577  6881 lineto
  2577  6871 lineto
  2582  6860 lineto
  2587  6855 lineto
  2597  6850 lineto
  2608  6850 lineto
  2618  6855 lineto
  2623  6860 lineto
  2628  6871 lineto
  2628  6881 lineto
  2623  6891 lineto
  2618  6896 lineto
  2608  6902 lineto
  2597  6902 lineto
  2587  6896 moveto
  2582  6886 lineto
  2582  6866 lineto
  2587  6855 lineto
  2618  6855 moveto
  2623  6866 lineto
  2623  6886 lineto
  2618  6896 lineto
  2623  6891 moveto
  2628  6896 lineto
  2639  6902 lineto
  2639  6896 lineto
  2628  6896 lineto
  2582  6860 moveto
  2577  6855 lineto
  2572  6845 lineto
  2572  6840 lineto
  2577  6830 lineto
  2592  6825 lineto
  2618  6825 lineto
  2633  6819 lineto
  2639  6814 lineto
  2572  6840 moveto
  2577  6835 lineto
  2592  6830 lineto
  2618  6830 lineto
  2633  6825 lineto
  2639  6814 lineto
  2639  6809 lineto
  2633  6799 lineto
  2618  6794 lineto
  2587  6794 lineto
  2572  6799 lineto
  2567  6809 lineto
  2567  6814 lineto
  2572  6825 lineto
  2587  6830 lineto
  2674  6891 moveto
  2767  6891 lineto
  2674  6860 moveto
  2767  6860 lineto
  2834  6937 moveto
  2818  6932 lineto
  2808  6917 lineto
  2803  6891 lineto
  2803  6876 lineto
  2808  6850 lineto
  2818  6835 lineto
  2834  6830 lineto
  2844  6830 lineto
  2859  6835 lineto
  2870  6850 lineto
  2875  6876 lineto
  2875  6891 lineto
  2870  6917 lineto
  2859  6932 lineto
  2844  6937 lineto
  2834  6937 lineto
  2834  6937 moveto
  2823  6932 lineto
  2818  6927 lineto
  2813  6917 lineto
  2808  6891 lineto
  2808  6876 lineto
  2813  6850 lineto
  2818  6840 lineto
  2823  6835 lineto
  2834  6830 lineto
  2844  6830 moveto
  2854  6835 lineto
  2859  6840 lineto
  2864  6850 lineto
  2870  6876 lineto
  2870  6891 lineto
  2864  6917 lineto
  2859  6927 lineto
  2854  6932 lineto
  2844  6937 lineto
  2916  6840 moveto
  2911  6835 lineto
  2916  6830 lineto
  2921  6835 lineto
  2916  6840 lineto
  2967  6937 moveto
  2957  6886 lineto
  2957  6886 moveto
  2967  6896 lineto
  2982  6902 lineto
  2998  6902 lineto
  3013  6896 lineto
  3023  6886 lineto
  3029  6871 lineto
  3029  6860 lineto
  3023  6845 lineto
  3013  6835 lineto
  2998  6830 lineto
  2982  6830 lineto
  2967  6835 lineto
  2962  6840 lineto
  2957  6850 lineto
  2957  6855 lineto
  2962  6860 lineto
  2967  6855 lineto
  2962  6850 lineto
  2998  6902 moveto
  3008  6896 lineto
  3018  6886 lineto
  3023  6871 lineto
  3023  6860 lineto
  3018  6845 lineto
  3008  6835 lineto
  2998  6830 lineto
  2967  6937 moveto
  3018  6937 lineto
  2967  6932 moveto
  2993  6932 lineto
  3018  6937 lineto
  1884  5251 moveto
  1874  5245 lineto
  1869  5240 lineto
  1864  5230 lineto
  1864  5220 lineto
  1869  5209 lineto
  1874  5204 lineto
  1884  5199 lineto
  1895  5199 lineto
  1905  5204 lineto
  1910  5209 lineto
  1915  5220 lineto
  1915  5230 lineto
  1910  5240 lineto
  1905  5245 lineto
  1895  5251 lineto
 stroke
     9 setlinewidth
  1895  5251 moveto
  1884  5251 lineto
  1874  5245 moveto
  1869  5235 lineto
  1869  5215 lineto
  1874  5204 lineto
  1905  5204 moveto
  1910  5215 lineto
  1910  5235 lineto
  1905  5245 lineto
  1910  5240 moveto
  1915  5245 lineto
  1926  5251 lineto
  1926  5245 lineto
  1915  5245 lineto
  1869  5209 moveto
  1864  5204 lineto
  1859  5194 lineto
  1859  5189 lineto
  1864  5179 lineto
  1879  5174 lineto
  1905  5174 lineto
  1920  5168 lineto
  1926  5163 lineto
  1859  5189 moveto
  1864  5184 lineto
  1879  5179 lineto
  1905  5179 lineto
  1920  5174 lineto
  1926  5163 lineto
  1926  5158 lineto
  1920  5148 lineto
  1905  5143 lineto
  1874  5143 lineto
  1859  5148 lineto
  1854  5158 lineto
  1854  5163 lineto
  1859  5174 lineto
  1874  5179 lineto
  1961  5240 moveto
  2054  5240 lineto
  1961  5209 moveto
  2054  5209 lineto
  2121  5286 moveto
  2105  5281 lineto
  2095  5266 lineto
  2090  5240 lineto
  2090  5225 lineto
  2095  5199 lineto
  2105  5184 lineto
  2121  5179 lineto
  2131  5179 lineto
  2146  5184 lineto
  2157  5199 lineto
  2162  5225 lineto
  2162  5240 lineto
  2157  5266 lineto
  2146  5281 lineto
  2131  5286 lineto
  2121  5286 lineto
  2121  5286 moveto
  2110  5281 lineto
  2105  5276 lineto
  2100  5266 lineto
  2095  5240 lineto
  2095  5225 lineto
  2100  5199 lineto
  2105  5189 lineto
  2110  5184 lineto
  2121  5179 lineto
  2131  5179 moveto
  2141  5184 lineto
  2146  5189 lineto
  2151  5199 lineto
  2157  5225 lineto
  2157  5240 lineto
  2151  5266 lineto
  2146  5276 lineto
  2141  5281 lineto
  2131  5286 lineto
  2203  5189 moveto
  2198  5184 lineto
  2203  5179 lineto
  2208  5184 lineto
  2203  5189 lineto
  2249  5266 moveto
  2254  5261 lineto
  2249  5256 lineto
  2244  5261 lineto
  2244  5266 lineto
  2249  5276 lineto
  2254  5281 lineto
  2269  5286 lineto
  2290  5286 lineto
  2305  5281 lineto
  2310  5276 lineto
  2316  5266 lineto
  2316  5256 lineto
  2310  5245 lineto
  2295  5235 lineto
  2269  5225 lineto
  2259  5220 lineto
  2249  5209 lineto
  2244  5194 lineto
  2244  5179 lineto
  2290  5286 moveto
  2300  5281 lineto
  2305  5276 lineto
  2310  5266 lineto
  2310  5256 lineto
  2305  5245 lineto
  2290  5235 lineto
  2269  5225 lineto
  2244  5189 moveto
  2249  5194 lineto
  2259  5194 lineto
  2285  5184 lineto
  2300  5184 lineto
  2310  5189 lineto
  2316  5194 lineto
  2259  5194 moveto
  2285  5179 lineto
  2305  5179 lineto
  2310  5184 lineto
  2316  5194 lineto
  2316  5204 lineto
  2357  5286 moveto
  2346  5235 lineto
  2346  5235 moveto
  2357  5245 lineto
  2372  5251 lineto
  2387  5251 lineto
  2403  5245 lineto
  2413  5235 lineto
  2418  5220 lineto
  2418  5209 lineto
  2413  5194 lineto
  2403  5184 lineto
  2387  5179 lineto
  2372  5179 lineto
  2357  5184 lineto
  2352  5189 lineto
  2346  5199 lineto
  2346  5204 lineto
  2352  5209 lineto
  2357  5204 lineto
  2352  5199 lineto
  2387  5251 moveto
  2398  5245 lineto
  2408  5235 lineto
 stroke
     9 setlinewidth
  2408  5235 moveto
  2413  5220 lineto
  2413  5209 lineto
  2408  5194 lineto
  2398  5184 lineto
  2387  5179 lineto
  2357  5286 moveto
  2408  5286 lineto
  2357  5281 moveto
  2382  5281 lineto
  2408  5286 lineto
  1600  6694 moveto
  1617  6694 lineto
  1634  6694 lineto
  1651  6693 lineto
  1668  6692 lineto
  1685  6691 lineto
  1702  6689 lineto
  1719  6688 lineto
  1736  6686 lineto
  1753  6684 lineto
  1770  6681 lineto
  1787  6678 lineto
  1804  6675 lineto
  1821  6672 lineto
  1838  6669 lineto
  1855  6665 lineto
  1872  6661 lineto
  1889  6657 lineto
  1906  6652 lineto
  1923  6647 lineto
  1940  6642 lineto
  1957  6637 lineto
  1974  6632 lineto
  1991  6626 lineto
  2008  6620 lineto
  2025  6614 lineto
  2042  6607 lineto
  2058  6601 lineto
  2075  6594 lineto
  2092  6587 lineto
  2109  6580 lineto
  2126  6572 lineto
  2143  6564 lineto
  2160  6556 lineto
  2177  6548 lineto
  2194  6540 lineto
  2211  6531 lineto
  2228  6522 lineto
  2245  6513 lineto
  2262  6504 lineto
  2279  6495 lineto
  2296  6485 lineto
  2313  6476 lineto
  2330  6466 lineto
  2347  6455 lineto
  2364  6445 lineto
  2381  6435 lineto
  2398  6424 lineto
  2415  6413 lineto
  2432  6402 lineto
  2449  6391 lineto
  2466  6380 lineto
  2483  6368 lineto
  2500  6357 lineto
  2517  6345 lineto
  2534  6333 lineto
  2551  6321 lineto
  2567  6309 lineto
  2584  6296 lineto
  2601  6284 lineto
  2618  6271 lineto
  2635  6259 lineto
  2652  6246 lineto
  2669  6233 lineto
  2686  6220 lineto
  2703  6207 lineto
  2720  6193 lineto
  2737  6180 lineto
  2754  6167 lineto
  2771  6153 lineto
  2788  6139 lineto
  2805  6125 lineto
  2822  6112 lineto
  2839  6098 lineto
  2856  6084 lineto
  2873  6069 lineto
  2890  6055 lineto
  2907  6041 lineto
  2924  6027 lineto
  2941  6012 lineto
  2958  5998 lineto
  2975  5983 lineto
  2992  5969 lineto
  3009  5954 lineto
  3026  5939 lineto
  3043  5925 lineto
  3060  5910 lineto
  3076  5895 lineto
  3093  5880 lineto
  3110  5865 lineto
  3127  5850 lineto
  3144  5835 lineto
  3161  5820 lineto
  3178  5805 lineto
  3195  5790 lineto
  3212  5775 lineto
  3229  5760 lineto
  3246  5744 lineto
  3263  5729 lineto
  3280  5714 lineto
  3297  5699 lineto
  3314  5684 lineto
  3331  5669 lineto
  3348  5653 lineto
  3365  5638 lineto
  3382  5623 lineto
  3399  5608 lineto
  3416  5592 lineto
  3433  5577 lineto
  3450  5562 lineto
  3467  5547 lineto
  3484  5532 lineto
  3501  5516 lineto
  3518  5501 lineto
  3535  5486 lineto
  3552  5471 lineto
  3569  5456 lineto
  3586  5441 lineto
  3602  5426 lineto
  3619  5410 lineto
  3636  5395 lineto
  3653  5380 lineto
  3670  5365 lineto
  3687  5350 lineto
  3704  5336 lineto
  3721  5321 lineto
  3738  5306 lineto
  3755  5291 lineto
  3772  5276 lineto
  3789  5261 lineto
  3806  5247 lineto
  3823  5232 lineto
  3840  5217 lineto
  3857  5203 lineto
  3874  5188 lineto
  3891  5174 lineto
  3908  5159 lineto
  3925  5145 lineto
  3942  5130 lineto
 stroke
     9 setlinewidth
  3942  5130 moveto
  3959  5116 lineto
  3976  5102 lineto
  3993  5087 lineto
  4010  5073 lineto
  4027  5059 lineto
  4044  5045 lineto
  4061  5031 lineto
  4078  5017 lineto
  4095  5003 lineto
  4111  4989 lineto
  4128  4975 lineto
  4145  4961 lineto
  4162  4947 lineto
  4179  4934 lineto
  4196  4920 lineto
  4213  4906 lineto
  4230  4893 lineto
  4247  4879 lineto
  4264  4866 lineto
  4281  4853 lineto
  4298  4839 lineto
  4315  4826 lineto
  4332  4813 lineto
  4349  4800 lineto
  4366  4786 lineto
  4383  4773 lineto
  4400  4760 lineto
  4417  4748 lineto
  4434  4735 lineto
  4451  4722 lineto
  4468  4709 lineto
  4485  4696 lineto
  4502  4684 lineto
  4519  4671 lineto
  4536  4659 lineto
  4553  4646 lineto
  4570  4634 lineto
  4587  4622 lineto
  4604  4609 lineto
  4621  4597 lineto
  4637  4585 lineto
  4654  4573 lineto
  4671  4561 lineto
  4688  4549 lineto
  4705  4537 lineto
  4722  4525 lineto
  4739  4513 lineto
  4756  4502 lineto
  4773  4490 lineto
  4790  4478 lineto
  4807  4467 lineto
  4824  4455 lineto
  4841  4444 lineto
  4858  4432 lineto
  4858  4432 moveto
  4875  4421 lineto
  4892  4410 lineto
  4909  4399 lineto
  4926  4388 lineto
  4943  4377 lineto
  4960  4365 lineto
  4977  4355 lineto
  4994  4344 lineto
  5011  4333 lineto
  5028  4322 lineto
  5045  4311 lineto
  5062  4301 lineto
  5079  4290 lineto
  5096  4279 lineto
  5113  4269 lineto
  5130  4259 lineto
  5147  4248 lineto
  5164  4238 lineto
  5180  4228 lineto
  5197  4217 lineto
  5214  4207 lineto
  5231  4197 lineto
  5248  4187 lineto
  5265  4177 lineto
  5282  4167 lineto
  5299  4157 lineto
  5316  4147 lineto
  5333  4138 lineto
  5350  4128 lineto
  5367  4118 lineto
  5384  4109 lineto
  5401  4099 lineto
  5418  4090 lineto
  5435  4080 lineto
  5452  4071 lineto
  5469  4062 lineto
  5486  4052 lineto
  5503  4043 lineto
  5520  4034 lineto
  5537  4025 lineto
  5554  4016 lineto
  5571  4007 lineto
  5588  3998 lineto
  5605  3989 lineto
  5622  3980 lineto
  5639  3971 lineto
  5656  3962 lineto
  5673  3954 lineto
  5689  3945 lineto
  5706  3936 lineto
  5723  3928 lineto
  5740  3919 lineto
  5757  3911 lineto
  5757  3911 moveto
  5774  3902 lineto
  5791  3894 lineto
  5808  3886 lineto
  5825  3877 lineto
  5842  3869 lineto
  5859  3861 lineto
  5876  3853 lineto
  5893  3845 lineto
  5910  3837 lineto
  5927  3829 lineto
  5944  3821 lineto
  5961  3813 lineto
  5978  3805 lineto
  5995  3797 lineto
  6012  3789 lineto
  6029  3781 lineto
  6046  3774 lineto
  6063  3766 lineto
  6080  3758 lineto
  6097  3751 lineto
  6114  3743 lineto
  6131  3736 lineto
  6148  3728 lineto
  6165  3721 lineto
  6182  3714 lineto
  6199  3706 lineto
  6215  3699 lineto
  6232  3692 lineto
  6249  3685 lineto
  6266  3677 lineto
  6283  3670 lineto
  6300  3663 lineto
  6317  3656 lineto
  6334  3649 lineto
  6351  3642 lineto
  6368  3635 lineto
  6385  3628 lineto
  6402  3622 lineto
  6419  3615 lineto
  6436  3608 lineto
 stroke
     9 setlinewidth
  6436  3608 moveto
  6453  3601 lineto
  6470  3595 lineto
  6487  3588 lineto
  6504  3581 lineto
  6521  3575 lineto
  6538  3568 lineto
  6555  3562 lineto
  6572  3555 lineto
  6589  3549 lineto
  6606  3542 lineto
  6623  3536 lineto
  6640  3530 lineto
  6657  3523 lineto
  6674  3517 lineto
  6691  3511 lineto
  1600  6862 moveto
  1617  6861 lineto
  1634  6860 lineto
  1651  6858 lineto
  1668  6857 lineto
  1685  6857 lineto
  1702  6856 lineto
  1719  6854 lineto
  1733  6853 lineto
  1865  6833 moveto
  1872  6832 lineto
  1889  6828 lineto
  1906  6825 lineto
  1923  6821 lineto
  1940  6816 lineto
  1957  6812 lineto
  1974  6807 lineto
  1991  6802 lineto
  1994  6801 lineto
  2119  6757 moveto
  2127  6754 lineto
  2143  6747 lineto
  2160  6740 lineto
  2177  6733 lineto
  2195  6725 lineto
  2212  6717 lineto
  2229  6709 lineto
  2240  6703 lineto
  2356  6639 moveto
  2365  6634 lineto
  2382  6624 lineto
  2400  6613 lineto
  2417  6602 lineto
  2433  6591 lineto
  2450  6580 lineto
  2466  6570 lineto
  2467  6569 lineto
  2573  6493 moveto
  2585  6483 lineto
  2602  6471 lineto
  2618  6458 lineto
  2637  6443 lineto
  2655  6429 lineto
  2673  6414 lineto
  2675  6413 lineto
  2772  6329 moveto
  2774  6328 lineto
  2791  6313 lineto
  2807  6298 lineto
  2824  6284 lineto
  2842  6267 lineto
  2860  6251 lineto
  2867  6244 lineto
  2959  6157 moveto
  2961  6155 lineto
  2980  6137 lineto
  2995  6122 lineto
  3011  6107 lineto
  3026  6092 lineto
  3046  6073 lineto
  3050  6069 lineto
  3139  5980 moveto
  3146  5973 lineto
  3162  5956 lineto
  3178  5940 lineto
  3196  5922 lineto
  3213  5904 lineto
  3227  5890 lineto
  3315  5799 moveto
  3319  5795 lineto
  3336  5778 lineto
  3353  5760 lineto
  3370  5742 lineto
  3388  5724 lineto
  3403  5709 lineto
  3491  5617 moveto
  3505  5602 lineto
  3521  5586 lineto
  3537  5569 lineto
  3554  5552 lineto
  3571  5534 lineto
  3579  5526 lineto
  3667  5436 moveto
  3683  5420 lineto
  3702  5400 lineto
  3709  5393 lineto
  3716  5386 lineto
  3723  5379 lineto
  3730  5372 lineto
  3737  5365 lineto
  3744  5358 lineto
  3751  5351 lineto
  3756  5347 lineto
  3845  5258 moveto
  3851  5252 lineto
  3859  5244 lineto
  3867  5236 lineto
  3875  5228 lineto
  3884  5219 lineto
  3892  5211 lineto
  3901  5203 lineto
  3909  5194 lineto
  3918  5186 lineto
  3927  5177 lineto
  3936  5169 lineto
  4027  5082 moveto
  4030  5079 lineto
  4040  5069 lineto
  4050  5060 lineto
  4060  5050 lineto
  4070  5040 lineto
  4081  5031 lineto
  4091  5021 lineto
  4102  5011 lineto
  4113  5001 lineto
  4120  4995 lineto
  4214  4909 moveto
  4216  4907 lineto
  4228  4896 lineto
  4240  4885 lineto
  4253  4874 lineto
  4265  4862 lineto
  4278  4851 lineto
  4291  4840 lineto
  4304  4828 lineto
  4309  4824 lineto
  4372  4769 moveto
  4387  4757 lineto
  4401  4744 lineto
  4416  4732 lineto
  4431  4719 lineto
  4446  4707 lineto
  4461  4694 lineto
  4470  4686 lineto
 stroke
     9 setlinewidth
  4570  4605 moveto
  4575  4601 lineto
  4593  4587 lineto
  4610  4573 lineto
  4628  4559 lineto
  4647  4544 lineto
  4665  4530 lineto
  4672  4524 lineto
  4776  4446 moveto
  4783  4440 lineto
  4804  4425 lineto
  4825  4409 lineto
  4847  4393 lineto
  4869  4377 lineto
  4881  4369 lineto
  4988  4294 moveto
  5012  4277 lineto
  5037  4260 lineto
  5063  4243 lineto
  5090  4225 lineto
  5096  4221 lineto
  5206  4149 moveto
  5231  4133 lineto
  5262  4114 lineto
  5293  4095 lineto
  5318  4080 lineto
  5431  4012 moveto
  5462  3995 lineto
  5498  3974 lineto
  5534  3953 lineto
  5546  3947 lineto
  5613  3910 moveto
  5652  3889 lineto
  5695  3866 lineto
  5730  3848 lineto
  5848  3788 moveto
  5876  3774 lineto
  5923  3751 lineto
  5967  3730 lineto
  6087  3674 moveto
  6137  3652 lineto
  6195  3626 lineto
  6209  3620 lineto
  6331  3569 moveto
  6383  3547 lineto
  6448  3521 lineto
  6454  3518 lineto
  6578  3470 moveto
  6595  3464 lineto
  6669  3436 lineto
  6703  3424 lineto
  6829  3380 moveto
  6837  3377 lineto
  6921  3347 lineto
  1600  6356 moveto
  1651  6353 lineto
  1702  6350 lineto
  1733  6347 lineto
  1865  6328 moveto
  1906  6319 lineto
  1957  6307 lineto
  1994  6296 lineto
  2120  6254 moveto
  2160  6238 lineto
  2212  6216 lineto
  2243  6202 lineto
  2360  6144 moveto
  2362  6143 lineto
  2365  6142 lineto
  2367  6140 lineto
  2370  6139 lineto
  2372  6138 lineto
  2375  6136 lineto
  2377  6135 lineto
  2380  6133 lineto
  2382  6132 lineto
  2385  6131 lineto
  2388  6129 lineto
  2390  6128 lineto
  2393  6126 lineto
  2395  6125 lineto
  2398  6123 lineto
  2401  6122 lineto
  2403  6121 lineto
  2406  6119 lineto
  2409  6118 lineto
  2412  6116 lineto
  2414  6114 lineto
  2417  6113 lineto
  2420  6111 lineto
  2422  6110 lineto
  2425  6108 lineto
  2428  6107 lineto
  2431  6105 lineto
  2434  6103 lineto
  2437  6102 lineto
  2439  6100 lineto
  2442  6098 lineto
  2445  6097 lineto
  2448  6095 lineto
  2451  6093 lineto
  2454  6092 lineto
  2457  6090 lineto
  2460  6088 lineto
  2463  6086 lineto
  2466  6085 lineto
  2469  6083 lineto
  2472  6081 lineto
  2474  6080 lineto
  2584  6010 moveto
  2587  6009 lineto
  2591  6006 lineto
  2595  6004 lineto
  2599  6001 lineto
  2603  5999 lineto
  2606  5996 lineto
  2610  5994 lineto
  2614  5991 lineto
  2618  5988 lineto
  2622  5986 lineto
  2626  5983 lineto
  2630  5980 lineto
  2635  5977 lineto
  2639  5974 lineto
  2643  5972 lineto
  2647  5969 lineto
  2651  5966 lineto
  2656  5963 lineto
  2660  5960 lineto
  2664  5957 lineto
  2669  5954 lineto
  2673  5951 lineto
  2677  5948 lineto
  2682  5945 lineto
  2686  5942 lineto
  2691  5938 lineto
  2692  5937 lineto
  2798  5862 moveto
  2801  5859 lineto
  2807  5855 lineto
  2812  5850 lineto
  2818  5846 lineto
  2824  5842 lineto
  2829  5838 lineto
  2835  5833 lineto
  2841  5829 lineto
  2847  5824 lineto
  2853  5820 lineto
  2859  5815 lineto
  2865  5810 lineto
 stroke
     9 setlinewidth
  2865  5810 moveto
  2871  5806 lineto
  2878  5801 lineto
  2884  5796 lineto
  2890  5791 lineto
  2897  5786 lineto
  2901  5783 lineto
  3002  5703 moveto
  3003  5702 lineto
  3010  5696 lineto
  3018  5690 lineto
  3026  5684 lineto
  3034  5677 lineto
  3042  5670 lineto
  3050  5664 lineto
  3059  5657 lineto
  3067  5650 lineto
  3076  5643 lineto
  3084  5636 lineto
  3084  5636 moveto
  3093  5629 lineto
  3102  5621 lineto
  3111  5614 lineto
  3120  5606 lineto
  3130  5599 lineto
  3139  5591 lineto
  3149  5583 lineto
  3158  5575 lineto
  3168  5567 lineto
  3178  5558 lineto
  3183  5554 lineto
  3282  5471 moveto
  3287  5467 lineto
  3299  5457 lineto
  3311  5447 lineto
  3323  5436 lineto
  3336  5425 lineto
  3348  5415 lineto
  3361  5404 lineto
  3374  5392 lineto
  3379  5388 lineto
  3477  5304 moveto
  3489  5294 lineto
  3505  5281 lineto
  3521  5267 lineto
  3537  5253 lineto
  3554  5239 lineto
  3571  5224 lineto
  3575  5221 lineto
  3673  5138 moveto
  3682  5130 lineto
  3702  5113 lineto
  3723  5095 lineto
  3723  5095 moveto
  3744  5078 lineto
  3766  5059 lineto
  3788  5041 lineto
  3811  5022 lineto
  3822  5013 lineto
  3921  4930 moveto
  3936  4919 lineto
  3963  4896 lineto
  3991  4874 lineto
  4020  4850 lineto
  4021  4849 lineto
  4123  4768 moveto
  4146  4750 lineto
  4180  4723 lineto
  4216  4696 lineto
  4225  4689 lineto
  4328  4610 moveto
  4331  4608 lineto
  4372  4577 lineto
  4416  4545 lineto
  4433  4533 lineto
  4539  4457 moveto
  4558  4443 lineto
  4610  4406 lineto
  4646  4382 lineto
  4754  4308 moveto
  4783  4289 lineto
  4847  4248 lineto
  4865  4237 lineto
  4976  4166 moveto
  4986  4160 lineto
  5063  4113 lineto
  5088  4098 lineto
  5202  4032 moveto
  5231  4015 lineto
  5317  3966 lineto
  5433  3904 moveto
  5534  3851 lineto
  5551  3842 lineto
  5669  3783 moveto
  5781  3729 lineto
  5789  3726 lineto
  5909  3670 moveto
  5923  3664 lineto
  6031  3617 lineto
  6153  3565 moveto
  6253  3524 lineto
  6276  3515 lineto
  6400  3467 moveto
  6448  3448 lineto
  6525  3420 lineto
  6650  3374 moveto
  6669  3367 lineto
  6776  3331 lineto
  6902  3288 moveto
  6921  3282 lineto
 stroke
 copypage
 erasepage

