%Paper: 
%From: RAJ@PHYS.TAMU.EDU
%Date: Tue, 25 Jan 1994 17:03:42 CST


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\begin{flushright}
CTP-TAMU-76/93\\
MRI-PHY-/12/93\\
LTH-326\\
\end{flushright}
\vglue 0.5cm
\begin{center}
{\Large\bf Two-Loop Neutrino Masses and the Solar Neutrino Problem\\}
\vglue 1cm
{\sc
Debajyoti Choudhury$^{(a)}$\footnote{debchou@iws166.mppmu.mpg.de,
  debchou@dmumpiwh.bitnet},
Raj Gandhi$^{(b),(c)}$\footnote{raj@tamphys.bitnet},
J.A. Gracey$^{(d)}$\footnote{jag@s-a.amtp.liv.ac.uk}
and Biswarup Mukhopadhyaya$^{(e)}$\footnote{biswarup@mri.ernet.in}
}
\vglue 0.4cm
{\em $^{(a)}$ Max-Planck-Inst. f\"{u}r Physik, Werner Heisenberg Inst.\\}
{\em F\"{o}hringer Ring 6, 80805 M\"{u}nchen, Germany\\}
{\em $^{(b)}$Center for Theoretical Physics, Department of Physics,
Texas A\&M University\\}
{\em College Station, TX 77843-4242, USA\\}
{\em $^{(c)}$Astroparticle Physics Group, Houston Advanced Research Center
(HARC)\\}
{\em The Woodlands, TX 77381, USA\\}
{\em $^{(d)}$ DAMTP, University of Liverpool,\\}
{\em P.O Box 147, Liverpool L69 3BX, U.K.\\}
{\em $^{(e)}$ Mehta Research Institute, 10 Kasturba Gandhi Marg,\\}
{\em Allahabad, 211002, India\\}
\vglue 1cm
{\tenrm \bf ABSTRACT}
\end{center}

\noindent The addition of $m$ singlet right-handed neutrinos to the Standard
Model leads to radiatively generated mass corrections for the $SU(2)_L $
doublet neutrinos. For those neutrinos which are massless at the tree level
after this addition, this implies a small mass generated at the two-loop
level via $W^{\pm}$ exchange. We calculate these mass corrections exactly
by obtaining an analytic form for the
general case of $n$ doublets and $m$ singlets. As a phenomenological
application, we consider the $m=1$ case and
 examine the masses and mixings of the doublet neutrinos
which arise as a result of the two-loop correction in the light of
 experimental data from two sources which may shed light on the question
of neutrino masses. These are
(a) the neutrino detectors reporting
a solar
neutrino deficit (and its resolution via Mikheyev-Smirnov-Wolfenstein matter
oscillations),
 and (b) the COBE satellite data on the non-zero
angular variations of the cosmic microwave background
temperature (and its possible implications for hot dark matter).
 Within the framework of the
extension
considered here, which leaves the gauge group structure of
the Standard Model intact, we  show that it is possible
for neutrinos to acquire  small masses naturally, with values
 which are compatible with
current theoretical bias and experimental data.
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\sect{Introduction}


It is fair to say that the problem of understanding the origin of
fermion masses is one of the most perplexing questions facing particle
physics today. The Standard model \cite{SM}
%   despite its spectacular successes,
can reproduce the observed fermion masses via electroweak
symmetry breaking and the Higgs mechanism, but provides no explanation
for their values.
%    are what they are.
 When such an understanding is obtained, one of the issues
that it must clarify is the smallness of neutrino masses (if, indeed, neutrinos
are massive) relative to those of the other fermions. An attractive
explanation for this observed feature of the fermion mass spectrum is the
see-saw mechanism \cite{ssaw}.
It postulates the existence of right-handed neutrinos with masses
of the order of the next energy threshold and uses this in
combination with the Higgs mechanism
to generate
light (Majorana) neutrino masses via an effective dimension five operator.

Given our present ignorance of the origins of mass and
the lack of experimental pointers towards any particular mechanism, it
is important to keep an open mind on the smallness of neutrino
mass. In this paper, we explore, via detailed calculation, the issue of
radiatively generated neutrino masses, since this is also a natural way
in which  masses small compared to those of other fermions may be generated.

Any such effort needs to invoke physics beyond the Standard model. In view of
the extraordinary and  demonstrated robustness of the model to
experimental tests over the last twenty years, we have thought it reasonable
to make the simplest possible extension to the standard theory and study
its effect on neutrino masses via radiative corrections,
 \ie the addition of $m$  $SU(2)_L \bigotimes U(1)_Y$ singlet right-handed
neutrinos.
 {\it A priori}, there is no connection
between their number and that of the doublet neutrinos, hence the simplest
case corresponds to $m=1$, \ie the addition of
one right-handed singlet neutrino
to the standard model \cite{JA}.

The gauge group structure of the weak sector remains unchanged as a
consequence of this extension, but  majorana mass terms incorporating the
scale of new physics are now allowed. We do not speculate on their origin, but
only note that it would require invoking an additional global symmetry
(such as a conserved lepton number) to set these
 to zero. The doublet neutrinos acquire radiative (and, in some cases,
tree-level) masses due to the presence of the singlets, as we discuss below.
The radiative masses arise (via mixing) due to a two-loop mechanism
\cite{PT,BM} involving the exchange
of $W^{\pm}$ bosons.
In Sections II and the Appendix, we calculate, exactly and in analytic form,
 the two-loop masses
 accquired by the initially massless doublet neutrinos.
Our calculation is general and valid for any number of doublet
and singlet neutrinos, but in order to obtain phenomenologically useful
information, we focus, in Section III, on the $m=1$  case.
Even this simplest extension
of the Standard model
introduces four new parameters into the theory. On the issue of neutrino
masses, it is non-accelerator
experiments that provide information on the cutting edge. Hence we have chosen
to examine
 the results for the $m=1$  case
  in the light of (a) the MSW {\cite{MSW}}
solution to the solar neutrino deficit seen by the  Kamiokande \cite{KII},
GALLEX \cite{GAL}, SAGE \cite{SA} and  Homestake \cite{Cl} neutrino detectors
and (b) the
implications for hot dark matter (neutrinos) from the recent COBE observations
on the anisotropy of the microwave background
\cite{COB}.
Invoking this experimental information restricts
the parameter space and consequently, in the context
considered in this paper, permits a handle on
the range of the mass scale
characterizing physics beyond the standard model. We show that doublet neutrino
masses compatible with both (a) and (b) above can result naturally from
such physics at the several hundred GeV scale.


\sect{Radiative Generation of Neutrino Masses }

In this section
we give a description of an exact general procedure for calculating
two-loop neutrino masses
 applicable
to any extension of the Standard model which incorporates singlet right-handed
neutrinos. (We remark below on the reason why a one-loop mass
does not arise in the sitiuation considered here, where {\it only}
right-handed handed neutrinos are added to the existing particle spectrum. )
After setting up the generic integral that needs to be
calculated we describe the procedure for evaluating it exactly in the Appendix.

The lepton sector of the extension considered here has, in general,
$n \:(\geq 3)$ doublet fields $[\nu'_{iL} \;l_{iL} ]^T $
 and $m$ singlet fields $(\nu'_{A L})^c = (\nu_A^{\prime c})_R
$. ( Here $i=1....n$, $A=1....m$ and $\nu^c \equiv C \bar\nu^T$ is the charge
conjugate spinor.)
In addition, one has the charged lepton $SU(2)_L$ singlet fields $l_{iR}$.
The primes on the neutrino fields denote weak
eigenstates as opposed to physical particle states.
Without any loss of generality,  we have  assumed that the
weak eigenstates $l_{i}$ are the same as the corresponding mass eigenstates,
\ie the charged lepton mass matrix is diagonal.

As noted in the Introduction,
in addition to the Dirac mass terms,
the most general Lagrangian consistent with
the gauge symmetry of the Standard model
 also contains possible Majorana mass terms for neutrinos
 of the form
$m_{AB} \overline{\left(\nu'_{A L}\right)^c} \nu'_{B L}$. In the minimal
model under consideration here,
such terms must
be bare mass terms, but in a more involved model they could arise,
for instance,
due to the  vacuum expectation value of a
singlet higgs.  To facilitate
discussion, we combine all
the left handed neutrinos into a $(n+m)$--dimensional vector in the flavour
space denoting it by $\nu'_{\alpha L}$.
The most general mass term is thus given by
\be
{\cal L}_{\rm m} = \sum_{i = 1}^n \mu_i \bar{l}_{iL} l_{iR}
    + \sum_{\alpha,\beta = 1}^{n+m}
          \overline{\left(\nu'_{\alpha L}\right)^c}
                   {\cal M}_{\alpha \beta} \nu'_{\beta L} + h.c
      \label{massinwk}
\ee
Here $\cal M$ is a complex
symmetric \footnote{That $\cal M$ has to be symmetric is evident from the
charge conjugation property of fermion bilinears.}
$(n+m) \times (n+m)$ matrix of the form
\be \displaystyle
{\cal M} = \pmatrix{0_{n \times n} & D_{n \times m} \cr
                        D^T_{m \times n} & M_{m \times m}\cr}
  \label{massmatrix}
\ee
with $D$ and $M$ denoting the Dirac and the Majorana  mass terms respectively.
The first block is identically zero in the absence of a non--trivial vacuum
expectation value for a $SU(2)_L$--triplet
higgs field. (This restriction is imposed not only by our philosophy of
minimal extension, but more importantly, by $m_W/m_Z$ --- the observed ratio
of the
gauge boson masses.) $\cal M$ can be diagonalized by
a biunitary transformation of the form
\be
V^T {\cal M} V =  \widehat{{\cal M}} = {\rm diagonal}(m_\alpha)
       \label{biunitary}
\ee
The mass eigenstates ( $\nu_\alpha$) are then easily identified to be
\be
\nu_L = V^\dagger \nu'_L
     \label{basischange}
\ee
The relevant piece of the  weak Lagrangian is then given by
\be
{\cal L}_{\rm wk} = J_\mu W^\mu
    \label{lagrangian}
\ee
where
\be
\barr{rcl}
J_\mu^+ & = & \displaystyle \frac{ig}{\sqrt{2}}
             \sum_{i = 1}^n \bar{l}_{i} \gamma_\mu P_L \nu'_i
 =  \displaystyle \frac{ig}{\sqrt{2}}
             \sum_{i = 1}^n \sum_{\alpha =1}^{n+m} K_{i\alpha}
             \bar{l}_{i} \gamma_\mu P_L \nu_i
     \\[2ex]
J_\mu^3 & = & \displaystyle \frac{ig}{2 c_W}
             \sum_{i = 1}^n \left(
                 \bar{l}_{i} \gamma_\mu P_L \l_i
                 + \bar{\nu'}_{i} \gamma_\mu P_L \nu'_i \right) \\[2ex]
& = & \displaystyle \frac{ig}{2 c_W} \left[
             \sum_{i = 1}^n  \bar{l}_{i} \gamma_\mu P_L \l_i
                 + \sum_{\alpha, \beta = 1}^{n+m}
                  (K^\dagger K)_{\alpha \beta}
                  \bar{\nu}_{\alpha} \gamma_\mu P_L \nu_\beta \right]
     \label{currents}
\earr
\ee
Here $c_W \equiv \cos \theta_W$, where $\theta_W$ is the Weinberg angle,
$g = e/\sin\theta_W$,
 $P_L \equiv (1 - \gamma_5)/2$ and
\be \displaystyle
 K = \pmatrix{I_{n \times n} & 0 \cr
              0 & 0\cr} V
    \label{CKM}
\ee
is the $(n+m)$--dimensional analog of the quark sector
Cabibbo--Kobayashi--Maskawa matrix. Note that though
$K K^\dagger = {\rm diag}(I_{n \times n},0)$,
$(K^\dagger K)_{\alpha \beta} \neq \delta_{\alpha \beta}$. Thus we do indeed
have flavour changing neutral currents (FCNC) in the neutrino sector.

Having set up the general formalism, let
 us now concentrate on the case where $n > m$.  There exist then $n - m$
neutrinos that are strictly massless at the tree level. We now
calculate the changes to such a spectrum accruing from
quantum corrections.



 Before proceeding,
in view of the  fact there exist massive neutrinos and also FCNC's in the
neutrino sector, it is appropriate at this point to remark on the possibility
of  one--loop graphs with a
$Z$ or a Higgs exchange  introducing a non--trivial correction
to the neutrino mass matrix. However, it can be easily seen that it is
possible to rotate the neutrino states such that only $m$ of them have
Yukawa couplings to the Higgs. Thus, only those doublet states that are
massive at tree level obtain a Higgs induced mass at the one-loop level.
In addition, since the flavor-changing $Z$ couplings have the same mixing
parameters as the flavor-changing Yukawa couplings, the one-loop
$Z$ exchange diagrams do not contribute to the masses of the $n-m$
neutrinos which are massless at the tree level. This reasoning
applies at all orders to any diagram where all virtual particles are
neutral.
Hence
the relevant diagram to compute is that given in Fig.(1).


We shall work in the weak interaction basis for the external neutrinos and the
mass basis for all the virtual particles. Furthermore, we shall concentrate
only on the first $n \times n$ block of ${\cal M}$, \ie on the
generation of Majorana mass terms for the doublet neutrinos.
In the unitary gauge,  the correction to the neutrino propagator
is then given by
\be
\barr{rcll}
i \Sigma^{(2)}_{ij} (p) &= &\displaystyle
                 \left(\frac{ig}{\sqrt{2}} \right)^4\;
                    \sum_{\alpha = 1}^{n + m}
                   \int
                 & \displaystyle \!\! \! \!
                   \frac{d^4 k}{(2 \pi)^4} \frac{d^4 q}{(2 \pi)^4}
                   \gamma_\mu P_R \frac{i}{\slp + \slq - \mu_i}
                   K^\dagger_{\alpha i} \gamma_\nu P_R
                        \\[1.5ex]
             &&& \hspace*{2ex} \displaystyle
   \frac{i}{\slp + \slq + \slk - m_\alpha}
                   K^\dagger_{\alpha j}
                   \gamma_\sigma P_L \frac{i}{\slp + \slk - \mu_j}
                   \gamma_\lambda P_L
                       \\[1.5ex]
             &&& \hspace*{2ex} \displaystyle
       \frac{ -i \left(g^{\mu \sigma} - q^\mu q^\sigma /m_W^2 \right)}
            {q^2 - m_W^2}\:
       \frac{ -i \left(g^{\nu \lambda} - k^\nu k^\lambda /m_W^2\right)}
            {k^2 - m_W^2}
\earr
     \label{2loopmass-1}
\ee
The mass correction is of course given by
${\cal M}^{(2)}_{ij} = \Sigma^{(2)}_{ij}(p = 0) $, and
after some algebra this leads to
\be
\barr{rcll}
{\cal M}^{(2)}_{ij} & = & \displaystyle
             \frac{g^4}{4}
                   \sum_{\alpha = 1}^{n + m} m_\alpha
                   K^\dagger_{\alpha j} K^\dagger_{\alpha i}
                   \int
                 & \displaystyle \! \! \! \!
                   \frac{d^4 k}{(2 \pi)^4} \frac{d^4 q}{(2 \pi)^4}
                     \frac{(k + q)^2\,k \cdot q }{ {\cal D}_{ij;\alpha}}
\\[2ex]
                 &&& \displaystyle
                   \left[ \left(4 + \frac{k^2 q^2}{m_W^4} \right)
                           - 4\,\frac{q^2 + k^2}{m_W^2} \right]
\earr
         \label{mass}
\ee
where
\be
{\cal D}_{ij;\alpha} = (k + q)^2 \left\{(k + q)^2 - m_\alpha^2 \right\}
                   (q^2 - \mu_i^2) (q^2 - m_W^2) (k^2 - m_W^2) (k^2 - \mu_j^2)
   \label{denom}
\ee
We see that the mass corrections would be identically zero if $m_\alpha =0,
\; \forall \alpha$. This ought to be so as any mass renormalization must be
proportional to the bare mass terms. The integral above has a naive degree of
 divergence of 4. However, note that
\be
\sum_{\alpha = 1}^{n + m} m_\alpha
                   K^\dagger_{\alpha j} K^\dagger_{\alpha i}
       = {\cal M}_{i j} = 0
\ee
and hence
\be
\sum_{\alpha = 1}^{n + m} m_\alpha K^\dagger_{\alpha i} K^\dagger_{\alpha j}
                  \frac{(k+q)^2}{(k+q)^2 - m_\alpha^2}
= \sum_{\alpha = 1}^{n + m}
                  \frac{ K^\dagger_{\alpha i} K^\dagger_{\alpha j} m_\alpha^3}
                       {(k+q)^2 - m_\alpha^2}
   \label{GIM-1}
\ee
This clearly is analogous to the GIM mechanism in the quark sector. Even on
substitution of eqn(\ref{GIM-1}) in eqn(\ref{mass}), the integral in the latter
is still formally divergent. Notice, however, that this is but
an artifact of the
unitary gauge and is not a real divergence \cite{div}.
In fact, by invoking identities
similar to eqn(\ref{GIM-1}) or equivalently, by working in the Feynman gauge,
one obtains \footnote{This has often been cited in the literature as a
GIM-like cancellation, but in our view the two are quite different.}
\be
{\cal M}^{(2)}_{ij}  =  \displaystyle
                   g^4
                   \sum_{\alpha = 1}^{n + m} m_\alpha^3
                   K^\dagger_{\alpha j} K^\dagger_{\alpha i}
                   \left[ 4 + 4 \frac{\mu_i^2 + \mu_j^2}{m_W^2}
                       + \frac{\mu_i^2 \mu_j^2}{m_W^4} \right]
                    \Lambda(\mu_i^2, m_W^2, m_\alpha^2,0,\mu_l^2, m_W^2 )
         \label{mass-final}
\ee
where
\begin{eqnarray}
&&  \Lambda(m_1^2,m_2^2,m_3^2,m_4^2,m_5^2,m_6^2) \equiv \nonumber \\
&&\!\!\!\!\!  \int \frac{d^4 k \;d^4 q\; k \cdot q}
               {   (q^2 + m_1^2) (q^2 + m_2^2)
                  \left\{(k + q)^2 + m_3^2\right\}
                   \left\{(k + q)^2 + m_4^2 \right\}
                   (k^2 + m_5^2) (k^2 + m_6^2)
               } \nonumber \\
    \label{Lambda-defn}
\end{eqnarray}
is an Euclidean integral evaluated in the Appendix.

The expression in eqn(\ref{mass-final}) thus represents the Majorana mass
generated for the doublet neutrino at the two--loop level. In operator
language, it arises from  terms of the form
\be
   \overline{(L_{iL})^c} L_{jL } \phi \phi S
       \label{operator}
\ee
where $L_{iL}$ represent the doublet lepton fields, $\phi$ is the usual
higgs field and $S$ represents the lepton number violating operator (whether
a singlet higgs or a bare mass term). We note that this five dimensional
effective operator for the radiative masses is the same as that for the
conventional see-saw mechanism. The difference between the two resides
in the scale of mass generation. Two-loop radiative masses compatible with the
solar and COBE data can arise from right-handed neutrinos at the several
hundred GeV scale, as we show below, whereas the see-saw mechanism generates
similar valued masses via heavy neutrinos at the  grand unified scale.

 We also note that though the corrections ostensibly are
proportional to $m_\alpha^3$ (eqn.(\ref{mass-final})),
 the actual dependence is linear (apart from logarithmic corrections) due to
suppressions hidden in $\Lambda$. As $m_\alpha$ becomes larger
and terms of the order of $(\mu_i/m_\alpha)^2$ become negligible, the
correction goes as $\Sigma_{\alpha} K^{\dagger}_{\alpha i} K^{\dagger}_{
\alpha j} m_{\alpha}$, which is simply the $(ij)^{\rm th}$
 element of the tree-level
mass matrix, and hence zero for the cases of interest here.

Finally, we remark
 that a complex ${\cal M}$ in eqn.(\ref{massinwk}) obviously leads
to a complex digonalizing matrix $V$ and hence possibly to $CP$--violating
processes. However, since there is no evidence as yet of any such
non-conservation in the leptonic sector, we have, in the interests of
 simplicity,
  chosen to perform all numerical calculations assuming a
real neutrino mass matrix.



\section{Application: The Solar Neutrino Deficit and COBE Data}

In order to make a connection to experiment and phenomenology, we
now
specialize to the $n=3$ and $m=1$ case and examine the two loop
mass corrections in the context of (a) the MSW solution \cite{MSW} to the solar
neutrino deficit reported by various detectors \cite{KII,GAL,SA,Cl} and
(b) recent COBE \cite{COB} data and its implications for neutrinos as dark
matter.

The solar deficit is the only long-standing possible evidence for physics
beyond the Standard model, and the MSW mechanism is its most popular
resolution. In its essence, the mechanism requires neutrinos to be
massive (and non-degenerate), allowing the interaction eigenstate
$\nu_e$ (assumed to comprise predominantly of the lightest mass eigenstate)
to oscillate to $\nu_{\mu}$ or $\nu_{\tau}$ due to the difference in the
forward scattering potential seen by the two states in their passage through
solar matter. It thus identifies a range of vacuum mixing angle and mass
squared difference  values which are compatible with the deficit observed
by the various detectors.
 Figure 2, excluding curves labelled (a), (b) and (c),
is taken from Ref. \cite{Hata} and shows the familiar two-flavor mixing
MSW solution space, where $\theta$ is the Cabibbo mixing angle and $\Delta m^2$
is the difference of the squares of the two neutrino masses, which, in the
present
context, are acquired at the two-loop level.

 COBE data
on the anisotropy of the microwave background, while not
making a definitive statement on the nature of dark matter, seem to suggest
that
it may have both hot and cold components, with the former
being a neutrino (since it is the only known hot dark matter candidate)
with a mass of $\approx 10 $ eV.

 We use both of the above considerations to restrict
the rather large parameter space available to us.

In the scenario with one additional singlet, we have two  massive and two
massless neutrinos at tree-level. The two massive ones acquire both
one-loop and two-loop corrections, which we neglect, and the massless
states acquire small masses at the two-loop level. The  two tree-level masses
 and all the radiative corrections are expressible in terms of four input mass
parameters for the matrix ${\cal M}$.
 For various plausible
 (fixed)  values
of $m_{\alpha}$, (the singlet mass, signifying the scale of new physics)
and the added constraint that the  other neutrino with a tree level mass
lie in the $10 $ eV range, we obtain a one parameter set of curves (see
Fig. $2$) which denotes the intersection of the "two-loop space" with
the MSW solution space. Note that restricting ourslves to
the two dimensional MSW space imposes
a third constraint, \ie that the $\nu_e$ mixes predominantly with
only one other state.
Curve (a) in Figure $2$ corresponds to a singlet mass
of $100 $ GeV and a $\nu_{\tau}$ mass of $\approx 8.6
$ eV. The two-loop masses  and mixings of $\nu_{e}$ and $\nu_{\mu}$
are then such that they span the MSW space as shown. Curve (b) corresponds
to a singlet mass of $ 400 $ GeV and a $ \nu_{\mu}$ mass of $\approx 7 $ eV.
$\nu_{\tau}$ and $\nu_{e}$ then acquire radiative masses
 and mixings that span the
solution space as shown. For sin$^22\theta$ greater than $\approx 3 \times
10^{-1}$,
$\nu_{\tau}$ becomes lighter than $\nu_{e}$, and MSW oscillations occur
between anti-neutrino rather than neutrino states, and are thus not relevant.
We note that (b) passes through the (small-angle, non-adiabatic)
 MSW region that is compatible with
all detectors and also represents a value of $m_{\nu_{\mu}}$ ($7$ eV)
that provides a very good fit to COBE data in the context of
a hot plus cold dark matter scenario.
 Finally, curve (c) represents a singlet mass of $1 $ TeV and a $\nu_{\mu}$
mass of $\approx 9.8 $ eV, and terminates where it does because for larger
mixing angles the $\nu_{e}$ becomes heavier than the $\nu_{\tau}$.
Note that the determination
of which flavor the $\nu_e$ oscillates to is made by examining the
mixing (diagonalizing) matrix of the full (\ie tree + loop ) mass matrix.
 A (reasonable) assumption
built into the results is that $\nu_e$ is the lightest state.

We stress that these curves represent a phenomenological
exercise more than anything else to demonstrate that our calculations can
 make connection with
experiment when the full parameter
space, which is quite large, is constrained by imposing physically and
empirically well-motivated restrictions.

We note that the singlet mass values chosen by us ($100$ GeV,
$400$ GeV and $1$ TeV) are
 not in conflict with accelerator \cite{RPP} or cosmological \cite{OL}
bounds on these particles.

Finally, we remark that a disparity between
the mass scales of the $\nu_{e}, \nu_{\mu} (\nu_{\tau})$ and that of
the $\nu_{\tau} (\nu_{\mu})$
seems to be required if we take both the solar and COBE implications
for neutrino masses seriously. In the simple model under consideration
here, such a disparity arises naturally since the neutrino which contributes
to dark matter has a tree level mass while the other two have loop masses.

\section{\bf Conclusions}

We have explicitly obtained an analytic
form for the radiative two-loop masses acquired by doublet neutrinos
in models where right-handed singlets are present. We have made an effort
to keep our calculation general and the expression for the mass correction
that we obtain may have applications in other models with right-handed
neutrinos. We have calculated
these masses (for the one singlet case) in the light of
experimental data from solar neutrino detectors
and from  COBE, within the confines of
the MSW solution to the solar deficit. By doing so we have made an effort
 to demonstrate
that intermediate scale physics  (\ie physics at $\leq 1$  TeV) can lead, in
a simple way, to naturally small masses for neutrinos which have physically
meaningful values, without requiring drastic changes in
the presently known particle spectrum
 or gauge group structure.



\appendix

\sect{\bf Appendix: Evaluation of $\Lambda_{123456}$}
In this section we discuss the exact evaluation of the fundamental finite
two loop four dimensional integral underlying the mechanism. As a first step,
though, we consider the more general two loop Euclidean space integral,
$\Lambda_{123456}$, defined by
\begin{equation}
\Lambda_{123456} =
\int_{p \, q} \frac{p \cdot q}{(p^2+m^2_1)(p^2+m^2_2)
((p+q)^2+m^2_3)((p+q)^2+m^2_4)(q^2+m^2_5)(q^2+m^2_6)} \nonumber \\
\end{equation}
which we will evaluate analytically and then specialize to the case we are
concerned with. For reasons which we explain below we choose to calculate eqn
(A.1) in $d$-dimensions where
\begin{equation}
\int_k ~=~ \frac{\mu^{4-d}}{(2\pi)^{d}} \int \, d^dk
\end{equation}
and $\mu$ is an arbitrary mass parameter introduced to ensure the coupling
constant remains dimensionless in our $d$-dimensional manipulations. The
subscripts on $\Lambda_{123456}$ correspond to the masses $m^2_i$ of the
integral and we note that the function has certain obvious symmetries,
$\Lambda_{123456}$ $=$ $\Lambda_{213456}$ $=$ $\Lambda_{563412}$, which ought
to be preserved in the final expression. The strategy to evaluate eqn (A.1)
is to use partial fractions to obtain a sum of $2$-loop integrals with three
propagators and then to substitute for the value of each of these
sub-integrals, which have been considered by other authors in different
contexts before, \cite{J0,J1,J2,J3}. For instance, if we define
\begin{equation}
J_{ijk} ~=~ \int_p \int_q \frac{p \cdot
q}{(p^2+m^2_i)(q^2+m^2_j)((p+q)^2+m^2_k)}
\end{equation}
then eqn (A.1) is built out of a sum of eight such integrals where its only
symmetry is $J_{ijk}$ $=$ $J_{jik}$. Rewriting the numerator of eqn (A.3) one
finds
\begin{equation}
J_{ijk} ~=~ \half [I_iI_j - I_jI_k - I_kI_i - (m^2_k-m^2_i-m^2_j)I_{ijk}]
\end{equation}
where
\begin{eqnarray}
I_i &=& \int_p \frac{1}{(p^2+m^2_i)} \\
I_{ijk} &=& \int_p \int_q \frac{1}{(p^2+m^2_i)((p+q)^2+m^2_j)(q^2+m^2_k)}
\end{eqnarray}
and the latter function is totally symmetric, corresponding to a two loop
vacuum
graph (ie zero external momentum). The integral $I_{ijk}$ has been considered
in \cite{J0,J1} and a single integral representation of it exists,
\cite{J2,J3,J4}. For our purposes, however, we have chosen to use the elegant
formula given in \cite{J4} since it is explictly symmetric in the masses.
Although $\Lambda_{123456}$ is itself ultraviolet finite the sub-integrals,
eqns (A.3) and (A.4), are divergent and therefore require regularization. In
\cite{J3,J4} dimensional regularization was introduced to control these
infinities, which is why we choose to calculate eqn (A.1) in $d$-dimensions, so
that $I_{ijk}$ involves double and simple poles in $\epsilon$ where $d$ $=$ $4$
$-$ $2\epsilon$. Therefore in the final result these must cancel for all
$m^2_i$. As a first step, it is trivial to observe that in the partial fraction
decomposition of eqn (A.1) the $I_iI_j$ type terms, which are also divergent,
formally cancel to leave only the $I_{ijk}$ terms. To proceed we recall the
important properties of $I_{ijk}$ which have been discussed in more detail in
\cite{J4}. In $d$-dimensions the exact value, for arbitrary $(\mbox{mass})^2$,
$x$, $y$ and $z$, is
\begin{equation}
I(x,y,z) ~=~ I(2a,0,0) + \Gamma^\prime [ F(\half c-y) + F(\half c-z)
- F(x-\half c) ]
\end{equation}
where
\begin{eqnarray}
I_{ijk} &=& I(m^2_i,m^2_j,m^2_k) \nonumber \\
\Gamma^\prime &=& \frac{(\mu^2)^{4-d}}{(4\pi)^d} \Gamma(2-\half d)
\Gamma(1 - \half d) \nonumber \\
a &=& \half [ x^2 + y^2 + z^2 - 2xy - 2yz - 2zx]^{1/2} \nonumber \\
c &=& x + y + z
\end{eqnarray}
and
\begin{equation}
F(w) ~=~ \int_a^w ds \, \frac{1}{(s^2-a^2)^{(4-d)/2}}
\end{equation}
The result (A.7) is valid in the region of $(x,y,z)$ space where $a^2$
$\geq$ $0$. For the case when $a^2$ $<$ $0$, then the solution is, with
$b^2$ $=$ $-$ $a^2$,
\begin{eqnarray}
I(x,y,z) &=& - \, I(2b,0,0) \sin (\half\pi d) \nonumber \\
&+& \Gamma^\prime [ G(\half c-x) + G(\half c-y) + G(\half c-z) ]
\end{eqnarray}
where
\begin{equation}
G(w) ~=~ \int_0^w ds \, \frac{1}{(s^2+b^2)^{(4-d)/2}}
\end{equation}
and, for example,
\begin{equation}
I(x,0,0) ~=~ \frac{\Gamma(2-\half d)\Gamma(3-d)\Gamma^2(\half d-1) x^{d-3}}
{(4\pi)^d\Gamma(\half d) (\mu^2)^{d-4}}
\end{equation}
which is clearly singular in four dimensions. To obtain the finite part of
$\Lambda_{123456}$ each part of $I(x,y,z)$ needs to be expanded in powers of
$\epsilon$ to the $O(1)$ term and the poles in $\epsilon$ cancelled. The
non-trivial part of this exercise is the $\epsilon$-expansion of the
$F(w)$ and $G(w)$ integrals. These have been given in \cite{J4} and we record
that to the $\epsilon$-finite term,
\begin{eqnarray}
(4\pi)^4 I(x,y,z) &=& - \, \frac{c}{2\epsilon^2} - \frac{1}{\epsilon} \left[
\frac{3c}{2} - L_1 \right] - \half [ L_2 - 6L_1 + \xi(x,y,z) \nonumber \\
&&+~ c(7+\zeta(2)) + (y+z-x)\overline{\ln}y \overline{\ln}z \nonumber \\
&&+~  (z+x-y)\overline{\ln}z \overline{\ln}x
+ (y+x-z)\overline{\ln}y\overline{\ln}x ]
\end{eqnarray}
where $\zeta(n)$ is the Riemann zeta function, $L_i$ $=$ $x\overline{\ln}^i x$
$+$ $y\overline{\ln}^i y$ $+$ $z\overline{\ln}^i z$, $\overline{\ln}x$ $=$
$\ln(x/\hat{\mu}^2)$, $\hat{\mu}^2$ $=$ $4\pi e^{-\gamma}\mu^2$ and $\gamma$
is Euler's constant, and for $a^2$ $>$ $0$,
\begin{equation}
\xi(x,y,z) ~=~ 8a [ M(\phi_z) + M(\phi_y) - M(-\phi_x)]
\end{equation}
where
\begin{equation}
M(t) ~=~ - \, \int_0^t d \phi \, \ln \sinh \phi
\end{equation}
and the angles $\phi_x$ are defined by
\begin{equation}
\phi_x ~=~ \coth^{-1} \left[ \frac{\half c - x}{a} \right]
\end{equation}
For $a^2$ $<$ $0$, then
\begin{equation}
\xi(x,y,z) ~=~ 8b[L(\theta_x)+L(\theta_y)+L(\theta_z) - \half \pi \ln 2]
\end{equation}
where the $\theta_x$ angles are given by
\begin{equation}
\theta_x ~=~ \tan^{-1} \left[ \frac{\half c - x}{b} \right]
\end{equation}
and $L(t)$ is the Lobachevskij function,
\begin{equation}
L(t) ~=~ - \, \int_0^t d \theta \, \ln \cos \theta
\end{equation}
Equation (A.17) can also be rewritten as
\begin{equation}
\xi(x,y,z) ~=~ 8b[\tilde{L}(\theta_z)+\tilde{L}(\theta_y)-\tilde{L}(-\theta_x)]
\end{equation}
where $\tilde{L}(t)$ $=$ $\int_t^{\pi/2} d\theta \, \ln \cos \theta$ in order
to make the obvious analytic continuation across $a^2$ $=$ $0$ more apparent.
It is worth noting that essentially eqn (A.1) has been reduced to a single
simple function, eqn (A.19), whose properties are well known. We have used the
following identities in order to write an efficient programme to calculate
$\Lambda_{123456}$ for a range of physical mass values. For instance,
\cite{J5},
\begin{eqnarray}
L(t) &=& - \, L(-t) \quad\quad\quad\quad\quad\quad\quad
\quad\quad\quad\quad\quad ~~~~~\mbox{for} ~
- \, \half\pi\leq t \leq \half \pi \nonumber \\
L(t) &=& L(\half\pi - t) + (t - \quarter \pi) \ln 2 - \half
L(\half\pi-2t) ~~~~\mbox{for} ~ 0 \leq t \leq \quarter \pi \nonumber \\
L(t) &=& \pm L(\pi\pm t) \mp \pi \ln 2
\end{eqnarray}
Therefore, when the argument of the Lobachevskij function is known, the
identities of eqn (A.21) mean that one need only write a routine to evaluate
$L(t)$ numerically in the range $[0,\half\pi)$. For example, if $0$ $\leq$
$\lambda$ $<$ $2\pi$ then for any integer $n$
\begin{equation}
L(2\pi n + \lambda) ~=~ 2\pi n \ln 2 + L(\lambda)
\end{equation}
and so on.

Returning to the partial fraction form of $\Lambda_{123456}$ with the result
for $I_{ijk}$, the $c$ and $L_i$ terms of the $\epsilon$ expansion cancel in
the final expression and we can therefore take the limit back to four
dimensions, $\epsilon$ $\rightarrow$ $0$. Consequently, we end up with the
following analytic expression:
\begin{eqnarray}
\Lambda_{123456} &=& - \, \frac{1}{4(4\pi)^4(m^2_1-m^2_2)(m^2_3-m^2_4)
(m^2_5-m^2_6)} \nonumber \\
&\times& \left[ (m^2_3-m^2_1-m^2_5) \left[ \xi_{135}
- m^2_1 \ln \left(\frac{m^2_1}{m^2_3}
\right) \ln \left(\frac{m^2_1}{m^2_5}\right) \right. \right. \nonumber \\
&&- \left. \left. m^2_3  \ln \left(\frac{m^2_3}{m^2_1}\right)
\ln \left(\frac{m^2_3}{m^2_5}\right)
- m^2_5 \ln \left(\frac{m^2_5}{m^2_1}\right)
\ln \left(\frac{m^2_5}{m^2_3}\right) \right] \right. \nonumber \\
&-& \left. (m^2_3-m^2_1-m^2_6)\left[ \xi_{136}
- m^2_1 \ln \left(\frac{m^2_1}{m^2_3}\right)
\ln \left(\frac{m^2_1}{m^2_6}\right) \right. \right. \nonumber \\
&&- \left. \left. m^2_3 \ln \left(\frac{m^2_3}{m^2_1}\right)
\ln \left(\frac{m^2_3}{m^2_6}\right)
- m^2_6 \ln \left(\frac{m^2_6}{m^2_1}\right)
\ln \left(\frac{m^2_6}{m^2_3}\right) \right] \right. \nonumber \\
&-& \left. (m^2_4-m^2_1-m^2_5) \left[ \xi_{145}
- m^2_1 \ln \left(\frac{m^2_1}{m^2_4}\right)
\ln \left( \frac{m^2_1}{m^2_5}\right) \right. \right. \nonumber \\
&&- \left. \left. m^2_4 \ln \left(\frac{m^2_4}{m^2_1}\right)
\ln \left(\frac{m^2_4}{m^2_5}\right)
- m^2_5 \ln\left(\frac{m^2_5}{m^2_1}\right)\ln\left(\frac{m^2_5}{m^2_4}\right)
\right] \right. \nonumber \\
&+& \left. (m^2_4-m^2_1-m^2_6)\left[ \xi_{146}
- m^2_1 \ln \left(\frac{m^2_1}{m^2_4}\right)
\ln \left(\frac{m^2_1}{m^2_6}\right) \right. \right. \nonumber \\
&&- \left. \left. m^2_4 \ln \left(\frac{m^2_4}{m^2_1}\right)
\ln \left(\frac{m^2_4}{m^2_6}\right)
- m^2_6 \ln \left(\frac{m^2_6}{m^2_1}\right)
\ln \left(\frac{m^2_6}{m^2_4}\right) \right] \right. \nonumber \\
&-& \left. (m^2_3-m^2_2-m^2_5)\left[ \xi_{235}
- m^2_2 \ln \left(\frac{m^2_2}{m^2_3}\right)
\ln \left(\frac{m^2_2}{m^2_5}\right) \right. \right. \nonumber \\
&&- \left. \left. m^2_3 \ln \left(\frac{m^2_3}{m^2_2}\right)
\ln \left(\frac{m^2_3}{m^2_5}\right)
+ m^2_5 \ln \left(\frac{m^2_5}{m^2_2}\right)
\ln \left(\frac{m^2_5}{m^2_3}\right) \right] \right. \nonumber \\
&+& \left.(m^2_3-m^2_2-m^2_6) \left[ \xi_{236}
- m^2_2 \ln \left(\frac{m^2_2}{m^2_3}\right)
\ln \left(\frac{m^2_2}{m^2_6}\right) \right. \right. \nonumber \\
&&- \left. \left. m^2_3 \ln \left(\frac{m^2_3}{m^2_2}\right)
\ln \left(\frac{m^2_3}{m^2_6}\right)
- m^2_6 \ln \left(\frac{m^2_6}{m^2_2}\right)
\ln \left(\frac{m^2_6}{m^2_3}\right) \right] \right. \nonumber \\
&+& \left.(m^2_4-m^2_2-m^2_5)\left[ \xi_{245}
- m^2_2 \ln \left(\frac{m^2_2}{m^2_4}\right)
\ln \left(\frac{m^2_2}{m^2_5}\right) \right. \right. \nonumber \\
&&- \left. \left. m^2_4 \ln \left(\frac{m^2_4}{m^2_2}\right)
\ln \left(\frac{m^2_4}{m^2_5}\right)
- m^2_5 \ln \left(\frac{m^2_5}{m^2_2}\right)
\ln \left( \frac{m^2_5}{m^2_4} \right) \right] \right. \nonumber \\
&-& \left. (m^2_4-m^2_2-m^2_6) \left[ \xi_{246}
- m^2_2\ln\left(\frac{m^2_2}{m^2_4}
\right) \ln\left(\frac{m^2_2}{m^2_6}\right) \right. \right. \nonumber \\
&&- \left. \left. m^2_4\ln\left(\frac{m^2_4}{m^2_2}\right)
\ln\left(\frac{m^2_4}{m^2_6} \right)
- m^2_6 \ln\left(\frac{m^2_6}{m^2_2}\right)\ln\left(\frac{m^2_6}{m^2_4}\right)
\! \right] \! \right]
\end{eqnarray}
where $\xi_{ijk}$ $=$ $\xi(m^2_i,m^2_j,m^2_k)$ and it is evaluated according to
eqns (A.14) or (A.17) depending on whether the particular $a^2$ is positive or
negative. A further check on our manipulations to obtain eqn (A.23) is the
absence of the arbitrary mass $\mu$ which was required at intermediate steps
to have logarithms whose arguments were dimensionless quantities.

Although it may appear that the final result is singular in certain cases
through denominator factors like $(m^2_1-m^2_2)$ when $m^2_1$ $=$ $m^2_2$, the
expression within the square brackets also vanishes. Moreover, if one sets
$m^2_2$ $=$ $m^2_1$ $+$ $\delta$, where $\delta$ is small, and expands in
powers of $\delta$ then in the limit as $\delta$ $\rightarrow$ $0$ a non-zero
non-singular function of the independent mass remains. Further, there is no
difficulty with singularities when one or more masses is zero. To illustrate
this point explicitly we consider the integral $\Lambda_{123056}$ where the
zero subscript means the corresponding mass of eqn (A.1) is zero. Its
form can readily be deduced from eqn (A.23) by taking the $m^2_4$ $\rightarrow$
$0$ limit. However, to do this the  behaviour of $\xi(x,y,z)$ in the
$z$ $\rightarrow$ $0$ limit is required since eqn (A.23) has terms like $\ln
m^2_4$ which are potentially infinite in the limit we require. It is is easy to
deduce from the explicit representation, eqn (A.14), that
\begin{equation}
\xi(x,y,z) ~ \sim ~ (x-y) \left[ 2 \mbox{Li}_2\left(1-\frac{y}{x}\right)
+ \ln \left(\frac{x}{y}\right) \ln \left(\frac{x}{z}\right)\right]
\end{equation}
as $z$ $\rightarrow$ $0$. Thus a little algebra leads to the compact
expression,
\begin{eqnarray}
\Lambda_{123056} &=& - \, \frac{1}{4(4\pi)^4(m^2_1-m^2_2)m^2_3(m^2_5-m^2_6)}
\nonumber \\
&\times& \left[(m^2_3-m^2_1-m^2_5)\xi_{135}
- (m^2_3-m^2_1-m^2_6)\xi_{136} \right. \nonumber \\
&-& \left. (m^2_3-m^2_2-m^2_5)\xi_{235}
+ (m^2_3-m^2_2-m^2_6)\xi_{236} \right. \nonumber \\
&-& \left. \rho(m^2_3,m^2_1,m^2_5) + \rho(m^2_3,m^2_1,m^2_6) \right.
\nonumber \\
&+& \left. \rho(m^2_3,m^2_2,m^2_5) - \rho(m^2_3,m^2_2,m^2_6) \right.
\nonumber \\
&+& \left. \lambda(m^2_1,m^2_5) - \lambda(m^2_1,m^2_6) - \lambda(m^2_1,m^2_5)
+ \lambda(m^2_2,m^2_6) \right]
\end{eqnarray}
with
\begin{eqnarray}
\rho(x,y,z) &=& (x-y-z) \left[ x \ln \left(\frac{x}{y}\right)
\ln\left(\frac{x}{z}\right)
+ y \ln \left(\frac{y}{x}\right) \ln\left(\frac{y}{z}\right) \right.
\nonumber \\
&& ~~~~~~~~~~~~~~~ \left. + ~ z \ln \left(\frac{z}{x}\right)
\ln\left(\frac{z}{y}\right) \right]
\end{eqnarray}
and
\begin{equation}
\lambda(x,y) ~=~ (x+y) \left[ 2(x-y) \mbox{Li}_2 \left( 1 - \frac{y}{x} \right)
- y \ln \left( \frac{x}{y} \right) \right]
\end{equation}
where $\mbox{Li}_2(t)$ is the dilogarithm function. Its properties have been
discussed extensively in \cite{J6} but we make use of the following ones here
\begin{eqnarray}
\mbox{Li}_2(-t) + \mbox{Li}_2(-1/t) &=& - \, \zeta(2) - \half \ln^2 t
\quad\quad ~~~~\mbox{for}~ t > 0 \nonumber \\
\mbox{Li}_2(t) + \mbox{Li}_2(1-t) &=& \zeta(2) - \ln t \ln (1-t)
\end{eqnarray}
and its integral representation is, \cite{J6},
\begin{equation}
\mbox{Li}_2(t) ~=~ - \, \int_0^t \frac{ds}{s} \ln (1-s)
\end{equation}
where $\mbox{Li}_2(1)$ $=$ $\zeta(2)$ $=$ $\pi^2/6$.

Finally, another check on our overall expression eqn (A.23) is the comparison
with the earlier result of \cite{BM} where only $m^2_3$ and $m^2_4$ are
non-zero, ie $\Lambda_{003400}$, which was evaluated by an independent method.
We can easily deduce an expression for $\Lambda_{003400}$ from eqn (A.25) by
using the relation (A.24) or by returning to the $I_{ijk}$ representation of
eqn (A.1) and taking the appropriate limits in that case. Useful for the former
approach are the properties of the dilogarithm function, \cite{J6}. Whilst in
the latter instance we made use of the Tyalor expansion of the $I_{ijk}$ about
zero mass and in particular,
\begin{equation}
\left. \frac{\partial^2I(x,y,z)}{\partial y \partial z} \right|_{~y~=~z~=~0}
{}~=~ \frac{\Gamma^2(\half d-2)\Gamma(4-\half d) \Gamma(5-d) x^{d-5}}
{(4\pi)^d(\mu^2)^{d-4} \Gamma(\half d)}
\end{equation}
whose $\epsilon$ expansion is easy to determine. Consequently, we find
\begin{equation}
\Lambda_{003400} ~=~ - \, \frac{1}{(4\pi)^4(m^2_3-m^2_4)} \ln \left(
\frac{m^2_3}{m^2_4} \right)
\end{equation}
This is in total agreement with the explicit calculation of \cite{BM} and is a
necessary non-trivial check that we have the overall normalization of our
integral correct, in terms of signs and factors of $2\pi$.

\vspace{1.0cm}
\noindent
{\bf Acknowledgements.} We are grateful to Naoya Hata and
and Paul Langacker for providing us with a computer-readable version of their
figure $12$ from Reference \cite{Hata}. RG would like to thank K.S. Babu
for very useful discussions on Ref. \cite{BM} and Bala Sundaram for discussions
and generous help.
 JAG thanks C. Ford and D.R.T. Jones for useful
discussions on \cite{J4} and DC thanks the Mehta Research Institute for
hospitality while this work was in progress.
\newpage

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\end{thebibliography}


\eject
\begin{center}
{\bf Figure Captions}
\end{center}
\vskip .5in
\noindent {\bf Figure $1$:} The two-loop diagram which gives rise to the
mass corrections considered in this paper.
\vskip .3in

\noindent{\bf Figure $2$:} The MSW solution space for the solar neutrino
deficit, from Ref. \cite{Hata}. Superposed on it are the $3$ curves (a),
(b) and (c) which represent sample calculations using our results. Each
curve shows the mass squared differences and mixings for the two light
neutrinos which acquire  masses radiatively, for fixed values of the masses
of the other two neutrinos which are massive at tree level.
Curve (a) in Figure $2$ corresponds to a singlet mass
of $100 $ GeV and a $\nu_{\tau}$ mass of $\approx 8.6
$ eV. Curve (b) corresponds
to a singlet mass of $ 400 $ GeV and a $ \nu_{\mu}$ mass of $7 $ eV.
 Finally, curve (c) represents a singlet mass of $1 $TeV and a $\nu_{\mu}$
mass of $\approx 9.8 $ eV.
\end{document}

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showpage
end


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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2042 1449 M 2024 1449 L
2042 1473 M 2024 1473 L
2042 1494 M 2024 1494 L
2042 1511 M 2024 1511 L
2042 1527 M 2024 1527 L
2042 1541 M 2006 1541 L
2042 1633 M 2024 1633 L
2042 1686 M 2024 1686 L
2042 1724 M 2024 1724 L
2042 1754 M 2024 1754 L
2042 1778 M 2024 1778 L
2042 1799 M 2024 1799 L
2042 1816 M 2024 1816 L
2042 1832 M 2024 1832 L
2042 1846 M 2006 1846 L
2042 1938 M 2024 1938 L
2042 1991 M 2024 1991 L
2042 2029 M 2024 2029 L
2042 2059 M 2024 2059 L
2042 2083 M 2024 2083 L
2042 2103 M 2024 2103 L
2042 2121 M 2024 2121 L
2042 2137 M 2024 2137 L
2042 2151 M 2006 2151 L
2042 2242 M 2024 2242 L
2042 2296 M 2024 2296 L
CS [] 0 setdash M
CS [6 12] 0 setdash M
 255 0 255 C
1012 1526 M 1023 1527 L
1034 1535 L
1045 1541 L
1056 1544 L
1067 1547 L
1078 1548 L
1089 1551 L
1100 1553 L
1111 1554 L
1123 1553 L
1134 1552 L
1145 1549 L
1156 1543 L
1167 1535 L
1178 1525 L
1189 1515 L
1200 1507 L
1211 1497 L
1222 1488 L
1233 1479 L
1244 1470 L
1255 1461 L
1255 1461 L
1266 1451 L
1277 1439 L
1285 1427 L
1287 1419 L
1277 1404 L
1266 1401 L
1255 1400 L
1244 1401 L
1233 1401 L
1222 1403 L
1211 1405 L
1200 1406 L
1189 1408 L
1178 1411 L
1167 1413 L
1156 1414 L
1145 1417 L
1134 1420 L
1123 1423 L
1111 1427 L
1100 1430 L
1089 1435 L
1078 1440 L
1067 1446 L
1056 1453 L
1045 1461 L
1034 1471 L
1023 1484 L
1013 1494 L
1012 1495 L
1009 1511 L
1012 1526 L
CS [] 0 setdash M
1271 1402 M 1279 1407 L
1262 1401 M 1284 1414 L
1255 1400 M 1287 1419 L
1249 1400 M 1286 1422 L
1243 1401 M 1286 1425 L
1238 1401 M 1284 1428 L
1232 1402 M 1283 1431 L
1227 1402 M 1281 1433 L
1222 1403 M 1279 1436 L
1217 1404 M 1277 1438 L
1213 1404 M 1275 1441 L
1208 1405 M 1273 1443 L
1203 1406 M 1271 1445 L
1198 1407 M 1269 1447 L
1194 1408 M 1267 1450 L
1189 1408 M 1265 1452 L
1185 1409 M 1262 1454 L
1180 1410 M 1260 1456 L
1176 1411 M 1257 1458 L
1171 1412 M 1255 1461 L
1167 1413 M 1252 1463 L
1161 1414 M 1250 1465 L
1156 1414 M 1247 1467 L
1152 1415 M 1245 1469 L
1148 1416 M 1242 1471 L
1143 1417 M 1240 1473 L
1139 1418 M 1237 1475 L
1135 1420 M 1235 1477 L
1131 1421 M 1232 1479 L
1127 1422 M 1230 1481 L
1123 1423 M 1227 1483 L
1119 1424 M 1224 1485 L
1115 1426 M 1222 1487 L
1111 1427 M 1219 1490 L
1107 1428 M 1217 1492 L
1103 1430 M 1214 1494 L
1099 1431 M 1212 1496 L
1096 1432 M 1209 1498 L
1092 1434 M 1207 1500 L
1089 1435 M 1205 1502 L
1085 1437 M 1202 1504 L
1082 1438 M 1200 1507 L
1079 1440 M 1197 1509 L
1075 1442 M 1195 1511 L
1072 1443 M 1192 1513 L
1069 1445 M 1190 1515 L
1066 1447 M 1187 1517 L
1063 1449 M 1185 1519 L
1060 1451 M 1182 1521 L
1057 1453 M 1180 1523 L
1055 1455 M 1177 1525 L
1052 1456 M 1175 1528 L
1049 1458 M 1172 1530 L
1046 1460 M 1170 1532 L
1044 1462 M 1168 1534 L
1041 1465 M 1165 1536 L
1039 1467 M 1162 1538 L
1037 1469 M 1160 1540 L
1034 1471 M 1157 1542 L
1032 1473 M 1154 1544 L
1030 1476 M 1151 1546 L
1028 1478 M 1148 1547 L
1026 1480 M 1144 1549 L
1024 1483 M 1140 1550 L
1022 1485 M 1136 1551 L
1020 1487 M 1132 1552 L
1017 1490 M 1127 1553 L
1015 1492 M 1122 1553 L
1013 1494 M 1116 1553 L
1012 1497 M 1109 1553 L
1011 1500 M 1102 1553 L
1011 1504 M 1094 1552 L
1010 1507 M 1085 1550 L
1010 1510 M 1075 1548 L
1010 1514 M 1067 1546 L
1011 1518 M 1058 1545 L
1012 1522 M 1021 1527 L
1033 1534 M 1046 1541 L
1012 1526 M 1013 1526 L
1920 1524 M 1919 1530 L
1920 1535 L
1920 1536 L
1921 1539 L
1922 1542 L
1923 1547 L
1923 1549 L
1925 1555 L
1925 1555 L
1926 1558 L
1928 1563 L
1931 1567 L
1933 1569 L
1936 1570 L
1939 1572 L
1941 1574 L
1942 1576 L
1942 1579 L
1944 1587 L
1947 1589 L
1949 1591 L
1952 1592 L
1955 1594 L
1956 1594 L
1956 1597 L
1955 1599 L
1955 1605 L
1957 1607 L
1960 1608 L
1963 1609 L
1966 1609 L
1968 1609 L
1971 1608 L
1974 1607 L
1976 1605 L
1979 1602 L
1979 1596 L
1979 1594 L
1979 1591 L
1979 1591 L
1981 1587 L
1984 1585 L
1987 1581 L
1989 1576 L
1989 1573 L
1989 1570 L
1990 1567 L
1990 1564 L
1991 1561 L
1992 1558 L
1992 1555 L
1993 1552 L
1992 1547 L
1992 1545 L
1991 1542 L
1991 1539 L
1991 1536 L
1991 1533 L
1990 1530 L
1990 1524 L
1990 1524 L
1989 1521 L
1987 1515 L
1987 1515 L
1986 1512 L
1984 1509 L
1983 1506 L
1981 1503 L
1979 1500 L
1976 1497 L
1974 1496 L
1971 1495 L
1968 1494 L
1966 1494 L
1963 1494 L
1960 1494 L
1957 1494 L
1955 1495 L
1952 1495 L
1949 1496 L
1947 1497 L
1944 1497 L
1941 1499 L
1939 1500 L
1936 1501 L
1933 1502 L
1931 1504 L
1928 1507 L
1925 1510 L
1923 1515 L
1923 1515 L
1921 1521 L
1920 1524 L
1920 1524 L
1970 1495 M 1979 1500 L
1962 1494 M 1983 1506 L
1957 1494 M 1985 1511 L
1952 1495 M 1987 1515 L
1948 1496 M 1988 1520 L
1944 1498 M 1989 1524 L
1940 1499 M 1990 1528 L
1937 1501 M 1990 1531 L
1934 1502 M 1991 1535 L
1931 1504 M 1991 1539 L
1928 1506 M 1991 1543 L
1927 1509 M 1992 1547 L
1925 1511 M 1992 1550 L
1923 1514 M 1992 1554 L
1922 1517 M 1992 1557 L
1921 1520 M 1991 1560 L
CS M
1920 1523 M 1990 1563 L
1920 1526 M 1990 1566 L
1919 1529 M 1989 1570 L
1920 1533 M 1989 1573 L
1921 1537 M 1988 1576 L
1922 1541 M 1987 1579 L
1922 1545 M 1986 1582 L
1924 1550 M 1984 1585 L
1925 1554 M 1982 1587 L
1926 1558 M 1980 1589 L
1928 1563 M 1979 1592 L
1940 1573 M 1979 1596 L
1942 1578 M 1979 1599 L
1943 1582 M 1978 1602 L
1944 1586 M 1976 1605 L
1956 1597 M 1974 1607 L
1955 1599 M 1970 1608 L
1955 1603 M 1965 1609 L
633 751 M 638 754 L
627 751 M 638 758 L
621 751 M 638 761 L
615 751 M 638 765 L
609 751 M 638 768 L
603 751 M 638 772 L
596 751 M 638 775 L
590 751 M 638 779 L
584 751 M 638 782 L
578 751 M 638 786 L
572 751 M 635 788 L
566 751 M 629 788 L
560 751 M 623 788 L
554 751 M 617 788 L
547 751 M 611 788 L
541 751 M 605 788 L
535 751 M 599 788 L
529 751 M 592 788 L
523 751 M 586 788 L
517 752 M 580 788 L
517 755 M 574 788 L
517 759 M 568 788 L
517 762 M 562 788 L
517 766 M 556 788 L
517 769 M 550 788 L
517 773 M 543 788 L
517 776 M 537 788 L
517 780 M 531 788 L
517 784 M 525 788 L
517 787 M 519 788 L
678 809 M CS [] 0 setdash M
678 774 M 706 804 M 708 799 L
708 809 L
706 804 L
703 808 L
698 809 L
694 809 L
689 808 L
686 804 L
684 801 L
683 796 L
683 788 L
684 783 L
686 779 L
689 776 L
694 774 L
698 774 L
703 776 L
706 779 L
708 783 L
694 809 M 691 808 L
688 804 L
686 801 L
684 796 L
684 788 L
686 783 L
688 779 L
691 776 L
694 774 L
728 798 M 723 796 L
719 793 L
718 788 L
718 784 L
719 779 L
723 776 L
728 774 L
731 774 L
736 776 L
740 779 L
741 784 L
741 788 L
740 793 L
736 796 L
731 798 L
728 798 L
724 796 L
721 793 L
719 788 L
719 784 L
721 779 L
724 776 L
728 774 L
731 774 M 734 776 L
738 779 L
740 784 L
740 788 L
738 793 L
734 796 L
731 798 L
754 798 M 754 774 L
756 798 M 756 774 L
756 793 M 760 796 L
764 798 L
768 798 L
773 796 L
774 793 L
774 774 L
768 798 M 771 796 L
773 793 L
773 774 L
774 793 M 778 796 L
783 798 L
786 798 L
791 796 L
793 793 L
793 774 L
786 798 M 789 796 L
791 793 L
791 774 L
749 798 M 756 798 L
749 774 M 761 774 L
768 774 M 779 774 L
786 774 M 798 774 L
810 809 M 810 774 L
811 809 M 811 774 L
811 793 M 815 796 L
818 798 L
821 798 L
826 796 L
830 793 L
831 788 L
831 784 L
830 779 L
826 776 L
821 774 L
818 774 L
815 776 L
811 779 L
821 798 M 825 796 L
828 793 L
830 788 L
830 784 L
828 779 L
825 776 L
821 774 L
805 809 M 811 809 L
845 809 M 843 808 L
845 806 L
846 808 L
845 809 L
845 798 M 845 774 L
846 798 M 846 774 L
840 798 M 846 798 L
840 774 M 851 774 L
863 798 M 863 774 L
865 798 M 865 774 L
865 793 M 868 796 L
873 798 L
876 798 L
881 796 L
883 793 L
883 774 L
876 798 M 880 796 L
881 793 L
881 774 L
858 798 M 865 798 L
858 774 M 869 774 L
876 774 M 888 774 L
898 788 M 918 788 L
918 791 L
916 794 L
915 796 L
911 798 L
906 798 L
901 796 L
898 793 L
896 788 L
896 784 L
898 779 L
901 776 L
906 774 L
910 774 L
915 776 L
918 779 L
916 788 M 916 793 L
915 796 L
906 798 M 903 796 L
900 793 L
898 788 L
898 784 L
900 779 L
903 776 L
906 774 L
948 809 M 948 774 L
950 809 M 950 774 L
948 793 M 945 796 L
941 798 L
938 798 L
933 796 L
930 793 L
928 788 L
928 784 L
930 779 L
933 776 L
938 774 L
941 774 L
945 776 L
948 779 L
938 798 M 935 796 L
931 793 L
930 788 L
930 784 L
931 779 L
935 776 L
938 774 L
943 809 M 950 809 L
948 774 M 955 774 L
993 809 M 993 774 L
995 809 M 995 774 L
1005 799 M 1005 786 L
988 809 M 1015 809 L
1015 799 L
1013 809 L
995 793 M 1005 793 L
988 774 M 1000 774 L
1026 809 M 1025 808 L
1026 806 L
1028 808 L
1026 809 L
1026 798 M 1026 774 L
1028 798 M 1028 774 L
1021 798 M 1028 798 L
1021 774 M 1033 774 L
1045 809 M 1045 781 L
1046 776 L
1050 774 L
1053 774 L
1056 776 L
1058 779 L
1046 809 M 1046 781 L
1048 776 L
1050 774 L
1040 798 M 1053 798 L
CS [] 0 setdash M
CS [6 12] 0 setdash M
 0 255 255 C
1186 1522 M 1192 1513 L
1214 1491 L
1236 1474 L
1259 1458 L
1280 1442 L
1303 1427 L
1325 1411 L
1347 1392 L
1369 1372 L
1391 1351 L
1413 1331 L
1435 1311 L
1457 1292 L
1479 1273 L
1501 1255 L
1523 1237 L
1545 1221 L
1567 1204 L
1589 1188 L
1611 1173 L
1634 1156 L
1656 1140 L
1678 1127 L
1700 1114 L
1722 1098 L
1738 1078 L
1744 1070 L
1766 1048 L
1788 1035 L
1810 1024 L
1832 1015 L
1854 1009 L
1876 1005 L
1898 1007 L
1920 1030 L
1922 1040 L
1923 1059 L
1922 1078 L
1923 1097 L
1922 1117 L
1922 1136 L
1922 1155 L
1920 1175 L
1920 1175 L
1914 1213 L
1910 1233 L
1903 1252 L
1898 1264 L
1882 1290 L
1876 1301 L
CS M
1854 1323 L
1832 1338 L
1810 1349 L
1788 1357 L
1766 1363 L
1744 1369 L
1722 1376 L
1700 1383 L
1678 1390 L
1656 1401 L
1656 1429 L
1678 1436 L
1700 1440 L
1722 1442 L
1744 1445 L
1766 1446 L
1788 1448 L
1810 1450 L
1832 1454 L
1854 1462 L
1857 1464 L
1876 1487 L
1879 1503 L
1882 1522 L
1880 1541 L
1876 1558 L
1867 1580 L
1854 1595 L
1832 1603 L
1810 1621 L
1788 1625 L
1766 1628 L
1744 1633 L
1722 1640 L
1700 1642 L
1678 1643 L
1656 1644 L
1634 1644 L
1611 1644 L
1589 1644 L
1567 1643 L
1545 1643 L
1523 1642 L
1501 1642 L
1479 1641 L
1457 1640 L
1435 1639 L
1413 1639 L
1391 1638 L
1369 1638 L
1347 1637 L
1325 1636 L
1303 1636 L
1280 1633 L
1259 1630 L
1236 1623 L
1228 1619 L
1217 1599 L
1214 1579 L
1192 1562 L
1191 1561 L
1178 1541 L
1186 1522 L
751 1831 M 773 1844 L
795 1850 L
817 1852 L
839 1854 L
861 1855 L
883 1855 L
905 1856 L
927 1856 L
950 1856 L
972 1856 L
994 1856 L
1016 1856 L
1038 1856 L
1060 1855 L
1082 1855 L
1104 1855 L
1126 1855 L
1148 1855 L
1170 1855 L
1192 1855 L
1214 1854 L
1236 1854 L
1259 1854 L
1280 1854 L
1303 1853 L
1325 1853 L
1347 1853 L
1369 1853 L
1391 1852 L
1413 1852 L
1435 1852 L
1457 1852 L
1479 1851 L
1501 1851 L
1523 1851 L
1545 1850 L
1567 1850 L
1589 1850 L
1611 1849 L
1634 1849 L
1656 1849 L
1678 1848 L
1700 1848 L
1722 1848 L
1744 1848 L
1766 1848 L
1788 1848 L
1810 1849 L
1832 1849 L
1854 1850 L
1876 1851 L
1898 1853 L
1920 1855 L
1942 1857 L
1964 1861 L
1987 1862 L
2009 1859 L
2024 1831 L
2027 1812 L
2030 1792 L
2031 1773 L
2031 1754 L
2032 1734 L
2032 1715 L
2032 1696 L
2032 1677 L
2032 1657 L
2032 1638 L
2031 1619 L
2031 1591 L
2030 1580 L
2028 1561 L
2028 1541 L
2027 1522 L
2025 1503 L
2023 1484 L
2021 1464 L
2019 1445 L
2014 1426 L
2009 1392 L
2008 1387 L
2008 1368 L
2009 1357 L
2012 1329 L
2015 1310 L
2018 1290 L
2021 1271 L
2023 1252 L
2024 1233 L
2024 1213 L
2026 1194 L
2027 1175 L
2027 1155 L
2027 1136 L
2028 1117 L
2028 1097 L
2029 1078 L
2029 1059 L
2029 1040 L
2030 1020 L
2031 1006 L
2031 982 L
2031 962 L
2031 936 L
2030 924 L
2028 904 L
2024 885 L
2009 854 L
1987 844 L
1964 844 L
1942 849 L
1920 856 L
1898 863 L
1876 872 L
1854 881 L
1832 891 L
1810 902 L
1788 913 L
1766 924 L
1744 935 L
1722 946 L
1700 958 L
1678 970 L
1656 983 L
1634 996 L
1611 1010 L
1589 1026 L
1567 1044 L
1545 1073 L
1527 1097 L
1523 1102 L
1501 1118 L
1479 1132 L
1457 1148 L
1435 1164 L
1413 1180 L
1391 1196 L
1369 1213 L
1347 1230 L
1325 1247 L
1303 1265 L
1280 1283 L
1259 1301 L
1236 1319 L
1214 1337 L
1192 1354 L
1170 1372 L
1148 1390 L
1126 1408 L
1104 1425 L
1082 1444 L
1060 1462 L
1038 1484 L
1038 1484 L
1016 1512 L
1016 1543 L
1038 1560 L
1040 1561 L
1058 1580 L
1045 1599 L
1038 1609 L
1016 1638 L
1016 1639 L
994 1658 L
972 1674 L
950 1688 L
927 1703 L
905 1717 L
883 1731 L
861 1745 L
839 1759 L
817 1773 L
795 1787 L
773 1802 L
751 1831 L
517 596 M 638 596 L
CS [] 0 setdash M
678 596 M CS [] 0 setdash M
678 579 M 686 614 M 686 579 L
688 614 M 688 579 L
708 614 M 708 579 L
709 614 M 709 579 L
681 614 M 693 614 L
703 614 M 715 614 L
688 597 M 708 597 L
681 579 M 693 579 L
703 579 M 715 579 L
733 602 M 728 600 L
724 597 L
723 592 L
723 589 L
724 584 L
728 580 L
733 579 L
736 579 L
741 580 L
745 584 L
746 589 L
746 592 L
745 597 L
741 600 L
736 602 L
733 602 L
730 600 L
726 597 L
724 592 L
724 589 L
726 584 L
730 580 L
733 579 L
736 579 M 740 580 L
743 584 L
745 589 L
745 592 L
743 597 L
740 600 L
736 602 L
760 602 M 760 579 L
761 602 M 761 579 L
761 597 M 765 600 L
769 602 L
773 602 L
778 600 L
779 597 L
779 579 L
773 602 M 776 600 L
778 597 L
778 579 L
779 597 M 783 600 L
788 602 L
791 602 L
796 600 L
798 597 L
798 579 L
791 602 M 795 600 L
796 597 L
796 579 L
754 602 M 761 602 L
754 579 M 766 579 L
CS M
773 579 M 785 579 L
791 579 M 803 579 L
813 592 M 833 592 L
833 595 L
831 599 L
830 600 L
826 602 L
821 602 L
816 600 L
813 597 L
811 592 L
811 589 L
813 584 L
816 580 L
821 579 L
825 579 L
830 580 L
833 584 L
831 592 M 831 597 L
830 600 L
821 602 M 818 600 L
815 597 L
813 592 L
813 589 L
815 584 L
818 580 L
821 579 L
860 599 M 861 602 L
861 595 L
860 599 L
858 600 L
855 602 L
848 602 L
845 600 L
843 599 L
843 595 L
845 593 L
848 592 L
856 589 L
860 587 L
861 585 L
843 597 M 845 595 L
848 593 L
856 590 L
860 589 L
861 587 L
861 582 L
860 580 L
856 579 L
850 579 L
846 580 L
845 582 L
843 585 L
843 579 L
845 582 L
875 614 M 875 585 L
876 580 L
880 579 L
883 579 L
886 580 L
888 584 L
876 614 M 876 585 L
878 580 L
880 579 L
869 602 M 883 602 L
900 599 M 900 597 L
898 597 L
898 599 L
900 600 L
903 602 L
910 602 L
913 600 L
915 599 L
916 595 L
916 584 L
918 580 L
920 579 L
915 599 M 915 584 L
916 580 L
920 579 L
921 579 L
915 595 M 913 593 L
903 592 L
898 590 L
896 587 L
896 584 L
898 580 L
903 579 L
908 579 L
911 580 L
915 584 L
903 592 M 900 590 L
898 587 L
898 584 L
900 580 L
903 579 L
933 614 M 933 579 L
935 614 M 935 579 L
951 602 M 935 585 L
943 592 M 953 579 L
941 592 M 951 579 L
928 614 M 935 614 L
946 602 M 956 602 L
928 579 M 940 579 L
946 579 M 956 579 L
966 592 M 986 592 L
986 595 L
985 599 L
983 600 L
980 602 L
975 602 L
970 600 L
966 597 L
965 592 L
965 589 L
966 584 L
970 580 L
975 579 L
978 579 L
983 580 L
986 584 L
985 592 M 985 597 L
983 600 L
975 602 M 971 600 L
968 597 L
966 592 L
966 589 L
968 584 L
971 580 L
975 579 L
CS [] 0 setdash M
CS [80 24] 0 setdash M
 0 0 255 C
919 1312 M 932 1297 L
959 1270 L
985 1247 L
1012 1225 L
1039 1204 L
1066 1182 L
1092 1162 L
1119 1141 L
1146 1120 L
1173 1099 L
1199 1078 L
1226 1057 L
1253 1037 L
1280 1018 L
1306 999 L
1333 979 L
1360 958 L
1386 936 L
1413 910 L
1440 886 L
1467 862 L
1493 838 L
1520 814 L
1547 791 L
1574 767 L
1600 742 L
1627 718 L
1654 699 L
1681 680 L
1707 661 L
1734 641 L
1761 621 L
1788 604 L
1814 590 L
1841 577 L
1868 566 L
1895 555 L
1921 546 L
1948 541 L
1975 542 L
2002 564 L
2009 581 L
2015 611 L
2018 642 L
2019 672 L
2018 703 L
2017 733 L
2014 764 L
2010 794 L
2005 825 L
2002 847 L
1994 886 L
1990 916 L
1988 947 L
1988 977 L
1990 1007 L
1994 1038 L
2002 1082 L
2004 1099 L
2009 1129 L
2013 1160 L
2018 1190 L
2017 1221 L
2019 1251 L
2019 1282 L
2019 1312 L
2015 1343 L
2010 1373 L
2002 1397 L
1975 1430 L
1948 1438 L
1921 1439 L
1895 1438 L
1868 1434 L
1841 1431 L
1814 1428 L
1788 1424 L
1761 1422 L
1734 1420 L
1707 1419 L
1681 1417 L
1654 1416 L
1627 1415 L
1600 1415 L
1574 1414 L
1547 1414 L
1520 1413 L
1493 1413 L
1467 1413 L
1440 1412 L
1413 1411 L
1386 1410 L
1360 1410 L
1333 1408 L
1306 1407 L
1280 1406 L
1253 1404 L
1226 1403 L
1199 1401 L
1173 1399 L
1146 1398 L
1119 1396 L
1092 1394 L
1066 1391 L
1039 1388 L
1012 1384 L
985 1379 L
959 1373 L
932 1362 L
917 1343 L
919 1312 L
1788 324 M 1761 349 L
1734 375 L
1707 394 L
1681 413 L
1654 430 L
1627 447 L
1600 464 L
1574 480 L
1547 496 L
1520 512 L
1493 527 L
1467 544 L
1440 559 L
1413 576 L
1386 597 L
1360 622 L
1333 648 L
1306 670 L
1280 691 L
1253 712 L
1226 737 L
1199 761 L
1173 784 L
1146 807 L
1119 829 L
1092 852 L
1066 874 L
1039 896 L
1012 918 L
985 940 L
959 962 L
932 984 L
905 1006 L
878 1028 L
852 1049 L
825 1071 L
798 1093 L
771 1114 L
745 1135 L
718 1156 L
691 1176 L
664 1196 L
638 1215 L
611 1234 L
584 1251 L
557 1270 L
531 1290 L
504 1309 L
477 1329 L
450 1350 L
477 1406 L
504 1419 L
531 1430 L
557 1444 L
579 1465 L
557 1486 L
531 1510 L
557 1550 L
584 1552 L
CS M
611 1558 L
638 1561 L
664 1562 L
691 1562 L
718 1563 L
745 1564 L
771 1565 L
798 1566 L
825 1567 L
852 1568 L
878 1569 L
905 1570 L
932 1571 L
959 1572 L
985 1574 L
1012 1576 L
1039 1583 L
1066 1592 L
1092 1600 L
1119 1608 L
1146 1615 L
1173 1621 L
1199 1625 L
1226 1629 L
1253 1631 L
1280 1634 L
1306 1636 L
1333 1640 L
1360 1644 L
1386 1649 L
1413 1659 L
1440 1670 L
1467 1693 L
1493 1739 L
1493 1739 L
1520 1770 L
1547 1796 L
1574 1811 L
1600 1825 L
1627 1837 L
1654 1850 L
1681 1863 L
1707 1878 L
1734 1898 L
1761 1928 L
1774 1952 L
1788 2006 L
1789 2013 L
1793 2044 L
1794 2074 L
1796 2105 L
1798 2135 L
517 687 M 638 687 L
CS [] 0 setdash M
678 687 M CS [] 0 setdash M
678 670 M 705 700 M 706 705 L
706 695 L
705 700 L
701 703 L
696 705 L
691 705 L
686 703 L
683 700 L
683 697 L
684 693 L
686 692 L
689 690 L
699 687 L
703 685 L
706 682 L
683 697 M 686 693 L
689 692 L
699 688 L
703 687 L
705 685 L
706 682 L
706 675 L
703 672 L
698 670 L
693 670 L
688 672 L
684 675 L
683 680 L
683 670 L
684 675 L
719 690 M 719 688 L
718 688 L
718 690 L
719 692 L
723 693 L
729 693 L
733 692 L
734 690 L
736 687 L
736 675 L
738 672 L
739 670 L
734 690 M 734 675 L
736 672 L
739 670 L
741 670 L
734 687 M 733 685 L
723 683 L
718 682 L
716 678 L
716 675 L
718 672 L
723 670 L
728 670 L
731 672 L
734 675 L
723 683 M 719 682 L
718 678 L
718 675 L
719 672 L
723 670 L
758 693 M 754 692 L
753 690 L
751 687 L
751 683 L
753 680 L
754 678 L
758 677 L
761 677 L
764 678 L
766 680 L
768 683 L
768 687 L
766 690 L
764 692 L
761 693 L
758 693 L
754 692 M 753 688 L
753 682 L
754 678 L
764 678 M 766 682 L
766 688 L
764 692 L
766 690 M 768 692 L
771 693 L
771 692 L
768 692 L
753 680 M 751 678 L
749 675 L
749 673 L
751 670 L
756 668 L
764 668 L
769 667 L
771 665 L
749 673 M 751 672 L
756 670 L
764 670 L
769 668 L
771 665 L
771 663 L
769 660 L
764 658 L
754 658 L
749 660 L
748 663 L
748 665 L
749 668 L
754 670 L
783 683 M 803 683 L
803 687 L
801 690 L
799 692 L
796 693 L
791 693 L
786 692 L
783 688 L
781 683 L
781 680 L
783 675 L
786 672 L
791 670 L
794 670 L
799 672 L
803 675 L
801 683 M 801 688 L
799 692 L
791 693 M 788 692 L
784 688 L
783 683 L
783 680 L
784 675 L
788 672 L
791 670 L
869 692 M 868 690 L
869 688 L
871 690 L
871 692 L
869 693 L
868 693 L
866 692 L
864 688 L
861 680 L
858 675 L
854 672 L
851 670 L
846 670 L
841 672 L
839 675 L
839 680 L
841 683 L
851 690 L
854 693 L
856 697 L
856 700 L
854 703 L
851 705 L
848 703 L
846 700 L
846 697 L
848 692 L
851 687 L
860 675 L
863 672 L
868 670 L
869 670 L
871 672 L
871 673 L
846 670 M 843 672 L
841 675 L
841 680 L
843 683 L
846 687 L
846 697 M 848 693 L
861 675 L
864 672 L
868 670 L
931 700 M 933 695 L
933 705 L
931 700 L
928 703 L
923 705 L
920 705 L
915 703 L
911 700 L
910 697 L
908 692 L
908 683 L
910 678 L
911 675 L
915 672 L
920 670 L
923 670 L
928 672 L
931 675 L
920 705 M 916 703 L
913 700 L
911 697 L
910 692 L
910 683 L
911 678 L
913 675 L
916 672 L
920 670 L
931 683 M 931 670 L
933 683 M 933 670 L
926 683 M 938 683 L
950 690 M 950 688 L
948 688 L
948 690 L
950 692 L
953 693 L
960 693 L
963 692 L
965 690 L
966 687 L
966 675 L
968 672 L
970 670 L
965 690 M 965 675 L
966 672 L
970 670 L
971 670 L
965 687 M 963 685 L
953 683 L
948 682 L
946 678 L
946 675 L
948 672 L
953 670 L
958 670 L
961 672 L
965 675 L
953 683 M 950 682 L
948 678 L
948 675 L
950 672 L
953 670 L
983 705 M 983 670 L
985 705 M 985 670 L
978 705 M 985 705 L
978 670 M 990 670 L
1001 705 M 1001 670 L
1003 705 M 1003 670 L
996 705 M 1003 705 L
996 670 M 1008 670 L
1018 683 M 1038 683 L
CS M
1038 687 L
1036 690 L
1035 692 L
1031 693 L
1026 693 L
1021 692 L
1018 688 L
1016 683 L
1016 680 L
1018 675 L
1021 672 L
1026 670 L
1030 670 L
1035 672 L
1038 675 L
1036 683 M 1036 688 L
1035 692 L
1026 693 M 1023 692 L
1020 688 L
1018 683 L
1018 680 L
1020 675 L
1023 672 L
1026 670 L
1050 693 M 1068 670 L
1051 693 M 1070 670 L
1070 693 M 1050 670 L
1046 693 M 1056 693 L
1063 693 M 1073 693 L
1046 670 M 1056 670 L
1063 670 M 1073 670 L
CS [] 0 setdash M
CS [6 12 32 12] 0 setdash M
 0 255 255 C
2002 1337 M 1986 1332 L
1976 1329 L
1952 1319 L
1951 1319 L
1925 1309 L
1913 1307 L
1899 1304 L
1873 1301 L
1847 1300 L
1821 1300 L
1796 1302 L
1770 1304 L
1744 1306 L
1743 1307 L
1718 1309 L
1692 1312 L
1666 1315 L
1641 1319 L
1641 1320 L
1615 1323 L
1589 1328 L
1570 1332 L
1563 1333 L
1537 1339 L
1518 1345 L
1511 1347 L
1485 1354 L
1475 1358 L
1460 1363 L
1440 1370 L
1434 1373 L
1411 1383 L
1408 1385 L
1387 1396 L
1382 1399 L
1368 1409 L
1357 1422 L
1356 1423 L
1350 1434 L
1348 1447 L
1350 1460 L
1355 1473 L
1356 1476 L
1361 1486 L
1362 1498 L
1356 1511 L
1356 1511 L
1356 1511 L
1366 1524 L
1382 1529 L
1408 1534 L
1423 1537 L
1434 1538 L
1460 1541 L
1485 1543 L
1511 1545 L
1537 1546 L
1563 1547 L
1589 1546 L
1615 1545 L
1641 1543 L
1666 1538 L
1672 1537 L
1685 1524 L
1677 1511 L
1692 1500 L
1695 1498 L
1718 1496 L
1744 1494 L
1770 1492 L
1796 1489 L
1821 1486 L
1847 1492 L
1859 1498 L
1873 1501 L
1899 1500 L
1905 1498 L
1925 1493 L
1942 1486 L
1951 1481 L
1962 1473 L
1976 1460 L
1977 1460 L
1986 1447 L
1993 1434 L
1999 1422 L
2002 1416 L
2009 1386 L
2002 1337 L
517 413 M 638 413 L
CS [] 0 setdash M
678 413 M CS [] 0 setdash M
678 396 M 686 431 M 686 396 L
688 431 M 688 396 L
698 421 M 698 407 L
681 431 M 708 431 L
708 421 L
706 431 L
688 414 M 698 414 L
681 396 M 708 396 L
708 406 L
706 396 L
719 419 M 738 396 L
721 419 M 740 396 L
740 419 M 719 396 L
716 419 M 726 419 L
733 419 M 743 419 L
716 396 M 726 396 L
733 396 M 743 396 L
771 414 M 769 412 L
771 411 L
773 412 L
773 414 L
769 417 L
766 419 L
761 419 L
756 417 L
753 414 L
751 409 L
751 406 L
753 401 L
756 397 L
761 396 L
764 396 L
769 397 L
773 401 L
761 419 M 758 417 L
754 414 L
753 409 L
753 406 L
754 401 L
758 397 L
761 396 L
786 431 M 786 396 L
788 431 M 788 396 L
781 431 M 788 431 L
781 396 M 793 396 L
804 419 M 804 401 L
806 397 L
811 396 L
814 396 L
819 397 L
823 401 L
806 419 M 806 401 L
808 397 L
811 396 L
823 419 M 823 396 L
824 419 M 824 396 L
799 419 M 806 419 L
818 419 M 824 419 L
823 396 M 829 396 L
858 431 M 858 396 L
860 431 M 860 396 L
858 414 M 854 417 L
851 419 L
848 419 L
843 417 L
839 414 L
838 409 L
838 406 L
839 401 L
843 397 L
848 396 L
851 396 L
854 397 L
858 401 L
848 419 M 844 417 L
841 414 L
839 409 L
839 406 L
841 401 L
844 397 L
848 396 L
853 431 M 860 431 L
858 396 M 864 396 L
874 409 M 895 409 L
895 412 L
893 416 L
891 417 L
888 419 L
883 419 L
878 417 L
874 414 L
873 409 L
873 406 L
874 401 L
878 397 L
883 396 L
886 396 L
891 397 L
895 401 L
893 409 M 893 414 L
891 417 L
883 419 M 879 417 L
876 414 L
874 409 L
874 406 L
876 401 L
879 397 L
883 396 L
924 431 M 924 396 L
926 431 M 926 396 L
924 414 M 921 417 L
918 419 L
914 419 L
909 417 L
906 414 L
904 409 L
904 406 L
906 401 L
909 397 L
914 396 L
918 396 L
921 397 L
924 401 L
914 419 M 911 417 L
908 414 L
906 409 L
906 406 L
908 401 L
911 397 L
914 396 L
919 431 M 926 431 L
924 396 M 931 396 L
988 419 M 986 414 L
983 411 L
978 409 L
976 409 L
971 411 L
968 414 L
966 419 L
966 421 L
968 426 L
971 429 L
976 431 L
979 431 L
985 429 L
988 426 L
989 421 L
989 411 L
988 404 L
986 401 L
983 397 L
978 396 L
973 396 L
969 397 L
968 401 L
968 402 L
969 404 L
971 402 L
969 401 L
976 409 M 973 411 L
969 414 L
968 419 L
968 421 L
969 426 L
973 429 L
976 431 L
979 431 M 983 429 L
986 426 L
988 421 L
988 411 L
986 404 L
985 401 L
981 397 L
978 396 L
1009 431 M 1004 429 L
1001 424 L
999 416 L
999 411 L
1001 402 L
CS M
1004 397 L
1009 396 L
1013 396 L
1018 397 L
1021 402 L
1023 411 L
1023 416 L
1021 424 L
1018 429 L
1013 431 L
1009 431 L
1006 429 L
1004 427 L
1003 424 L
1001 416 L
1001 411 L
1003 402 L
1004 399 L
1006 397 L
1009 396 L
1013 396 M 1016 397 L
1018 399 L
1020 402 L
1021 411 L
1021 416 L
1020 424 L
1018 427 L
1016 429 L
1013 431 L
1063 431 M 1033 396 L
1041 431 M 1044 427 L
1044 424 L
1043 421 L
1040 419 L
1036 419 L
1033 422 L
1033 426 L
1034 429 L
1038 431 L
1041 431 L
1044 429 L
1050 427 L
1055 427 L
1059 429 L
1063 431 L
1056 407 M 1053 406 L
1051 402 L
1051 399 L
1055 396 L
1058 396 L
1061 397 L
1063 401 L
1063 404 L
1059 407 L
1056 407 L
1123 426 M 1125 421 L
1125 431 L
1123 426 L
1120 429 L
1115 431 L
1111 431 L
1106 429 L
1103 426 L
1101 422 L
1100 417 L
1100 409 L
1101 404 L
1103 401 L
1106 397 L
1111 396 L
1115 396 L
1120 397 L
1123 401 L
1125 404 L
1111 431 M 1108 429 L
1105 426 L
1103 422 L
1101 417 L
1101 409 L
1103 404 L
1105 401 L
1108 397 L
1111 396 L
1138 399 M 1136 397 L
1138 396 L
1140 397 L
1138 399 L
1155 431 M 1155 396 L
1156 431 M 1156 396 L
1149 431 M 1161 431 L
1149 396 M 1175 396 L
1175 406 L
1173 396 L
1185 399 M 1183 397 L
1185 396 L
1186 397 L
1185 399 L
CS [] 0 setdash M
CS [] 0 setdash M
 0 255 0 C
754 1831 M 771 1808 L
798 1790 L
825 1774 L
852 1758 L
878 1742 L
905 1726 L
932 1709 L
959 1692 L
985 1675 L
1012 1657 L
1039 1639 L
1066 1620 L
1092 1601 L
1119 1582 L
1146 1563 L
1173 1544 L
1199 1525 L
1226 1507 L
1253 1489 L
1280 1471 L
1306 1454 L
1333 1438 L
1360 1422 L
1386 1403 L
1413 1377 L
1440 1345 L
1467 1317 L
1493 1292 L
1520 1268 L
1547 1245 L
1574 1222 L
1600 1200 L
1627 1178 L
1654 1155 L
1681 1133 L
1707 1109 L
1734 1079 L
1761 1058 L
1788 1042 L
1814 1030 L
1841 1020 L
1868 1012 L
1895 1009 L
1921 1016 L
1935 1038 L
1937 1068 L
1935 1099 L
1931 1129 L
1925 1160 L
1921 1173 L
1906 1221 L
1895 1248 L
1871 1282 L
1868 1286 L
1841 1308 L
1814 1321 L
1788 1331 L
1761 1339 L
1734 1347 L
1707 1353 L
1681 1360 L
1654 1367 L
1627 1375 L
1600 1384 L
1574 1394 L
1574 1437 L
1600 1445 L
1627 1450 L
1654 1453 L
1681 1456 L
1707 1458 L
1734 1459 L
1761 1461 L
1788 1463 L
1814 1466 L
1841 1473 L
1868 1507 L
1895 1546 L
1900 1556 L
1921 1587 L
1929 1617 L
1931 1648 L
1931 1678 L
1931 1709 L
1925 1739 L
1921 1750 L
1895 1787 L
1868 1803 L
1841 1812 L
1814 1819 L
1788 1824 L
1761 1827 L
1734 1830 L
1707 1833 L
1681 1835 L
1654 1837 L
1627 1838 L
1600 1840 L
1574 1841 L
1547 1842 L
1520 1843 L
1493 1844 L
1467 1845 L
1440 1846 L
1413 1846 L
1386 1847 L
1360 1848 L
1333 1848 L
1306 1849 L
1280 1849 L
1253 1850 L
1226 1850 L
1199 1850 L
1173 1851 L
1146 1851 L
1119 1851 L
1092 1852 L
1066 1852 L
1039 1852 L
1012 1852 L
985 1853 L
959 1853 L
932 1853 L
905 1853 L
878 1853 L
852 1852 L
825 1851 L
798 1848 L
771 1837 L
754 1831 L
CS [] 0 setdash M
 0 255 0 C
450 1896 M 477 1898 L
504 1900 L
531 1901 L
557 1904 L
584 1905 L
611 1905 L
638 1906 L
664 1906 L
691 1906 L
718 1907 L
745 1907 L
771 1907 L
798 1907 L
825 1908 L
852 1908 L
878 1908 L
905 1908 L
932 1908 L
959 1908 L
985 1908 L
1012 1909 L
1039 1909 L
1066 1909 L
1092 1909 L
1119 1910 L
1146 1910 L
1173 1911 L
1199 1911 L
1226 1911 L
1253 1912 L
1280 1913 L
1306 1914 L
1333 1914 L
1360 1916 L
1386 1917 L
1413 1918 L
1440 1919 L
1467 1921 L
1493 1924 L
1520 1926 L
1547 1930 L
1574 1933 L
1600 1938 L
1627 1943 L
1654 1950 L
1681 1958 L
1707 1969 L
1734 1983 L
1761 2002 L
1788 2032 L
1796 2044 L
1814 2082 L
1823 2105 L
1831 2135 L
450 1761 M 477 1746 L
504 1730 L
531 1714 L
557 1697 L
584 1680 L
611 1663 L
638 1645 L
664 1627 L
691 1608 L
718 1589 L
745 1570 L
771 1549 L
798 1528 L
825 1507 L
852 1486 L
878 1465 L
905 1442 L
932 1420 L
959 1398 L
CS M
985 1377 L
1012 1356 L
1039 1335 L
1066 1312 L
1092 1291 L
1119 1269 L
1146 1248 L
1173 1225 L
1199 1202 L
1226 1179 L
1253 1156 L
1280 1131 L
1306 1109 L
1333 1080 L
1360 1050 L
1386 1029 L
1413 1011 L
1440 995 L
1467 979 L
1493 964 L
1520 948 L
1547 933 L
1574 918 L
1600 903 L
1627 887 L
1654 870 L
1681 854 L
1707 836 L
1734 817 L
1761 794 L
1788 768 L
1789 764 L
1814 718 L
1818 703 L
1824 672 L
1831 642 L
1834 611 L
1841 566 L
1843 550 L
1845 520 L
1846 489 L
1848 459 L
1849 428 L
1851 398 L
1851 367 L
1851 337 L
517 505 M 638 505 L
CS [] 0 setdash M
678 505 M CS [] 0 setdash M
678 487 M 686 522 M 686 487 L
688 522 M 688 487 L
709 522 M 688 500 L
696 507 M 709 487 L
694 507 M 708 487 L
681 522 M 693 522 L
703 522 M 713 522 L
681 487 M 693 487 L
703 487 M 713 487 L
723 522 M 723 487 L
724 522 M 724 487 L
718 522 M 729 522 L
718 487 M 729 487 L
741 522 M 741 487 L
743 522 M 743 487 L
736 522 M 748 522 L
736 487 M 748 487 L
813 509 M 811 507 L
813 505 L
814 507 L
814 509 L
813 510 L
811 510 L
809 509 L
808 505 L
805 497 L
801 492 L
798 489 L
794 487 L
789 487 L
784 489 L
783 492 L
783 497 L
784 500 L
794 507 L
798 510 L
799 514 L
799 517 L
798 520 L
794 522 L
791 520 L
789 517 L
789 514 L
791 509 L
794 504 L
803 492 L
806 489 L
811 487 L
813 487 L
814 489 L
814 490 L
789 487 M 786 489 L
784 492 L
784 497 L
786 500 L
789 504 L
789 514 M 791 510 L
805 492 L
808 489 L
811 487 L
855 522 M 855 487 L
856 522 M 856 487 L
850 522 M 861 522 L
850 487 M 861 487 L
873 522 M 873 487 L
875 522 M 875 487 L
868 522 M 879 522 L
868 487 M 879 487 L
891 522 M 891 487 L
893 522 M 893 487 L
886 522 M 898 522 L
886 487 M 898 487 L
CS [] 0 setdash M
 255 255 0 C
CS M 12 setlinewidth
/P { moveto 0 12.05 rlineto stroke } def
CS [] 0 setdash M
477 1047 M 497 1047 L
517 1047 L
537 1047 L
557 1047 L
578 1048 L
598 1048 L
618 1048 L
638 1048 L
658 1048 L
678 1048 L
698 1048 L
718 1048 L
738 1048 L
758 1048 L
778 1048 L
798 1048 L
818 1048 L
838 1048 L
858 1048 L
878 1049 L
898 1049 L
918 1049 L
939 1049 L
959 1049 L
979 1049 L
999 1049 L
1019 1049 L
1039 1049 L
1059 1049 L
1079 1049 L
1099 1049 L
1119 1049 L
1139 1049 L
1159 1049 L
1179 1049 L
1199 1049 L
1219 1049 L
1239 1049 L
1259 1049 L
1280 1049 L
1300 1050 L
1320 1050 L
1340 1050 L
1360 1050 L
1380 1050 L
1400 1051 L
1420 1051 L
1440 1051 L
1460 1052 L
1480 1052 L
1500 1052 L
1520 1053 L
1540 1053 L
1560 1054 L
1580 1054 L
1600 1055 L
1621 1055 L
1641 1056 L
1661 1056 L
1681 1057 L
1701 1057 L
1721 1058 L
1741 1059 L
1761 1060 L
1781 1062 L
1801 1064 L
1821 1067 L
1841 1071 L
1861 1075 L
1881 1079 L
1901 1085 L
1921 1091 L
1941 1096 L
1961 1104 L
1981 1118 L
2002 1141 L
2022 1170 L
2042 1202 L
603 1078 M CS M 6 setlinewidth
/P { moveto 0 6.05 rlineto stroke } def
CS [] 0 setdash M
603 1078 M 621 1119 M 618 1116 L
615 1111 L
611 1104 L
610 1096 L
610 1089 L
611 1081 L
615 1074 L
618 1069 L
621 1066 L
618 1116 M 615 1109 L
613 1104 L
611 1096 L
611 1089 L
613 1081 L
615 1076 L
618 1069 L
635 1098 M 635 1096 L
633 1096 L
633 1098 L
635 1099 L
638 1101 L
645 1101 L
648 1099 L
650 1098 L
651 1094 L
651 1083 L
653 1079 L
655 1078 L
650 1098 M 650 1083 L
651 1079 L
655 1078 L
656 1078 L
650 1094 M 648 1093 L
638 1091 L
633 1089 L
631 1086 L
631 1083 L
633 1079 L
638 1078 L
643 1078 L
646 1079 L
650 1083 L
638 1091 M 635 1089 L
633 1086 L
633 1083 L
635 1079 L
638 1078 L
665 1119 M 668 1116 L
671 1111 L
675 1104 L
676 1096 L
676 1089 L
675 1081 L
671 1074 L
668 1069 L
665 1066 L
668 1116 M 671 1109 L
673 1104 L
675 1096 L
675 1089 L
673 1081 L
671 1076 L
668 1069 L
CS [] 0 setdash M
CS M 12 setlinewidth
/P { moveto 0 12.05 rlineto stroke } def
CS [] 0 setdash M
736 1482 M 744 1482 L
752 1482 L
760 1482 L
768 1482 L
776 1482 L
784 1482 L
792 1482 L
800 1482 L
809 1482 L
817 1482 L
825 1482 L
833 1482 L
841 1482 L
849 1482 L
857 1482 L
865 1482 L
873 1482 L
881 1482 L
889 1482 L
897 1482 L
905 1482 L
913 1482 L
921 1482 L
929 1482 L
937 1482 L
945 1482 L
953 1482 L
961 1482 L
969 1482 L
977 1482 L
985 1482 L
993 1482 L
1001 1483 L
1009 1483 L
1017 1483 L
1025 1483 L
1033 1483 L
1041 1483 L
1049 1483 L
1057 1484 L
CS M
1065 1484 L
1073 1484 L
1081 1484 L
1089 1484 L
1097 1485 L
1105 1485 L
1113 1485 L
1121 1485 L
1129 1485 L
1137 1486 L
1145 1486 L
1154 1486 L
1162 1486 L
1170 1486 L
1178 1486 L
1186 1486 L
1194 1486 L
1202 1486 L
1210 1487 L
1218 1487 L
1226 1487 L
1234 1487 L
1242 1487 L
1250 1486 L
1258 1486 L
1266 1486 L
1274 1486 L
1282 1486 L
1290 1486 L
1298 1486 L
1306 1485 L
1314 1485 L
1322 1485 L
1330 1485 L
1338 1484 L
1346 1484 L
1354 1483 L
1362 1483 L
1370 1483 L
1378 1482 L
1386 1482 L
1394 1481 L
1402 1481 L
1410 1480 L
1418 1480 L
1426 1480 L
1434 1479 L
1442 1479 L
1450 1479 L
1458 1479 L
1466 1478 L
1474 1478 L
1482 1478 L
1490 1478 L
1499 1478 L
1507 1478 L
1515 1478 L
1523 1479 L
1531 1479 L
1539 1479 L
1547 1480 L
1555 1480 L
1563 1481 L
1571 1482 L
1579 1483 L
1587 1484 L
1595 1485 L
1603 1486 L
1611 1487 L
1619 1489 L
1627 1490 L
1635 1492 L
1643 1493 L
1651 1494 L
1659 1494 L
1667 1494 L
1675 1493 L
1683 1490 L
1691 1487 L
1699 1483 L
1707 1479 L
1715 1473 L
1723 1467 L
1731 1460 L
1739 1452 L
1747 1444 L
1755 1435 L
1763 1426 L
1771 1416 L
1779 1406 L
1787 1395 L
1795 1385 L
1803 1373 L
1811 1362 L
1819 1350 L
1827 1339 L
1835 1327 L
736 1421 M CS M 6 setlinewidth
/P { moveto 0 6.05 rlineto stroke } def
CS [] 0 setdash M
736 1421 M 755 1463 M 751 1459 L
748 1454 L
745 1448 L
743 1439 L
743 1433 L
745 1424 L
748 1418 L
751 1413 L
755 1409 L
751 1459 M 748 1453 L
746 1448 L
745 1439 L
745 1433 L
746 1424 L
748 1419 L
751 1413 L
768 1456 M 768 1421 L
770 1456 M 770 1421 L
770 1439 M 773 1443 L
776 1444 L
780 1444 L
785 1443 L
788 1439 L
790 1434 L
790 1431 L
788 1426 L
785 1423 L
780 1421 L
776 1421 L
773 1423 L
770 1426 L
780 1444 M 783 1443 L
786 1439 L
788 1434 L
788 1431 L
786 1426 L
783 1423 L
780 1421 L
763 1456 M 770 1456 L
800 1463 M 803 1459 L
806 1454 L
810 1448 L
811 1439 L
811 1433 L
810 1424 L
806 1418 L
803 1413 L
800 1409 L
803 1459 M 806 1453 L
808 1448 L
810 1439 L
810 1433 L
808 1424 L
806 1419 L
803 1413 L
CS [] 0 setdash M
CS M 12 setlinewidth
/P { moveto 0 12.05 rlineto stroke } def
CS [] 0 setdash M
483 2151 M 578 2151 L
657 2151 L
748 2151 L
820 2151 L
870 2151 L
1135 2151 L
1360 2151 L
1479 2151 L
1573 2156 L
CS M 6 setlinewidth
/P { moveto 0 6.05 rlineto stroke } def
578 2181 M CS [] 0 setdash M
578 2181 M 597 2223 M 593 2219 L
590 2214 L
587 2208 L
585 2199 L
585 2193 L
587 2184 L
590 2178 L
593 2173 L
597 2169 L
593 2219 M 590 2213 L
588 2208 L
587 2199 L
587 2193 L
588 2184 L
590 2179 L
593 2173 L
627 2199 M 625 2198 L
627 2196 L
628 2198 L
628 2199 L
625 2203 L
622 2204 L
617 2204 L
612 2203 L
608 2199 L
607 2194 L
607 2191 L
608 2186 L
612 2183 L
617 2181 L
620 2181 L
625 2183 L
628 2186 L
617 2204 M 613 2203 L
610 2199 L
608 2194 L
608 2191 L
610 2186 L
613 2183 L
617 2181 L
638 2223 M 642 2219 L
645 2214 L
648 2208 L
650 2199 L
650 2193 L
648 2184 L
645 2178 L
642 2173 L
638 2169 L
642 2219 M 645 2213 L
647 2208 L
648 2199 L
648 2193 L
647 2184 L
645 2179 L
642 2173 L
CS [] 0 setdash M
CS M 6 setlinewidth
/P { moveto 0 6.05 rlineto stroke } def
CS [] 0 setdash M
1099 150 M 1130 179 M 1132 184 L
1132 174 L
1130 179 L
1127 182 L
1123 184 L
1113 184 L
1108 182 L
1106 179 L
1106 174 L
1108 172 L
1113 170 L
1125 165 L
1130 162 L
1132 160 L
1106 177 M 1108 174 L
1113 172 L
1125 167 L
1130 165 L
1132 162 L
1132 155 L
1130 153 L
1125 150 L
1115 150 L
1111 153 L
1108 155 L
1106 160 L
1106 150 L
1108 155 L
1152 201 M 1149 199 L
1152 196 L
1154 199 L
1152 201 L
1152 184 M 1152 150 L
1154 184 M 1154 150 L
1144 184 M 1154 184 L
1144 150 M 1161 150 L
1178 184 M 1178 150 L
1180 184 M 1180 150 L
1180 177 M 1185 182 L
1192 184 L
1197 184 L
1204 182 L
1207 177 L
1207 150 L
1197 184 M 1202 182 L
1204 177 L
1204 150 L
1171 184 M 1180 184 L
1171 150 M 1188 150 L
1197 150 M 1214 150 L
1225 213 M 1226 212 L
1225 211 L
1223 212 L
1223 213 L
1225 216 L
1226 218 L
1230 219 L
1236 219 L
1241 218 L
1242 216 L
1243 213 L
1243 211 L
1242 208 L
1238 205 L
1230 202 L
1228 200 L
1225 198 L
1223 193 L
1223 189 L
1236 219 M 1239 218 L
1241 216 L
1242 213 L
1242 211 L
1241 208 L
1236 205 L
1230 202 L
1223 192 M 1225 193 L
1228 193 L
1235 190 L
1239 190 L
1242 192 L
1243 193 L
1228 193 M 1235 189 L
CS M
1241 189 L
1242 190 L
1243 193 L
1243 196 L
1296 191 M 1298 189 L
1296 187 L
1294 189 L
1294 191 L
1296 196 L
1298 199 L
1306 201 L
1315 201 L
1322 199 L
1325 196 L
1327 191 L
1327 187 L
1325 182 L
1318 177 L
1306 172 L
1301 170 L
1296 165 L
1294 158 L
1294 150 L
1315 201 M 1320 199 L
1322 196 L
1325 191 L
1325 187 L
1322 182 L
1315 177 L
1306 172 L
1294 155 M 1296 158 L
1301 158 L
1313 153 L
1320 153 L
1325 155 L
1327 158 L
1301 158 M 1313 150 L
1322 150 L
1325 153 L
1327 158 L
1327 162 L
1361 201 M 1354 199 L
1349 191 L
1347 187 L
1344 179 L
1342 167 L
1342 158 L
1344 153 L
1349 150 L
1354 150 L
1361 153 L
1366 160 L
1368 165 L
1371 172 L
1373 184 L
1373 194 L
1371 199 L
1366 201 L
1361 201 L
1356 199 L
1351 191 L
1349 187 L
1347 179 L
1344 167 L
1344 158 L
1347 153 L
1349 150 L
1354 150 M 1359 153 L
1363 160 L
1366 165 L
1368 172 L
1371 184 L
1371 194 L
1368 199 L
1366 201 L
1347 177 M 1368 177 L
CS [] 0 setdash M
CS [] 0 setdash M
287 1234 M 237 1258 M 287 1239 L
237 1258 M 287 1277 L
244 1258 M 287 1275 L
285 1241 M 285 1275 L
287 1239 M 287 1277 L
254 1294 M 287 1294 L
254 1297 M 287 1297 L
261 1297 M 256 1301 L
254 1309 L
254 1314 L
256 1321 L
261 1323 L
287 1323 L
254 1314 M 256 1318 L
261 1321 L
287 1321 L
261 1323 M 256 1328 L
254 1335 L
254 1340 L
256 1347 L
261 1350 L
287 1350 L
254 1340 M 256 1345 L
261 1347 L
287 1347 L
254 1287 M 254 1297 L
287 1287 M 287 1304 L
287 1314 M 287 1330 L
287 1340 M 287 1357 L
224 1367 M 226 1369 L
227 1367 L
226 1366 L
224 1366 L
221 1367 L
220 1369 L
218 1373 L
218 1379 L
220 1383 L
221 1385 L
224 1386 L
227 1386 L
230 1385 L
233 1380 L
236 1373 L
237 1370 L
240 1367 L
244 1366 L
249 1366 L
218 1379 M 220 1382 L
221 1383 L
224 1385 L
227 1385 L
230 1383 L
233 1379 L
236 1373 L
246 1366 M 244 1367 L
244 1370 L
247 1377 L
247 1382 L
246 1385 L
244 1386 L
244 1370 M 249 1377 L
249 1383 L
247 1385 L
244 1386 L
242 1386 L
CS [] 0 setdash M
stroke
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