%Paper: 
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%%%%%%%%%%%% Level-Crossing in the Instanton-Anti-Instanton Valley
%%%%%%%%%%%% by  R. Guida, K. Konishi and  N. Magnoli
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\begin{document}

\begin{titlepage}
\begin{center}
{\Large
Level-Crossing in the Instanton-Anti-Instanton Valley}
\end{center}

\vspace{1em}
\begin{center}
{\large
Riccardo Guida, Kenichi Konishi and Nicodemo Magnoli }
\end{center}

\vspace{1em}
\begin{center}
{\it Dipartimento di Fisica -- Universit\`a di Genova\\
     Istituto Nazionale di Fisica Nucleare -- sez. di Genova\\
     Via Dodecaneso, 33 -- 16146 Genova (Italy)\\
     E-mail: Decnet 32655; Bitnet KONISHI@GENOVA.INFN.IT\\}
\end{center}

\vspace{7em}
{\bf ABSTRACT:}
We study the level crossing of the fermion system described by the
 euclidean Dirac Hamiltonian in the valley background.
 One chiral fermion level is shown to cross twice the zero value in the
case  of
well-separated  instanton-anti-instanton background.
 Below a critical separation, however, level crossings
are  absent.  The phenomenon can be interpreted
as the transition  to a gauge field configuration of purely perturbative
nature,
 below a  critical instanton-anti-instanton
separation.   In the context of high-energy
electroweak interactions,  our findings seem to definitely invalidate
some  optimistic
argument concerning the  observability of baryon number violation
 based on the use of the optical theorem in conjunction with
the valley fields.
\vspace{2em}
\begin{flushleft}
GEF-Th-14/1993~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
{}~~~~~~~~~ June 1993 \end{flushleft}
\end{titlepage}

%%%%%%%%%%%%%%%% latex definitions
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\newcommand{\eeq}{\end{equation}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% definitions

\def\dirac{{\cal D}}
\def\dplus{{\cal D_{+}}}
\def\dminus{{\cal D_{-}}}
\def\dbar{\bar{D}}
\def\H{\cal{H}}
\def\de{\partial}
\def\si{\sigma}
\def\sb{{\bar \sigma}}
\def\rn{{\bf R}^n}
\def\r4{{\bf R}^4}
\def\s4{{\bf S}^4}
\def\ker{\hbox{\rm ker}}
\def\dim{\hbox{\rm dim}}
\def\sup{\hbox{\rm sup}}
\def\inf{\hbox{\rm inf}}
\def\infi{\infty}
\def\nrm{\parallel}
\def\nrmi{\parallel_\infty}
\def\teo{\noindent{\bf Theorem}\ }
\def\tt{\tilde T}
\def\st{\tilde S}
\def\om{\Omega}
\def\Tr{ \hbox{\rm tr}}

\section{Introduction.}
The purpose of the present paper is to gain a deeper insight in the
behaviour of chiral fermions in non-Abelian gauge theories  in the background
of the so-called valley fields (i.e., a particular class of
gauge backgrounds of instanton-anti-instanton type),
 by analyzing the spectral flow of the
corresponding euclidean Dirac Hamiltonian.

The interest in this issue arose from  the problem of the observability
of the anomaly-induced fermion-number violation at high energies
in the standard electroweak theory,
from the study of the vacuum structure in QCD,
and also from the questions related to the large-order behaviour of
perturbation theory in non-Abelian
gauge theories in general.

The problems involved, in particular the one
 related to unitarity and chiral anomaly,
have been studied by two of the present authors in Ref.\cite{guida}.
 It was shown there
that the
 forward elastic four-fermion amplitude computed in the valley backgrounds
indeed satisfies unitarity by having an anomalous piece, proportional
to the products of the standard fermion zero modes,\footnote{i.e,
of pure instanton or of pure anti-instanton background. Recall that no
four-dimensional zero modes exist in the valley
background.\protect\cite{guida}}
  in spite of the fact that
the relevant gauge field is a topologically (globally) trivial one.
Furthermore, it was found that
 such an anomalous contribution is there as long as the instanton
anti-instanton  ($i - a $) separation is large enough.

On the other hand, at small $i - a $  separation the amplitude reduces to
a perturbative one.  In fact, on the basis of the behaviour of the integrated
topological density,
\beq C(x_4) = - \int_{-\infi}^{x_4}\int d^3x {g^2\over 16 \pi^2}
\Tr\,F_{\mu \nu}{\tilde F}_{\mu \nu} ={\cal N}_{CS}(x_4)- {\cal
N}_{CS}(-\infi),
 \label{top}      \eeq
where
${\cal N}_{CS}(x_4)\equiv - {g^2\over 16 \pi^2}\int d^3x \epsilon^{4ijk}
\Tr (F_{ij} A_k - {2\over 3}A_i A_j A_k) $
 is the Chern-Simons number,
the same authors argued that such a transition to purely perturbative
 amplitude should  takes place abruptly at around the unit $i - a $
separation,  $ R/\rho \simeq 1 $.
 However, the studies
of Ref.\cite{guida}
 left unresolved the question exactly at which value of the $i - a $
separation this occurs, and did not quite
 clarify the nature of this transition.

 In this note we close that gap
 by studying the zero level crossing (spectral flow) of the chiral fermion
system described by the euclidean
 Dirac Hamiltonian in the valley background.
Our numerical analysis clearly
reveals a transition in the behaviour of the spectral flow at
a  critical value of separation, $R = R_c \simeq \sqrt{4\over 3} \rho$,
$\rho$ being the common size of the instantons.
 For $R>R_c$
we find two level crossings,
located near the positions of instanton and anti-instanton,
while for $R<R_c$  no level crosses zero.


\section{Spectral flow}
Let us consider an $SU(2)$ gauge theory with  $N_F$ massless left-handed
 doublets.  We consider in particular the fermions in
a fixed external gauge field,
\bea
A_{\mu}^{(valley)}& =&-{i\over g}(\sigma_{\mu} {\bar \sigma_{\nu}}
-\delta_{\mu \nu} )\,[\, {(x-x_a)_{\nu} \over (x-x_a)^2 +\rho^2 }   \non \\
& &
+
{(x-x_i)_{\nu} \rho^2 \over (x-x_i)^2 ((x-x_i)^2 +\rho^2) }
 + {(x-x_i +y)_{\nu} \over
(x-x_i + y )^2  } -  {(x-x_i)_{\nu} \over
(x-x_i)^2  }\,]
\label{valley} \eea
where
\beq
 y^{\mu} = -R^{\mu}/(z-1);\,\,
 z = (R^2 + 2 \rho^2 + \sqrt{R^4 + 4\rho^2 R^2}) /2\rho^2;\,\,
R^{\mu} = (x_i - x_a)^{\mu}. \label{parameters} \eeq
$x_i^{\mu}$ and $x_a^{\mu}$ are the centers of the instanton and
 anti-instanton, $\rho$ is their (common) size.  The gauge fields of Eq.(\ref
{valley})
 has been  termed the valley (or streamline) trajectory
(\cite{Yung}).

$N_F$ must be an even number in order for the theory to be well defined.
  These fermions can be  rearranged (in an $SU(2)$ gauge
theory) to  ${N_F \over 2}$ Dirac
fermions by appropriately combining  pairs of  left-handed fermions and their
righthanded anti-fermions; the resulting Dirac fermions are coupled
 vectorially to the gauge boson. In this case the axial charge will
be anomalous, corresponding to the  fermion number violation
 of the original model.

For simplicity of notation, let us restrict below
 to $N_F=2$, equivalent to the case of a
single Dirac
doublet. The generalisation to the case with $N_F$ chiral fermions is
straightforward.
 The euclidean  Dirac  operator is $i\gamma_\mu D_\mu$,
where $D_\mu=(\de-igA)_\mu$ is the usual covariant derivative.
In the Lorentz gauge
 of Eq.(\ref{valley}) the gauge field can be expressed as:
\beq \label{field}
A_\mu =-{i\over g} (\si _\mu \sb _\nu -\delta_{\mu \nu})
\, F_\nu = \bar{\eta}_{\mu\nu}^a F_\nu \si ^a\eeq
\beq
F_\mu(x)\equiv {1\over 2}\,\de_\mu\log L(x);\;\;\;\;\;\;
L(x) \equiv {(x-x_a)^2+\rho^2\over (x-x_i)^2+\rho^2}\,(x-x_i+y)^2.
\label{elle}\eeq
The instanton and anti-instanton location will be chosen at:
$x_i=(-{R\over 2},0,0,0)$ and $x_a=({R\over 2},0,0,0)$; as a consequence
the vector $y$ above (\ref{parameters}) is in the time direction
$y=(y,0,0,0)$.

Decomposing
$
i\gamma _\mu D_\mu=i\gamma_0 (D_0 + \H)
$
as
$$
\cal{H}= \pmatrix {H_{+} & 0\cr
                 0 &H_{-}},$$
and passing to a noncovariant formalism,
\footnote{Usually the spectral flow is studied in the axial gauge where
$A_0=0$. The Hamiltonian in that gauge
would be related to (\ref{hamiltonian}) by a time-dependent gauge
transformation of spatial components
of the gauge field that however  preserve the spectrum of the Hamiltonian and
correspondingly spectral flow.}
 we find:
\beq
 H_{+}=-H_{-}=H
\label{condition}\eeq
 with
\beq
H=+i\vec{\si}\cdot\vec{\nabla}
-\vec{\si}\wedge \vec{\tau}\cdot\vec{F}-F_0 \vec{\si}\cdot\vec{\tau}
\label{hamiltonian}\eeq
 where
$$x_\mu\equiv(t,\vec{x});\;\;\;\vec{F}=\vec{x}f(r,t);\;\;\; F_0=F_0(r,t).$$
$r=\sqrt{{\vec{x}}^2}$;  $\vec{\si} $ are spin operators, and
$\vec{\tau } $ are isospin one.

Clearly $\H$ is a self-adjoint operator with entirely essential spectrum,
that is,  eventual (normalisable) eigenvalues
are imbedded into the continuum, which
covers  the real axis of energies.

To have a picture of well separated levels, some of which crossing zero
at certain values of the euclidean time\footnote{Note that
 $\H$ depends explicitly on $t$.}, one
must consider the Dirac operator on a compactified space, $S^4$ (sphere)
or $T^4$ (torus), but one must make sure that the compactification preserves
the zero modes present in the continuum theory.
We shall bypass the whole
 problem, just studying the eventual zero modes
of $\H$ for some values of $t$ as the  signal  of level crossing, and
 disregarding the  more complex problem
of the  full analysis of time evolution of levels.

 The  condition (\ref{condition})
implies that zero modes always appear in pair of different chirality.
We assume that for each chirality  zero modes are non degenerate, so that
we must look for singlets of the total angular momentum (which is a symmetry
of $\H$),  $\vec{J}=\vec{x}\wedge -i\vec{\nabla}+\vec{\si}+\vec{\tau}$.
As noted in \cite{guida} the most general form of
the singlet is:
\beq
\eta(\vec{x},t)=\si _2 S(r,t)-i \vec{x}\vec{\si} \si _2 T(r,t).
\eeq


The equation $H\eta =0$ then reads:
\footnote{ We take into account the fact
 that the only spatial vector is $\vec{x}$.}
\bea
3 F_0 S-3T-\vec{x}\vec{\nabla}T+2 \vec{F} \vec{x} T&=&0, \\
-F_0 r^2 T +\vec{x}\vec{\nabla} S+2\vec{F} S&=&0.
\eea

Making the substitution
\beq
S={1\over L}\st, \qquad
T={L\over r^3} \tt,
\eeq
 we get a  simplified system:
\beq \label{system}
\tt '(r;t)=3F_0 {r^2 \over L^{2}} \st ; \qquad
\st '(r;t)=F_0 {L^2\over r^2} \tt
,\eeq
where the prime means differentiation with respect to $r$ and the
time $t$ appears as a
 parameter.
Normalisability of the solution implies:
\beq \label{normalisation}
\infi >\int_0^{\infi}d\!r r^2  |S|^2=\int_0^{\infi}d\!r {r^2\over L^2}
|\st|^2 ;\qquad
\infi >\int_0^{\infi}d\!r r^4 |T|^2=\int_0^{\infi}d\!r {L^2\over r^2}
|\tt|^2.\eeq
Recall that with our choice of parameters we have:
\bea\label{functions}
L &\equiv& {t_a^2+r^2+\rho^2\over t_i^2+r^2+\rho^2}\,(t_y^2+r^2)\\
F_0&\equiv&{t_a\over  t_a^2+r^2+\rho^2}-{t_i\over  t_i^2+r^2+\rho^2}
+{t_y\over  t_y^2+r^2+\rho^2}
\eea
where
$
t_a\equiv t+{R\over 2};\;\;t_i\equiv t-{R\over 2};\;\; t_y\equiv t-
{R\over 2}+y.$
We want to know for which range of the parameter $R$ and for which values
of $t$ - if any - (the other
parameter $\rho$ just fixes the scale) the system Eq.(\ref{system}) has
a normalisable solution.
Clearly, normalisability
enforces the following initial condition (at a given $t$ )
\footnote{Note the qualitative
 change in the behaviour of $L$ which occurs at $t_y=0$.}:
\bea
&&\st (0)=1;\;\; \tt (0)=0;\;\;\hbox{\rm if}\;\; t_y\neq 0\label{initial1}\\
&&\st (0)=0;\;\; \tt (0)=1;\;\;\hbox{\rm if}\;\; t_y = 0. \label{initial2}
\eea


The  problem turns out to be
 too hard  to be
 treated exactly: it just suffices to note that the system is equivalent to
a second order linear equation for $\tt$  with $12$ regular fuchsian points!
We thus proceeded to a numerical method for finding solutions of the problem
(\ref{system}), (\ref{normalisation}) and (\ref{initial1}) or (\ref{initial2}).

In passing we recall \cite{Kiskis}
 that for the case of a pure anti-instanton
(or an instanton)
a normalisable solution can be found easily. For instance,
in the case of an  anti-instanton (centered at $x_{\mu}=0$)
 the gauge field has the
same form as (\ref{field}), with
$
F_\nu={x_\nu /( x^2+\rho^2)}$
so that
\beq F_0={t\over t^2+r^2+\rho^2}, \qquad
     L=t^2+r^2+\rho^2.\eeq
The normalisation condition (\ref{normalisation}) requires in this case
that
\beq
\st (0)=1;\;\; \tt (0)=0.\eeq
It is clear that for $t=0$ (corresponding to the time position of
the anti-instanton)
$F_0=0$ for all $r$ and the system decouples.  It follows
that
$ \st(r)=1;\;\; \tt(r)=0$,\,\, i.e.,
$$S(r,0) = {1\over r^2 + \rho^2};\;\;\;\;T(r,0) = 0 $$
is the desired  normalisable solution.  Furthermore, it can be
shown that $t=0$ is the only value
of the parameter for which such a solution exists.

\section{Numerical solution of the system.}

Solving the system by a power series in $r$ (method of Frobenius), we find
the asymptotic behaviour of two independent solutions of
(\ref{system}), which is reported
\footnote{Disregarding the numerical coefficients.}
 in Table 1 for  $r\sim 0$ and in Table 2 for  $r\rightarrow \infi $.
We treated separately the special cases,
 $t={R\over 2}-y$ (corresponding to $t_y=0$),
 $t = t_0$ (where $t_0$ is by definition a zero of $F_0(r=0)$ ),
$t=t_1$ (where $t_1=-{R\over 2}-y$).

Clearly  the solution satisfying (\ref{initial1}) or
(\ref{initial2}) can be easily selected
near $r=0$; the problem is
how this solution evolves at $r\rightarrow \infi$.
To see this an accurate numerical analysis is required. We have made such a
numerical analysis, passing to a compactified variable
$p=r/(1+r)$, and using as initial conditions at a point very near
\footnote{Note that the coefficients of the system are singular at $r=0$, so
that numerical evolution must start a little after zero, typically $r=10^{-9}$
(in $\rho$ units).}
$p=0$ (i.e. $r=0$) the values of $\st$ and $\tt$ which
are  obtained as a power-series approximation of  the adequate
solution of Table 1. The common normalisation  of $\st$ and $\tt$
  at this point can be
arbitrarily set to unity.
Then we studied whether the solution evolves towards $r = \infi$
in such a way that the normalisation
condition (\ref{normalisation}) is satisfied.


First, we find that for  $t_y=0$ the solution
satisfying the initial condition (\ref{initial2}), is not normalisable.

For  $t_y\neq 0$,  we ask whether  $\tt (r=\infi)=0$,
i.e., whether
 the solution of the initial condition problem is  normalisable.
In Fig. 1a  $\tt (r=\infi)$ is plotted as a  function of $t$  for
 $R=10$: it is seen that the curve intersects  zero twice.
Note that the two values of
 $t$  at which the solution
is normalisable are very close to the (time) positions of the instanton and
anti-instanton.

   The singularity in Fig. 1a is located at $t_y=0$ ($t\sim 5.1$), and is
due to the fact that there are no solutions of the system for this value
of $t$ that
 satisfy the condition (\ref{initial1}):
see Table 1.

%In Figure 2 (and Figure 3) the behaviour in $r$ of solutions corresponding to
%different values of time near the zero crossings is reported ($R=10$).

We find furthermore that the situation is similar for  all values of
 the instanton-anti-instanton separation above a critical value, i.e.,
 $R>R_c$.
(See also Fig. 1b  where $R=1.5$).  The critical separation turns out to be
 \footnote{The value $\sqrt{4\over 3}$ is a
fit to our numerical result.}
\beq{R_c\over \rho} \simeq 1.15470....\simeq \sqrt{{4\over 3}}.\eeq
At   $R= R_c$ the two  level-crossing points  coalesce
into one.

Below the critical distance, $R < R_c$,  on the contrary,
no values of $\,t\,$ are found for
 which  the system Eq.(\ref{system}) has a normalisable solution. (See
Fig. 1c that reports the situation just below the threshold, for $R=1.0$).


For large instanton anti-instanton
 separations, our calculations thus  indicates
 the  presence of level crossings
near the positions of instanton and anti-instanton centers, that seems to
reproduce locally the situation mentioned above
for a single (anti)-instanton field.
This means that the lowest  level $H_{+}$ (in a
 compactified space), for instance,
starts from a  positive value at  $t= -\infi $ and crosses
zero to  negative,   then crosses  zero back to positive, and
returns to the original value at $t =\infi$. A level of opposite chirality
also crosses zero twice but in the specular manner with respect to
the $E=0$ line.

At the separation below the critical value $R_c$ there is  a qualitative
 change of behaviour: no levels cross zero.
 The valley field
thus appears to lose the topologically non-trivial  aspect and degenerate into
a field configuration of purely perturbative nature. (At $R=0$ the valley field
Eq.(\ref{valley}) is indeed a gauge transform of $A_{\mu} = 0 $.) In other
words, the
instanton-anti-instanton pair "melt" at $R\simeq \sqrt{(4/3)}\rho$.

Before closing this section we observe
that the  critical value of $R$, $R = R_c\simeq\sqrt{4\over 3}\rho $
corresponds to the value of the conformally
invariant variable $z$ (see Eq.(\ref{parameters})),
  very close to $z = 3$.   And this corresponds \footnote{We thanks
 P.Provero for pointing  this out to us.} to
 the maximum
of the integrated topological density,
equal to  $C(x_4=0)={1\over 2}$ (see
Eq. (4.3) of  \cite{guida}
 for the explicit expression in terms of $z$).
 The situation is very similar to what happens in the
Schwinger  model on the cylinder (QED on  $R^1 \otimes S^1$)
 (see \cite{shifman}), where the level
crossings disappear at the same value of the analogous topological quantity.
It is interesting that this value of  $C(x_4=0)$
  just corresponds to
the top of the hill separating the two adjacent minima of the gauge
field action.

\section{Discussion}

We  propose here a physical interpretation of the mathematical analysis
made above, on the level crossing of the euclidean Dirac Hamiltonian.

Consider the (euclidean) time evolution of a given state from $t=-\infi$
 to $t=\infi$.  For definiteness first consider  the vacuum-to-vacuum
transition. In the
first quantized
approach, the  wave
function is given   at $t=-\infi$  by the Dirac sea
with all negative energy levels filled.  Then we follow its (euclidean)
 time evolution in the
Schr\" odinger picture.
Suppose \footnote{The adiabatic approximation suppresses the unavoidable
perturbative creation of particle-anti-particle pairs of
same chirality (hence with no net chirality change)
that is present for every time-dependent background,
allowing  us to concentrate on the
 nonperturbative phenomenon related to the level crossings.}
that the
time evolution   is  adiabatic.  This means that an
eigenfunction of $H(t)$ with eigenvalue $E_n(t)$ evolves at a successive
 instant $t'$
into an eigenfunction of $H(t')$
with eigenvalue $E_n(t')$, corresponding to  the original one.
 If one level crosses $0$ during the
 evolution, for example from negative to positive,
 the state we started with will   no longer be  the vacuum of the theory
 at the instant $t'$
but corresponds to the state with one positive energy level filled, that is
to a one-particle state (where "particle" means here "quantum of energy", see
\cite{labonte} for a possible, more accurate, definition).
 Analogous situation for creation of anti-particles,
corresponding to empty negative levels (holes) with respect to the new
 vacuum.

In Section 3 we found that in the valley background with
$R > R_c$ one left level crosses  zero twice (and so does a right level).
 This suggests \footnote{We assume on  physical grounds
 that  in the valley the level crossing near the  instanton location is such
that a left (right) positive (negative) level crosses zero downwards
(upwards) as in the case of  a single instanton field; and an
opposite situation  near the anti-instanton.}
 that during the time evolution every quantum
state acquires an extra lefthanded fermion
and an antiparticle of the righthanded fermion
 near  the anti-instanton position, hence with
the net chirality change, $-2$.\footnote{A more precise statement
is  $\Delta Q_5 \equiv \{Q_R - Q_L\}|_{t_a+} - \{Q_R - Q_L\}|_{t_a-}=-2
 $: whether this means the creation of two lefthanded particles, the
annihilation of two righthanded particles, or e.g., the creation of one left
particle and the annihilation of a right particle, depends on the particular
process considered and is a matter of no importance.}
 An analogous net chirality change $+2$ occurs  near the instanton center.
In the case of vacuum-to-vacuum transition this would mean the propagation
of two lefthanded fermions between
 the instanton centers, signalling the instability of the vacuum.
The vacuum-to-vacuum amplitude would get an imaginary part, corresponding to
 intermediate states with $Q_5 =  2$ or $ Q_5 = -2.$

Although the physical interpretation of the level-crossing similar to this
 is commonly used (\cite{gross},\cite{shifman}),  a rigorous universally
accepted interpretation seems to be lacking in mathematical physics
 literature
\cite{particle}:
we do not pretend to go beyond our simple picture here.


In the case of elastic four fermion amplitude (two fermion-
two fermion transition) as the one considered in Ref.\cite{guida}
(relevant to the problem of high energy fermion number violation in the
electroweak theory),
the presence of the level-crossings for the backgrounds with well-separated
instanton  and anti-instanton ($R>R_c$) precisely corresponds to the
anomalous term in the amplitude,\cite{guida}
  with an imaginary part associated with
 intermediate states without the initial fermions.

The disappearance of the level crossings at and below
the critical $i-a$ separation
$R= R_c \simeq \sqrt{4\over 3} \rho$, on the other hand, means that
 the anomalous,
nonperturbative contribution to the amplitude  is absent in these backgrounds.
This confirms  the idea that the valley background Eq.(\ref{valley}) ceases to
be topologically significant long before $R$ reaches $0$ where $A_{\mu}$ is the
vacuum field.\footnote {As is well known, the
form of the valley field is not unique, and depends on the choice of
the so-called weight function. However they all share the common
property of reducing to the vacuum field at $R=0$. We therefore expect
the presence of a critical separation for any such field, below which
the level crossings disappear.}


It is important, though it might appear surprising at first sight,
 that the gauge background,
Eq.(\ref{valley}),
which is perfectly smooth as a function of $R$ and $\rho$, leads to a
discontinuous physical result at $R = R_c$.\footnote{This conclusion is not
really surprising.  As is well known from the theory of phase
transitions, physical results in
 a system with an infinite number
 of degrees of freedom can have  a nonanalytic dependence on the parameters
of the system, even if the Hamiltonian is analytic in these.
  A relativistic system of fermions  we are
concerned with here, is just such a system.}
   This conclusion is also  corroborated by (and independently implied by)
 the known
 behaviour of
the action \cite{Khoze}  and  of the integrated
 topological density \cite{guida}  $C(0)$, Eq.(\ref{top}),  as functions
 of $R$.  Indeed,  both of them behave as
$\sim R^2$    for  $R/\rho \ll 1$,
which is clearly and simply a reflection of the perturbative, quadratic
fluctuations  around $A_{\mu}=0$.\footnote {This is evident in the
"clever" gauge used in \protect\cite{guida} in which $A_{\mu}\propto R \,$ for
small $R$. }


In the physics context of Tev-region electroweak interactions, the outcome
 of our analysis is that the valley configuration at $R < R_c$, hence with
action  $S < S_c ={16 \pi^2\over g^2}(0.5960...), $  has no relation at all
to
the instanton-induced fermion number violating processes.  The argument
made in some literature \cite{Khoze}
 that  such processes become unsuppressed, on the basis of the behaviour
of the valley field at $R=0$,  thus seems to be unfounded.\footnote {We
agree on this point with a conjecture made in \protect\cite{maggiore}.}
  The valley field
is relevant only up to
  an energy where
 the cross section  is still exponentially suppressed
($\sigma_c = \exp - {16 \pi^2\over g^2}(0.3184...)$).



As for the problem of the QCD vacuum,  a
 series of works in Ref. \cite{liquid}
have led to the picture of  the physical vacuum
(with broken chiral symmetry ) as a sort of
instanton liquid,  with the average  instanton-anti-instanton distance
about
three  times their mean sizes.  Our result of  the minimum
separation $R_c/ \rho \simeq
\sqrt{4/3} $ for the instanton anti-instanton
 configuration to be non-perturbative,
 seems to support their assumptions.


Eventual implications of our work to the study of
 the large-order behaviour of perturbative series in non-Abelian
gauge theories, are still to be worked out.




\bigskip
{\bf Acknowledgments} The authors are grateful to
S. Forte, C. Imbimbo, G. Morchio and P. Provero
for interesting discussions.

\begin{thebibliography}{30}
\bibitem{guida}
         R. Guida and K.Konishi, University of Genoa preprint GEF-Th-8/1993;
         Phys. Lett. {\bf B298} (1993) 371.

\bibitem {Yung} I. Balitsky and A. Yung, Phys. Lett. {\bf B168} (1986) 113;
                A. Yung, Nucl. Phys. {\bf B297} (1988) 47;
                V.V. Khoze and A. Ringwald, Nucl.Phys. {\bf B355} (1991) 351;
                CERN-TH-6082/91.

\bibitem{Kiskis} J. Kiskis,  Phys. Rev. {\bf D18} (1978) 3690.
\bibitem{shifman}
 M.Shifman,  Phys. Rep. {\bf 209} (1991) 341.
\bibitem{maggiore} M. Maggiore and M. Shifman, Phys. Rev. {\bf D 47}
(1993)  3610.
\bibitem{labonte} G. Labont\'e,  Can.J.Phys. {\bf 53} (1975) 1533.
\bibitem{gross} C.G. Callan, R. Dashen and D.J. Gross, Phys. Rev. {\bf D17}
                 (1978) 2717; N.H. Christ, Phys. Rev. {\bf D21} (1980) 1591;
                J. Ambj\o rn, J. Greensite and C. Peterson, Nucl. Phys.
                {\bf B 221} (1983) 381.
\bibitem{particle}
    H. Fierz and G. Scharf, Helv. Phys. Acta {\bf 52} (1979) 437;
    P. Falkensteiner and H. Grosse Nucl. Phys. {\bf B305} (1988) 126.
\bibitem {Khoze} V.V. Khoze, J. Kripfganz and A. Ringwald, Phys. Lett.
        {\bf B275} (1992)381; {\bf B277} (1992) 496; {\bf B279} (1992) 429E.
\bibitem{liquid} E.V. Shuryak, "The  QCD vacuum, hadrons and the superdense
 matter", World Scientific (1988).
\end{thebibliography}


{\bf Figure Captions}

\begin{description}

\item
{Fig. 1.} Behaviour of $\tt (r=\infi )$ in function of time, for
a) $R/\rho=10$, b) $R/\rho=1.5$, c) $R/\rho=1.0$.

\end{description}

\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TABELLA 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%% thanks to Prof. S.Frixione for explaining me how to do this table
%%%%%%%%%% and the other one.

 \begin{table}
 \begin{center}
 \begin{tabular}[bht]{|c||c|c|c|c|}  \hline
   & $t$ generic & $t=t_0$ & $t=t_1$ & $t={R\over 2}-y$ \\
 \hline\hline
   $\st_1$
 & $ 1+r^2$
 & $ 1+r^4 $
 & $ 1+r^2$
 & $ 1+r^2$
 \\
   $\tt_1$
 & $ r^3$
 & $ r^5$
 & $ r^3$
 & $ r^{-1}$
 \\
   $\st_2$
 & $ r^{-1}$
 & $ r$
 & $ r^{-1}$
 & $ r^3$
 \\
   $\tt_2$
 & $1+r^2$
 & $1+r^6$
 & $1+r^2$
 & $1+r^2$
\\
  \hline
 \end{tabular}
 \vskip .3cm
 \end{center}
 {
 \leftskip 1cm
 \rightskip 1cm
 {\bf Table 1}: Behaviour of the two linearly indipendent
 solutions  near  $r= 0$.

 }
 \end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TABELLA 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%% thanks to Prof. S.Frixione for explaining me how to do this table
%%%%%%%%%% and the other one.

 \begin{table}
 \begin{center}
 \begin{tabular}[bht]{|c||c|c|c|c|}  \hline
   & $t$ generic & $t=t_0$ & $t=t_1$ & $t={R\over 2}-y$ \\
 \hline\hline
  $\st_3$
 & $ r$
 & $ r $
 & $ r^{-1}$
 & $ r$
 \\
 $\tt_3$
 & $ 1+r^{-2}$
 & $ 1+r^{-2}$
 & $ 1+r^{-6}$
 & $ 1+r^{-2}$
 \\
 $\st_4$
 & $ 1+r^{-2}$
 & $ 1+r^{-2}$
 & $ 1+r^{-6}$
 & $ 1$
 \\
 $\tt_4$
 & $r^{-3}$
 & $r^{-3}$
 & $r^{-5}$
 & $r^{-3}$
 \\ \hline
 \end{tabular}
 \vskip .3cm
 \end{center}
 {
 \leftskip 1cm
 \rightskip 1cm
 {\bf Table 2}: Behaviour of the two linearly indipendent
solutions, at  $r\rightarrow\infi$.

 }
 \end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\end{document}



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1815 736 M
1838 736 D
S
N
1815 651 M
1883 651 D
S
N
1815 566 M
1838 566 D
S
N
1815 480 M
1838 480 D
S
N
1761 2412 M
1761 2397 D
S
N
1739 2382 M
1739 2367 D
S
N
1739 2375 M
1784 2375 D
1776 2382 D
S
N
1739 2345 M
1739 2330 D
1761 2322 D
1784 2330 D
1784 2345 D
1761 2352 D
1739 2345 D
S
N
1761 1965 M
1761 1950 D
S
N
1746 1935 M
1739 1928 D
1739 1913 D
1746 1905 D
1761 1905 D
1769 1913 D
1769 1928 D
1761 1935 D
1784 1935 D
1784 1905 D
S
N
1739 1511 M
1739 1496 D
1761 1489 D
1784 1496 D
1784 1511 D
1761 1519 D
1739 1511 D
S
N
1746 1092 M
1739 1085 D
1739 1070 D
1746 1062 D
1761 1062 D
1769 1070 D
1769 1085 D
1761 1092 D
1784 1092 D
1784 1062 D
S
N
1739 686 M
1739 671 D
S
N
1739 679 M
1784 679 D
1776 686 D
S
N
1739 649 M
1739 634 D
1761 626 D
1784 634 D
1784 649 D
1761 656 D
1739 649 D
S
N
2355 2527 M
2378 2527 D
S
N
2355 2442 M
2378 2442 D
S
N
2310 2357 M
2378 2357 D
S
N
2355 2271 M
2378 2271 D
S
N
2355 2186 M
2378 2186 D
S
N
2355 2101 M
2378 2101 D
S
N
2355 2015 M
2378 2015 D
S
N
2310 1930 M
2378 1930 D
S
N
2355 1845 M
2378 1845 D
S
N
2355 1760 M
2378 1760 D
S
N
2355 1674 M
2378 1674 D
S
N
2355 1589 M
2378 1589 D
S
N
2310 1504 M
2378 1504 D
S
N
2355 1419 M
2378 1419 D
S
N
2355 1333 M
2378 1333 D
S
N
2355 1248 M
2378 1248 D
S
N
2355 1163 M
2378 1163 D
S
N
2310 1077 M
2378 1077 D
S
N
2355 992 M
2378 992 D
S
N
2355 907 M
2378 907 D
S
N
2355 822 M
2378 822 D
S
N
2355 736 M
2378 736 D
S
N
2310 651 M
2378 651 D
S
N
2355 566 M
2378 566 D
S
N
2355 480 M
2378 480 D
S
N
1822 2527 M
1822 2505 D
S
N
1856 2527 M
1856 2505 D
S
N
1891 2527 M
1891 2505 D
S
N
1925 2527 M
1925 2460 D
S
N
1959 2527 M
1959 2505 D
S
N
1994 2527 M
1994 2505 D
S
N
2028 2527 M
2028 2505 D
S
N
2062 2527 M
2062 2505 D
S
N
2097 2527 M
2097 2460 D
S
N
2131 2527 M
2131 2505 D
S
N
2166 2527 M
2166 2505 D
S
N
2200 2527 M
2200 2505 D
S
N
2234 2527 M
2234 2505 D
S
N
2269 2527 M
2269 2460 D
S
N
2303 2527 M
2303 2505 D
S
N
2338 2527 M
2338 2505 D
S
N
2372 2527 M
2372 2505 D
S
N
1925 2704 M
1925 2689 D
S
N
1902 2667 M
1902 2652 D
1925 2644 D
1947 2652 D
1947 2667 D
1925 2674 D
1902 2667 D
S
N
1902 2629 M
1910 2629 D
1910 2622 D
1902 2622 D
1902 2629 D
S
N
1940 2607 M
1947 2599 D
1947 2584 D
1940 2577 D
1925 2577 D
1910 2607 D
1902 2607 D
1902 2577 D
S
N
2074 2667 M
2074 2652 D
2097 2644 D
2119 2652 D
2119 2667 D
2097 2674 D
2074 2667 D
S
N
2074 2629 M
2082 2629 D
2082 2622 D
2074 2622 D
2074 2629 D
S
N
2074 2599 M
2074 2584 D
2097 2577 D
2119 2584 D
2119 2599 D
2097 2607 D
2074 2599 D
S
N
2246 2667 M
2246 2652 D
2269 2644 D
2291 2652 D
2291 2667 D
2269 2674 D
2246 2667 D
S
N
2246 2629 M
2254 2629 D
2254 2622 D
2246 2622 D
2246 2629 D
S
N
2284 2607 M
2291 2599 D
2291 2584 D
2284 2577 D
2269 2577 D
2254 2607 D
2246 2607 D
2246 2577 D
S
N
1822 503 M
1822 480 D
S
N
1856 503 M
1856 480 D
S
N
1891 503 M
1891 480 D
S
N
1925 548 M
1925 480 D
S
N
1959 503 M
1959 480 D
S
N
1994 503 M
1994 480 D
S
N
2028 503 M
2028 480 D
S
N
2062 503 M
2062 480 D
S
N
2097 548 M
2097 480 D
S
N
2131 503 M
2131 480 D
S
N
2166 503 M
2166 480 D
S
N
2200 503 M
2200 480 D
S
N
2234 503 M
2234 480 D
S
N
2269 548 M
2269 480 D
S
N
2303 503 M
2303 480 D
S
N
2338 503 M
2338 480 D
S
N
2372 503 M
2372 480 D
S
N
2097 2357 M
2097 651 D
S
N
1590 2527 M
1028 2527 D
S
N
1028 2527 M
1028 480 D
S
N
1028 480 M
1590 480 D
S
N
1590 480 M
1590 2527 D
S
N
1212 2354 M
1212 2357 D
1210 2357 D
1210 2354 D
1212 2354 D
S
N
1212 2337 M
1212 2340 D
1210 2340 D
1210 2337 D
1212 2337 D
S
N
1212 2320 M
1212 2323 D
1209 2323 D
1209 2320 D
1212 2320 D
S
N
1211 2303 M
1211 2305 D
1209 2305 D
1209 2303 D
1211 2303 D
S
N
1211 2286 M
1211 2288 D
1209 2288 D
1209 2286 D
1211 2286 D
S
N
1211 2269 M
1211 2271 D
1208 2271 D
1208 2269 D
1211 2269 D
S
N
1210 2252 M
1210 2254 D
1208 2254 D
1208 2252 D
1210 2252 D
S
N
1210 2235 M
1210 2237 D
1207 2237 D
1207 2235 D
1210 2235 D
S
N
1209 2218 M
1209 2220 D
1207 2220 D
1207 2218 D
1209 2218 D
S
N
1209 2201 M
1209 2203 D
1206 2203 D
1206 2201 D
1209 2201 D
S
N
1208 2184 M
1208 2186 D
1206 2186 D
1206 2184 D
1208 2184 D
S
N
1207 2167 M
1207 2169 D
1205 2169 D
1205 2167 D
1207 2167 D
S
N
1207 2150 M
1207 2152 D
1205 2152 D
1205 2150 D
1207 2150 D
S
N
1206 2133 M
1206 2135 D
1204 2135 D
1204 2133 D
1206 2133 D
S
N
1205 2116 M
1205 2118 D
1203 2118 D
1203 2116 D
1205 2116 D
S
N
1205 2099 M
1205 2101 D
1202 2101 D
1202 2099 D
1205 2099 D
S
N
1204 2081 M
1204 2084 D
1202 2084 D
1202 2081 D
1204 2081 D
S
N
1203 2064 M
1203 2067 D
1201 2067 D
1201 2064 D
1203 2064 D
S
N
1202 2047 M
1202 2050 D
1200 2050 D
1200 2047 D
1202 2047 D
S
N
1201 2030 M
1201 2033 D
1199 2033 D
1199 2030 D
1201 2030 D
S
N
1200 2013 M
1200 2015 D
1197 2015 D
1197 2013 D
1200 2013 D
S
N
1198 1996 M
1198 1998 D
1196 1998 D
1196 1996 D
1198 1996 D
S
N
1197 1979 M
1197 1981 D
1195 1981 D
1195 1979 D
1197 1979 D
S
N
1196 1962 M
1196 1964 D
1193 1964 D
1193 1962 D
1196 1962 D
S
N
1194 1945 M
1194 1947 D
1192 1947 D
1192 1945 D
1194 1945 D
S
N
1193 1928 M
1193 1930 D
1190 1930 D
1190 1928 D
1193 1928 D
S
N
1191 1911 M
1191 1913 D
1189 1913 D
1189 1911 D
1191 1911 D
S
N
1189 1894 M
1189 1896 D
1187 1896 D
1187 1894 D
1189 1894 D
S
N
1188 1877 M
1188 1879 D
1185 1879 D
1185 1877 D
1188 1877 D
S
N
1186 1860 M
1186 1862 D
1184 1862 D
1184 1860 D
1186 1860 D
S
N
1184 1843 M
1184 1845 D
1182 1845 D
1182 1843 D
1184 1843 D
S
N
1182 1826 M
1182 1828 D
1180 1828 D
1180 1826 D
1182 1826 D
S
N
1181 1809 M
1181 1811 D
1179 1811 D
1179 1809 D
1181 1809 D
S
N
1180 1792 M
1180 1794 D
1178 1794 D
1178 1792 D
1180 1792 D
S
N
1179 1774 M
1179 1777 D
1177 1777 D
1177 1774 D
1179 1774 D
S
N
1179 1757 M
1179 1760 D
1177 1760 D
1177 1757 D
1179 1757 D
S
N
1180 1740 M
1180 1743 D
1178 1743 D
1178 1740 D
1180 1740 D
S
N
1182 1723 M
1182 1726 D
1180 1726 D
1180 1723 D
1182 1723 D
S
N
1185 1706 M
1185 1708 D
1183 1708 D
1183 1706 D
1185 1706 D
S
N
1189 1689 M
1189 1691 D
1187 1691 D
1187 1689 D
1189 1689 D
S
N
1194 1672 M
1194 1674 D
1192 1674 D
1192 1672 D
1194 1672 D
S
N
1200 1655 M
1200 1657 D
1198 1657 D
1198 1655 D
1200 1655 D
S
N
1206 1638 M
1206 1640 D
1204 1640 D
1204 1638 D
1206 1638 D
S
N
1213 1621 M
1213 1623 D
1210 1623 D
1210 1621 D
1213 1621 D
S
N
1218 1604 M
1218 1606 D
1216 1606 D
1216 1604 D
1218 1604 D
S
N
1224 1587 M
1224 1589 D
1221 1589 D
1221 1587 D
1224 1587 D
S
N
1228 1570 M
1228 1572 D
1226 1572 D
1226 1570 D
1228 1570 D
S
N
1231 1553 M
1231 1555 D
1229 1555 D
1229 1553 D
1231 1553 D
S
N
1233 1536 M
1233 1538 D
1231 1538 D
1231 1536 D
S
N
1231 1536 M
1233 1536 D
S
N
1234 1519 M
1234 1521 D
1232 1521 D
1232 1519 D
1234 1519 D
S
N
1234 1502 M
1234 1504 D
1232 1504 D
1232 1502 D
1234 1502 D
S
N
1233 1484 M
1233 1487 D
1231 1487 D
1231 1484 D
1233 1484 D
S
N
1232 1467 M
1232 1470 D
1229 1470 D
1229 1467 D
1232 1467 D
S
N
1230 1450 M
1230 1453 D
1227 1453 D
1227 1450 D
1230 1450 D
S
N
1227 1433 M
1227 1436 D
1225 1436 D
1225 1433 D
1227 1433 D
S
N
1223 1416 M
1223 1419 D
1221 1419 D
1221 1416 D
1223 1416 D
S
N
1218 1399 M
1218 1401 D
1216 1401 D
1216 1399 D
1218 1399 D
S
N
1210 1382 M
1210 1384 D
1208 1384 D
1208 1382 D
1210 1382 D
S
N
1195 1365 M
1195 1367 D
1193 1367 D
1193 1365 D
1195 1365 D
S
N
1163 1348 M
1163 1350 D
1161 1350 D
1161 1348 D
1163 1348 D
S
N
1077 1331 M
1077 1333 D
1075 1333 D
1075 1331 D
1077 1331 D
S
N
1546 1246 M
1546 1248 D
1543 1248 D
1543 1246 D
1546 1246 D
S
N
1408 1229 M
1408 1231 D
1405 1231 D
1405 1229 D
1408 1229 D
S
N
1347 1212 M
1347 1214 D
1345 1214 D
1345 1212 D
1347 1212 D
S
N
1314 1195 M
1314 1197 D
1312 1197 D
1312 1195 D
1314 1195 D
S
N
1294 1177 M
1294 1180 D
1292 1180 D
1292 1177 D
1294 1177 D
S
N
1280 1160 M
1280 1163 D
1278 1163 D
1278 1160 D
1280 1160 D
S
N
1271 1143 M
1271 1146 D
1269 1146 D
1269 1143 D
1271 1143 D
S
N
1264 1126 M
1264 1129 D
1261 1129 D
1261 1126 D
1264 1126 D
S
N
1258 1109 M
1258 1111 D
1256 1111 D
1256 1109 D
1258 1109 D
S
N
1254 1092 M
1254 1094 D
1252 1094 D
1252 1092 D
1254 1092 D
S
N
1250 1075 M
1250 1077 D
1248 1077 D
1248 1075 D
1250 1075 D
S
N
1247 1058 M
1247 1060 D
1245 1060 D
1245 1058 D
1247 1058 D
S
N
1245 1041 M
1245 1043 D
1243 1043 D
1243 1041 D
1245 1041 D
S
N
1243 1024 M
1243 1026 D
1241 1026 D
1241 1024 D
1243 1024 D
S
N
1241 1007 M
1241 1009 D
1239 1009 D
1239 1007 D
1241 1007 D
S
N
1239 990 M
1239 992 D
1237 992 D
1237 990 D
1239 990 D
S
N
1238 973 M
1238 975 D
1236 975 D
1236 973 D
1238 973 D
S
N
1237 956 M
1237 958 D
1235 958 D
1235 956 D
1237 956 D
S
N
1236 939 M
1236 941 D
1233 941 D
1233 939 D
1236 939 D
S
N
1235 922 M
1235 924 D
1232 924 D
1232 922 D
1235 922 D
S
N
1234 905 M
1234 907 D
1232 907 D
1232 905 D
1234 905 D
S
N
1233 888 M
1233 890 D
1231 890 D
1231 888 D
1233 888 D
S
N
1232 870 M
1232 873 D
1230 873 D
1230 870 D
1232 870 D
S
N
1232 853 M
1232 856 D
1229 856 D
1229 853 D
1232 853 D
S
N
1231 836 M
1231 839 D
1229 839 D
1229 836 D
1231 836 D
S
N
1230 819 M
1230 822 D
1228 822 D
1228 819 D
1230 819 D
S
N
1230 802 M
1230 804 D
1228 804 D
1228 802 D
1230 802 D
S
N
1229 785 M
1229 787 D
1227 787 D
1227 785 D
1229 785 D
S
N
1229 768 M
1229 770 D
1227 770 D
1227 768 D
1229 768 D
S
N
1229 751 M
1229 753 D
1226 753 D
1226 751 D
1229 751 D
S
N
1228 734 M
1228 736 D
1226 736 D
1226 734 D
1228 734 D
S
N
1228 717 M
1228 719 D
1226 719 D
1226 717 D
1228 717 D
S
N
1228 700 M
1228 702 D
1225 702 D
1225 700 D
1228 700 D
S
N
1227 683 M
1227 685 D
1225 685 D
1225 683 D
1227 683 D
S
N
1227 666 M
1227 668 D
1225 668 D
1225 666 D
1227 666 D
S
N
1227 649 M
1227 651 D
1224 651 D
1224 649 D
1227 649 D
S
N
1028 2527 M
1095 2527 D
S
N
1028 2459 M
1050 2459 D
S
N
1028 2391 M
1050 2391 D
S
N
1028 2323 M
1050 2323 D
S
N
1028 2254 M
1050 2254 D
S
N
1028 2186 M
1095 2186 D
S
N
1028 2118 M
1050 2118 D
S
N
1028 2050 M
1050 2050 D
S
N
1028 1981 M
1050 1981 D
S
N
1028 1913 M
1050 1913 D
S
N
1028 1845 M
1095 1845 D
S
N
1028 1777 M
1050 1777 D
S
N
1028 1708 M
1050 1708 D
S
N
1028 1640 M
1050 1640 D
S
N
1028 1572 M
1050 1572 D
S
N
1028 1504 M
1095 1504 D
S
N
1028 1436 M
1050 1436 D
S
N
1028 1367 M
1050 1367 D
S
N
1028 1299 M
1050 1299 D
S
N
1028 1231 M
1050 1231 D
S
N
1028 1163 M
1095 1163 D
S
N
1028 1094 M
1050 1094 D
S
N
1028 1026 M
1050 1026 D
S
N
1028 958 M
1050 958 D
S
N
1028 890 M
1050 890 D
S
N
1028 822 M
1095 822 D
S
N
1028 753 M
1050 753 D
S
N
1028 685 M
1050 685 D
S
N
1028 617 M
1050 617 D
S
N
1028 549 M
1050 549 D
S
N
1028 480 M
1095 480 D
S
N
974 2562 M
974 2547 D
S
N
974 2532 M
981 2525 D
981 2510 D
974 2502 D
959 2502 D
951 2510 D
951 2525 D
959 2532 D
989 2532 D
996 2525 D
996 2510 D
989 2502 D
S
N
974 2221 M
974 2206 D
S
N
951 2169 M
996 2169 D
966 2191 D
966 2161 D
S
N
974 1880 M
974 1865 D
S
N
989 1850 M
996 1843 D
996 1828 D
989 1820 D
974 1820 D
959 1850 D
951 1850 D
951 1820 D
S
N
951 1511 M
951 1496 D
974 1489 D
996 1496 D
996 1511 D
974 1519 D
951 1511 D
S
N
989 1178 M
996 1170 D
996 1155 D
989 1148 D
974 1148 D
959 1178 D
951 1178 D
951 1148 D
S
N
951 814 M
996 814 D
966 837 D
966 807 D
S
N
974 495 M
981 488 D
981 473 D
974 465 D
959 465 D
951 473 D
951 488 D
959 495 D
989 495 D
996 488 D
996 473 D
989 465 D
S
N
1523 2527 M
1590 2527 D
S
N
1568 2459 M
1590 2459 D
S
N
1568 2391 M
1590 2391 D
S
N
1568 2323 M
1590 2323 D
S
N
1568 2254 M
1590 2254 D
S
N
1523 2186 M
1590 2186 D
S
N
1568 2118 M
1590 2118 D
S
N
1568 2050 M
1590 2050 D
S
N
1568 1981 M
1590 1981 D
S
N
1568 1913 M
1590 1913 D
S
N
1523 1845 M
1590 1845 D
S
N
1568 1777 M
1590 1777 D
S
N
1568 1708 M
1590 1708 D
S
N
1568 1640 M
1590 1640 D
S
N
1568 1572 M
1590 1572 D
S
N
1523 1504 M
1590 1504 D
S
N
1568 1436 M
1590 1436 D
S
N
1568 1367 M
1590 1367 D
S
N
1568 1299 M
1590 1299 D
S
N
1568 1231 M
1590 1231 D
S
N
1523 1163 M
1590 1163 D
S
N
1568 1094 M
1590 1094 D
S
N
1568 1026 M
1590 1026 D
S
N
1568 958 M
1590 958 D
S
N
1568 890 M
1590 890 D
S
N
1523 822 M
1590 822 D
S
N
1568 753 M
1590 753 D
S
N
1568 685 M
1590 685 D
S
N
1568 617 M
1590 617 D
S
N
1568 549 M
1590 549 D
S
N
1523 480 M
1590 480 D
S
N
1039 2527 M
1039 2460 D
S
N
1075 2527 M
1075 2505 D
S
N
1111 2527 M
1111 2505 D
S
N
1146 2527 M
1146 2505 D
S
N
1182 2527 M
1182 2505 D
S
N
1218 2527 M
1218 2460 D
S
N
1253 2527 M
1253 2505 D
S
N
1289 2527 M
1289 2505 D
S
N
1325 2527 M
1325 2505 D
S
N
1361 2527 M
1361 2505 D
S
N
1396 2527 M
1396 2460 D
S
N
1432 2527 M
1432 2505 D
S
N
1468 2527 M
1468 2505 D
S
N
1503 2527 M
1503 2505 D
S
N
1539 2527 M
1539 2505 D
S
N
1575 2527 M
1575 2460 D
S
N
1039 2623 M
1039 2608 D
S
N
1017 2593 M
1017 2578 D
S
N
1017 2586 M
1062 2586 D
1054 2593 D
S
N
1195 2586 M
1195 2571 D
1218 2563 D
1240 2571 D
1240 2586 D
1218 2593 D
1195 2586 D
S
N
1374 2593 M
1374 2578 D
S
N
1374 2586 M
1419 2586 D
1411 2593 D
S
N
1590 2593 M
1597 2586 D
1597 2571 D
1590 2563 D
1575 2563 D
1560 2593 D
1552 2593 D
1552 2563 D
S
N
1039 548 M
1039 480 D
S
N
1075 503 M
1075 480 D
S
N
1111 503 M
1111 480 D
S
N
1146 503 M
1146 480 D
S
N
1182 503 M
1182 480 D
S
N
1218 548 M
1218 480 D
S
N
1253 503 M
1253 480 D
S
N
1289 503 M
1289 480 D
S
N
1325 503 M
1325 480 D
S
N
1361 503 M
1361 480 D
S
N
1396 548 M
1396 480 D
S
N
1432 503 M
1432 480 D
S
N
1468 503 M
1468 480 D
S
N
1503 503 M
1503 480 D
S
N
1539 503 M
1539 480 D
S
N
1575 548 M
1575 480 D
S
N
1218 2357 M
1218 651 D
S
N
803 2527 M
241 2527 D
S
N
241 2527 M
241 480 D
S
N
241 480 M
803 480 D
S
N
803 480 M
803 2527 D
S
N
520 2354 M
520 2357 D
518 2357 D
518 2354 D
520 2354 D
S
N
520 2337 M
520 2340 D
518 2340 D
518 2337 D
520 2337 D
S
N
520 2320 M
520 2323 D
517 2323 D
517 2320 D
520 2320 D
S
N
519 2303 M
519 2305 D
517 2305 D
517 2303 D
519 2303 D
S
N
519 2286 M
519 2288 D
517 2288 D
517 2286 D
519 2286 D
S
N
519 2269 M
519 2271 D
517 2271 D
517 2269 D
519 2269 D
S
N
519 2252 M
519 2254 D
517 2254 D
517 2252 D
519 2252 D
S
N
519 2235 M
519 2237 D
517 2237 D
517 2235 D
519 2235 D
S
N
519 2218 M
519 2220 D
516 2220 D
516 2218 D
519 2218 D
S
N
518 2201 M
518 2203 D
516 2203 D
516 2201 D
518 2201 D
S
N
518 2184 M
518 2186 D
516 2186 D
516 2184 D
518 2184 D
S
N
518 2167 M
518 2169 D
516 2169 D
516 2167 D
518 2167 D
S
N
518 2150 M
518 2152 D
515 2152 D
515 2150 D
518 2150 D
S
N
517 2133 M
517 2135 D
515 2135 D
515 2133 D
517 2133 D
S
N
517 2116 M
517 2118 D
515 2118 D
515 2116 D
517 2116 D
S
N
517 2099 M
517 2101 D
515 2101 D
515 2099 D
517 2099 D
S
N
517 2081 M
517 2084 D
514 2084 D
514 2081 D
517 2081 D
S
N
516 2064 M
516 2067 D
514 2067 D
514 2064 D
516 2064 D
S
N
516 2047 M
516 2050 D
514 2050 D
514 2047 D
516 2047 D
S
N
515 2030 M
515 2033 D
513 2033 D
513 2030 D
515 2030 D
S
N
515 2013 M
515 2015 D
513 2015 D
513 2013 D
515 2013 D
S
N
515 1996 M
515 1998 D
512 1998 D
512 1996 D
515 1996 D
S
N
514 1979 M
514 1981 D
512 1981 D
512 1979 D
514 1979 D
S
N
514 1962 M
514 1964 D
511 1964 D
511 1962 D
514 1962 D
S
N
513 1945 M
513 1947 D
511 1947 D
511 1945 D
513 1945 D
S
N
512 1928 M
512 1930 D
510 1930 D
510 1928 D
512 1928 D
S
N
512 1911 M
512 1913 D
509 1913 D
509 1911 D
512 1911 D
S
N
511 1894 M
511 1896 D
509 1896 D
509 1894 D
511 1894 D
S
N
510 1877 M
510 1879 D
508 1879 D
508 1877 D
510 1877 D
S
N
509 1860 M
509 1862 D
507 1862 D
507 1860 D
509 1860 D
S
N
509 1843 M
509 1845 D
506 1845 D
506 1843 D
509 1843 D
S
N
508 1826 M
508 1828 D
506 1828 D
506 1826 D
508 1826 D
S
N
507 1809 M
507 1811 D
505 1811 D
505 1809 D
507 1809 D
S
N
506 1792 M
506 1794 D
504 1794 D
504 1792 D
506 1792 D
S
N
505 1774 M
505 1777 D
503 1777 D
503 1774 D
505 1774 D
S
N
505 1757 M
505 1760 D
502 1760 D
502 1757 D
505 1757 D
S
N
504 1740 M
504 1743 D
502 1743 D
502 1740 D
504 1740 D
S
N
504 1723 M
504 1726 D
501 1726 D
501 1723 D
504 1723 D
S
N
503 1706 M
503 1708 D
501 1708 D
501 1706 D
503 1706 D
S
N
504 1689 M
504 1691 D
501 1691 D
501 1689 D
504 1689 D
S
N
504 1672 M
504 1674 D
502 1674 D
502 1672 D
504 1672 D
S
N
505 1655 M
505 1657 D
503 1657 D
503 1655 D
505 1655 D
S
N
507 1638 M
507 1640 D
505 1640 D
505 1638 D
507 1638 D
S
N
509 1621 M
509 1623 D
506 1623 D
506 1621 D
509 1621 D
S
N
511 1604 M
511 1606 D
508 1606 D
508 1604 D
511 1604 D
S
N
513 1587 M
513 1589 D
510 1589 D
510 1587 D
513 1587 D
S
N
515 1570 M
515 1572 D
512 1572 D
512 1570 D
515 1570 D
S
N
516 1553 M
516 1555 D
514 1555 D
514 1553 D
516 1553 D
S
N
518 1536 M
518 1538 D
516 1538 D
516 1536 D
S
N
516 1536 M
518 1536 D
S
N
519 1519 M
519 1521 D
516 1521 D
516 1519 D
519 1519 D
S
N
519 1502 M
519 1504 D
517 1504 D
517 1502 D
519 1502 D
S
N
518 1484 M
518 1487 D
516 1487 D
516 1484 D
518 1484 D
S
N
517 1467 M
517 1470 D
515 1470 D
515 1467 D
517 1467 D
S
N
515 1450 M
515 1453 D
513 1453 D
513 1450 D
515 1450 D
S
N
511 1433 M
511 1436 D
509 1436 D
509 1433 D
511 1433 D
S
N
505 1416 M
505 1419 D
503 1419 D
503 1416 D
505 1416 D
S
N
494 1399 M
494 1401 D
492 1401 D
492 1399 D
494 1399 D
S
N
473 1382 M
473 1384 D
471 1384 D
471 1382 D
473 1382 D
S
N
426 1365 M
426 1367 D
424 1367 D
424 1365 D
426 1365 D
S
N
290 1348 M
290 1350 D
288 1350 D
288 1348 D
290 1348 D
S
N
758 1263 M
758 1265 D
756 1265 D
756 1263 D
758 1263 D
S
N
663 1246 M
663 1248 D
660 1248 D
660 1246 D
663 1246 D
S
N
618 1229 M
618 1231 D
616 1231 D
616 1229 D
618 1229 D
S
N
593 1212 M
593 1214 D
591 1214 D
591 1212 D
593 1212 D
S
N
577 1195 M
577 1197 D
575 1197 D
575 1195 D
577 1195 D
S
N
567 1177 M
567 1180 D
565 1180 D
565 1177 D
567 1177 D
S
N
560 1160 M
560 1163 D
557 1163 D
557 1160 D
560 1160 D
S
N
554 1143 M
554 1146 D
552 1146 D
552 1143 D
554 1143 D
S
N
550 1126 M
550 1129 D
548 1129 D
548 1126 D
550 1126 D
S
N
547 1109 M
547 1111 D
544 1111 D
544 1109 D
547 1109 D
S
N
544 1092 M
544 1094 D
542 1094 D
542 1092 D
544 1092 D
S
N
542 1075 M
542 1077 D
539 1077 D
539 1075 D
542 1075 D
S
N
540 1058 M
540 1060 D
538 1060 D
538 1058 D
540 1058 D
S
N
538 1041 M
538 1043 D
536 1043 D
536 1041 D
538 1041 D
S
N
537 1024 M
537 1026 D
535 1026 D
535 1024 D
537 1024 D
S
N
536 1007 M
536 1009 D
533 1009 D
533 1007 D
536 1007 D
S
N
535 990 M
535 992 D
532 992 D
532 990 D
535 990 D
S
N
534 973 M
534 975 D
532 975 D
532 973 D
534 973 D
S
N
533 956 M
533 958 D
531 958 D
531 956 D
533 956 D
S
N
532 939 M
532 941 D
530 941 D
530 939 D
532 939 D
S
N
532 922 M
532 924 D
530 924 D
530 922 D
532 922 D
S
N
531 905 M
531 907 D
529 907 D
529 905 D
531 905 D
S
N
531 888 M
531 890 D
528 890 D
528 888 D
531 888 D
S
N
530 870 M
530 873 D
528 873 D
528 870 D
530 870 D
S
N
530 853 M
530 856 D
528 856 D
528 853 D
530 853 D
S
N
529 836 M
529 839 D
527 839 D
527 836 D
529 836 D
S
N
529 819 M
529 822 D
527 822 D
527 819 D
529 819 D
S
N
529 802 M
529 804 D
527 804 D
527 802 D
529 802 D
S
N
529 785 M
529 787 D
526 787 D
526 785 D
529 785 D
S
N
528 768 M
528 770 D
526 770 D
526 768 D
528 768 D
S
N
528 751 M
528 753 D
526 753 D
526 751 D
528 751 D
S
N
528 734 M
528 736 D
526 736 D
526 734 D
528 734 D
S
N
528 717 M
528 719 D
525 719 D
525 717 D
528 717 D
S
N
527 700 M
527 702 D
525 702 D
525 700 D
527 700 D
S
N
527 683 M
527 685 D
525 685 D
525 683 D
527 683 D
S
N
527 666 M
527 668 D
525 668 D
525 666 D
527 666 D
S
N
527 649 M
527 651 D
525 651 D
525 649 D
527 649 D
S
N
241 2527 M
308 2527 D
S
N
241 2459 M
263 2459 D
S
N
241 2391 M
263 2391 D
S
N
241 2323 M
263 2323 D
S
N
241 2254 M
263 2254 D
S
N
241 2186 M
308 2186 D
S
N
241 2118 M
263 2118 D
S
N
241 2050 M
263 2050 D
S
N
241 1981 M
263 1981 D
S
N
241 1913 M
263 1913 D
S
N
241 1845 M
308 1845 D
S
N
241 1777 M
263 1777 D
S
N
241 1708 M
263 1708 D
S
N
241 1640 M
263 1640 D
S
N
241 1572 M
263 1572 D
S
N
241 1504 M
308 1504 D
S
N
241 1436 M
263 1436 D
S
N
241 1367 M
263 1367 D
S
N
241 1299 M
263 1299 D
S
N
241 1231 M
263 1231 D
S
N
241 1163 M
308 1163 D
S
N
241 1094 M
263 1094 D
S
N
241 1026 M
263 1026 D
S
N
241 958 M
263 958 D
S
N
241 890 M
263 890 D
S
N
241 822 M
308 822 D
S
N
241 753 M
263 753 D
S
N
241 685 M
263 685 D
S
N
241 617 M
263 617 D
S
N
241 549 M
263 549 D
S
N
241 480 M
308 480 D
S
N
187 2562 M
187 2547 D
S
N
187 2532 M
194 2525 D
194 2510 D
187 2502 D
172 2502 D
164 2510 D
164 2525 D
172 2532 D
202 2532 D
209 2525 D
209 2510 D
202 2502 D
S
N
187 2221 M
187 2206 D
S
N
164 2169 M
209 2169 D
179 2191 D
179 2161 D
S
N
187 1880 M
187 1865 D
S
N
202 1850 M
209 1843 D
209 1828 D
202 1820 D
187 1820 D
172 1850 D
164 1850 D
164 1820 D
S
N
164 1511 M
164 1496 D
187 1489 D
209 1496 D
209 1511 D
187 1519 D
164 1511 D
S
N
202 1178 M
209 1170 D
209 1155 D
202 1148 D
187 1148 D
172 1178 D
164 1178 D
164 1148 D
S
N
164 814 M
209 814 D
179 837 D
179 807 D
S
N
187 495 M
194 488 D
194 473 D
187 465 D
172 465 D
164 473 D
164 488 D
172 495 D
202 495 D
209 488 D
209 473 D
202 465 D
S
N
736 2527 M
803 2527 D
S
N
781 2459 M
803 2459 D
S
N
781 2391 M
803 2391 D
S
N
781 2323 M
803 2323 D
S
N
781 2254 M
803 2254 D
S
N
736 2186 M
803 2186 D
S
N
781 2118 M
803 2118 D
S
N
781 2050 M
803 2050 D
S
N
781 1981 M
803 1981 D
S
N
781 1913 M
803 1913 D
S
N
736 1845 M
803 1845 D
S
N
781 1777 M
803 1777 D
S
N
781 1708 M
803 1708 D
S
N
781 1640 M
803 1640 D
S
N
781 1572 M
803 1572 D
S
N
736 1504 M
803 1504 D
S
N
781 1436 M
803 1436 D
S
N
781 1367 M
803 1367 D
S
N
781 1299 M
803 1299 D
S
N
781 1231 M
803 1231 D
S
N
736 1163 M
803 1163 D
S
N
781 1094 M
803 1094 D
S
N
781 1026 M
803 1026 D
S
N
781 958 M
803 958 D
S
N
781 890 M
803 890 D
S
N
736 822 M
803 822 D
S
N
781 753 M
803 753 D
S
N
781 685 M
803 685 D
S
N
781 617 M
803 617 D
S
N
781 549 M
803 549 D
S
N
736 480 M
803 480 D
S
N
244 2527 M
244 2505 D
S
N
279 2527 M
279 2505 D
S
N
313 2527 M
313 2505 D
S
N
348 2527 M
348 2460 D
S
N
382 2527 M
382 2505 D
S
N
417 2527 M
417 2505 D
S
N
452 2527 M
452 2505 D
S
N
486 2527 M
486 2505 D
S
N
521 2527 M
521 2460 D
S
N
555 2527 M
555 2505 D
S
N
590 2527 M
590 2505 D
S
N
625 2527 M
625 2505 D
S
N
659 2527 M
659 2505 D
S
N
694 2527 M
694 2460 D
S
N
728 2527 M
728 2505 D
S
N
763 2527 M
763 2505 D
S
N
798 2527 M
798 2505 D
S
N
348 2623 M
348 2608 D
S
N
363 2593 M
370 2586 D
370 2571 D
363 2563 D
348 2563 D
333 2593 D
325 2593 D
325 2563 D
S
N
498 2586 M
498 2571 D
521 2563 D
543 2571 D
543 2586 D
521 2593 D
498 2586 D
S
N
709 2593 M
716 2586 D
716 2571 D
709 2563 D
694 2563 D
679 2593 D
671 2593 D
671 2563 D
S
N
244 503 M
244 480 D
S
N
279 503 M
279 480 D
S
N
313 503 M
313 480 D
S
N
348 548 M
348 480 D
S
N
382 503 M
382 480 D
S
N
417 503 M
417 480 D
S
N
452 503 M
452 480 D
S
N
486 503 M
486 480 D
S
N
521 548 M
521 480 D
S
N
555 503 M
555 480 D
S
N
590 503 M
590 480 D
S
N
625 503 M
625 480 D
S
N
659 503 M
659 480 D
S
N
694 548 M
694 480 D
S
N
728 503 M
728 480 D
S
N
763 503 M
763 480 D
S
N
798 503 M
798 480 D
S
N
521 2357 M
521 651 D
S
N
488 2902 M
533 2902 D
533 2872 D
S
N
511 2880 M
511 2902 D
S
N
488 2857 M
488 2842 D
S
N
488 2850 M
518 2850 D
518 2857 D
S
N
496 2805 M
488 2812 D
488 2820 D
496 2827 D
511 2827 D
518 2820 D
518 2812 D
511 2805 D
S
N
518 2805 M
481 2805 D
473 2812 D
S
N
488 2790 M
496 2790 D
496 2782 D
488 2782 D
488 2790 D
S
N
488 2767 M
488 2752 D
S
N
488 2760 M
533 2760 D
526 2767 D
S
N
496 2715 M
488 2722 D
488 2730 D
496 2737 D
511 2737 D
518 2730 D
518 2722 D
511 2715 D
S
N
488 2700 M
496 2692 D
526 2692 D
533 2700 D
S
N
1275 2902 M
1320 2902 D
1320 2872 D
S
N
1298 2880 M
1298 2902 D
S
N
1275 2857 M
1275 2842 D
S
N
1275 2850 M
1305 2850 D
1305 2857 D
S
N
1283 2805 M
1275 2812 D
1275 2820 D
1283 2827 D
1298 2827 D
1305 2820 D
1305 2812 D
1298 2805 D
S
N
1305 2805 M
1268 2805 D
1260 2812 D
S
N
1275 2790 M
1283 2790 D
1283 2782 D
1275 2782 D
1275 2790 D
S
N
1275 2767 M
1275 2752 D
S
N
1275 2760 M
1320 2760 D
1313 2767 D
S
N
1275 2737 M
1320 2737 D
S
N
1298 2737 M
1305 2730 D
1305 2722 D
1298 2715 D
1283 2715 D
1275 2722 D
1275 2730 D
1283 2737 D
S
N
1275 2700 M
1283 2692 D
1313 2692 D
1320 2700 D
S
N
2063 2902 M
2108 2902 D
2108 2872 D
S
N
2085 2880 M
2085 2902 D
S
N
2063 2857 M
2063 2842 D
S
N
2063 2850 M
2093 2850 D
2093 2857 D
S
N
2070 2805 M
2063 2812 D
2063 2820 D
2070 2827 D
2085 2827 D
2093 2820 D
2093 2812 D
2085 2805 D
S
N
2093 2805 M
2055 2805 D
2048 2812 D
S
N
2063 2790 M
2070 2790 D
2070 2782 D
2063 2782 D
2063 2790 D
S
N
2063 2767 M
2063 2752 D
S
N
2063 2760 M
2108 2760 D
2100 2767 D
S
N
2093 2737 M
2093 2722 D
2085 2715 D
2063 2715 D
S
N
2070 2715 M
2063 2722 D
2063 2730 D
2070 2737 D
2078 2737 D
2085 2730 D
2085 2722 D
2078 2715 D
S
N
2063 2700 M
2070 2692 D
2100 2692 D
2108 2700 D
S
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