%Paper: hep-th/9312114
%From: DATTANASIO@vaxpr.pr.infn.it
%Date: Tue, 14 Dec 1993 12:27:21 +0100 (WET)
%Date (revised): Tue, 8 Mar 1994 12:40:23 +0200 (MET)


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\begin{document}
\begin{titlepage}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\begin{flushright}
     UPRF 93-388 \\
     October 1993
\end{flushright}
\par \vskip 10mm
\begin{center}
{\Large \bf
Renormalization group flow for $SU(2)$ Yang-Mills\\
theory and gauge invariance\footnote{
Research supported in part by MURST, Italy}}
\end{center}
\par \vskip 2mm
\begin{center}
        {\bf M.\ Bonini, M.\ D'Attanasio and G.\ Marchesini} \\
        Dipartimento di Fisica, Universit\`a di Parma and\\
        INFN, Gruppo Collegato di Parma, Italy
        \end{center}
\par \vskip 2mm
\begin{center} {\large \bf Abstract} \end{center}
\begin{quote}
We study the formulation of the Wilson renormalization group (RG)
method for a non-Abelian gauge theory.
We analyze the simple case of $SU(2)$ and show that
the local gauge symmetry can be
implemented by suitable boundary conditions for the RG flow.
Namely we require that the relevant couplings present in the
physical effective action, \ie the coefficients of the field monomials
with dimension not larger than four, are fixed to satisfy the
Slavnov-Taylor identities. The full action obtained from the RG
equation should then satisfy the same identities.
This procedure is similar to the one we used in QED. In this way we
avoid the cospicuous fine tuning problem which arises if one gives
instead the couplings of the bare Lagrangian.
To show the practical character of this formulation we deduce
the perturbative expansion for the vertex functions in terms
of the physical coupling $g$ at the subtraction point $\mu$
and perform one loop calculations.
In particular we analyze to this order some ST identities and compute
the nine bare couplings.
We give a schematic proof of perturbative renormalizability.
\end{quote}
\end{titlepage}

\section{Introduction}
An elegant treatment of local gauge theories is provided by
dimensional regularization \cite{eps} which preserves BRS invariance.
However, this method is necessarily bounded to the perturbative
regime. Moreover, in the case of chiral gauge theories, even this
regularization conflicts \cite{chi} with the local symmetry. The
reason for this is that the usual definition of $\g_5$
is not consistent in complex dimension. Therefore it is useful to study
and exploit alternative formulations which work in four dimensions and
are in principle nonperturbative.

The exact renormalization group (RG) formulation \cite{W} works in the
physical space-time and provides a deep physical meaning of the
ultraviolet divergences. The original motivation for this method was
the study of the flow of Lagrangians at various scales. Once the
action $S_{\L_0}$ at the ultraviolet (UV) scale $\L_0$ is given, one
computes the Wilsonian action $S_\L$ at a lower scale $\L$, by integrating the
fields with frequencies $p$ between $\L$ and $\L_0$ ($\L^2 <p^2
<\L_0^2$). A very simple way of constructing the RG equations has been
written few years ago by Polchinski \cite{P}. See also
\cite{G}-\cite{oth}.
This method not only allows the study of the evolution in $\L$ of
$S_\L$, but provides a practical way of computing Green's functions.
For instance the iterative solution of RG equations gives the usual
loop expansion.

The RG formulation was not considered suitable for local gauge
theories since the presence of a momentum scale $\L$ implies that the BRS
invariance is completely lost. As a consequence the bare couplings of
the UV action $S_{\L_0}$ are not constrained by BRS invariance.
Obviously the various bare couplings in $S_{\L_0}$ are anyway
related by the fact that the physical Green's functions must
satisfy Slavnov-Taylor (ST) identities. Even assuming that this is
possible, the bare couplings would be constrained by a tremendous
fine tuning process \cite{romani}.
Recently Becchi \cite{B}, by studying the ``quantum action principle''
for the UV action in $SU(2)$ Yang-Mills (YM) case was able to extend
the BRS transformations and obtain a functional fine tuning equation
for the bare couplings. See also \cite{Riv}.

In this paper we present an attempt of giving a RG formulation for
gauge theories which avoids the fine tuning problem.
In a previous paper \cite{QED} we discussed the Abelian case while
here we analyze the non-Abelian case of $SU(2)$ Yang-Mills theory.
The method is quite general, uses some well known procedures but organized
in a rather unusual way. Therefore it is useful
to summarize the main steps.
\newline{\it 1) Definition of the cutoff effective action.}
Consider the physical effective action $\G[\Phi]$ where $\Phi(x)$
denotes all the fields of the theory. For QED $\Phi=(A_\mu,\bar
\psi,\psi)$, \ie the photon and electron fields; for SU(2) YM case
$\Phi=(A_\mu^a,c^a,\bc^a,u_\mu^a,v^a)$, \ie the gauge, ghost,
antighost fields and the sources for the BRS transformations of
$A_\mu^a$ and $c_a$ respectively. $\G[\Phi]$ must satisfy Ward or ST
identities. We consider now the ``cutoff effective action'' $\G[\Phi;\L]$,
obtained by putting an infrared (IR) cutoff $\L$ and an UV cutoff
$\L_0$ for all propagators in the vertices, \ie we set to zero each
propagator if its frequency is lower than $\L$ and larger than $\L_0$.
In the \UV limit $\G[\Phi;\L=0]$ is just the physical effective action,
while for $\L\ne 0$ it does not satisfy Ward or ST
identities. The functional $\G[\Phi;\L]$ is related to the Wilsonian
action $S_\L$ and we introduced it only because of technical
reasons. Actually $\G[\Phi;\L=\L_0]$ is the UV action $S_{\L_0}$.
\newline{\it 2) RG equation for} $\G[\Phi;\L]$. The RG equation for
$S_\L$ can be formulated directly for $\G[\Phi;\L]$ and, as we shall
discuss, takes the form (see \cite{P})
\beq\nome{1}
\L\partial_\L \G[\Phi;\L]=I[\Phi;\L]
\eeq
where the functional $I[\Phi;\L]$ is given (non linearly) in terms of
$\G[\Phi;\L]$. Once the boundary conditions are given, the RG equation
\re{1} allows one to obtain $\G[\Phi;\L]$ for any $\L$ and in particular,
for $\L=0$, the physical effective action.
As one expects, the boundary conditions for the various vertices
depend on dimensional counting.
\newline{\it 3) Relevant and irrelevant parts of} $\G[\Phi;\L]$.
One distinguishes irrelevant vertices, which have negative mass
dimension, and relevant couplings with non-negative dimension. One can
therefore write
\beq\nome{2}
\G[\Phi;\L]=\G_{rel}[\Phi;\s_i(\L)] + \G_{irr}[\Phi;\L]\,,
\eeq
where $\G_{irr}[\Phi;\L]$ involves only the irrelevant vertices and
$\s_i(\L)$ are the relevant couplings: seven for QED and nine for the
$SU(2)$ case. Therefore $\G_{rel}$ is given by a polynomial in the
fields. The relevant couplings are defined as the values of some
vertex functions or their derivatives at a given normalization point.
We are ready now to discuss the boundary conditions for the evolution
equation in \re{1}.
\newline{\it 4) Boundary conditions for $\G_{irr}[\Phi;\L]$}.
For $\L=\L_0\to\infty$ one assumes
\beq\nome{3}
\G[\Phi;\L_0]=\G_{rel}[\Phi;\s_i(\L_0)]\,,
\eeq
\ie the irrelevant vertices vanish in the UV region. Notice that this
hypothesis is essential to have perturbative renormalizability.
The parameters $\s_i(\L_0)$ are the bare parameters of the UV action.
In the usual procedure one gives these bare parameters and,
by using \re{1}, evaluates $\G[\Phi;\L=0]$.
The requirement that this resulting effective action  satisfies the ST
identities gives rise to the fine tuning problem, which we want to
avoid.
\newline{\it 5) Boundary conditions for  $\G_{rel}[\Phi;\s_i(\L)]$}.
Instead of fixing the boundary conditions for the
relevant parameters $\s_i(\L)$ at $\L=\L_0\to\infty$, we set them at
the physical point $\L=0$. The values $\s_i(0)$ are then the physical
masses, wave function costants and couplings. Moreover, for a gauge
theory we must fix these couplings in such a way that
$\G_{rel}[\Phi;\s_i(0)]$ satisfies Ward or ST identities.
This is then the place where we implement the symmetry of the theory.
For the Abelian case, studied in the previuos paper, we showed that
one can simply assume $\G_{rel}[\Phi;\s_i(0)]=S_{cl}[\Phi]$, where
$S_{cl}[\Phi]$ is the QED classical action including the gauge fixing
term. Then $\G_{rel}[\Phi;\s_i(0)]$ satisfies Ward identities.
For the non-Abelian theory, as we shall illustrate, the situation is more
complex.
In this case one has two features which are new
with respect to QED.
The first difference is that in the YM case one can construct relevant
monomials, such as for instance $(A_\mu^a A^a_\mu)^2$,
which are not present in the classical Lagrangian.
The second important difference between QED and YM theories is that
Ward identities are linear while ST identities are not linear.
In QED, Ward identities do not couple relevant and irrelevant
couplings, so that the relevant part of the effective action
satisfies Ward identities by itself.
In the YM case instead the ST identities couple the relevant and
irrelevant part of the effective action. Therefore the requirement
that $\G_{rel}[\Phi;\s_i(0)]$ satisfies ST identities implies thas
some of the nine relevant couplings are fixed in terms of irrelevant
vertices. This will be discussed in detail in sect.~3.
\newline{\it 6) Analysis of the ST identities}.
These boundary conditions and the RG equation \re{1} completely
determine $\G[\Phi;\L]$, at least perturbatively. The fundamental
point is whether the physical effective action $\G[\Phi;\L=0]$ does
satisfy Ward or ST identities. Recall that these identities have been
implemented only for the relevant part of $\G[\Phi;\L=0]$.
In the Abelian case we have been able to show \cite{QED} that the
renormalized Green's functions indeed satisfy Ward identities to all
order in perturbation theory.
The method was based on the properties of RG flow and the use
of the above boundary conditions. However the proof still required the
analysis of graphs. The same method could be extended to
the non-Abelian case but it would be quite more complicated.
In this paper we limit ourself
to the analysis of some ST identities to one loop. In this way we
illustrate the key points needed to obtain these identities.
A general proof to all loops should use more synthetic techniques.

The paper is organized as follows.
In sect.~2
we define the cutoff effective action for the $SU(2)$ YM theory
for which we write the RG equation.
In sect.~3
we define the relevant couplings and deduce in detail their values at
$\L=0$ to satisfy ST identities.
In sect.~4
we convert the differential functional equation into an integral
equation which embodies the boundary conditions.
In sect.~5
we show that the iterative solution
gives the loop expansion for the vertex functions. This perturbative
expansion involves only the physical parameter $g$ defined at the
subtraction point $\mu$.
To one-loop order we explicitly derive:
the various vertex functions,
the bare parameters in the UV Lagrangian, \ie the relevant couplings
$\s_i(\L)$ at $\L=\L_0$, and
the usual one-loop beta function.
We check the ST identities at one loop
for the vector propagator and for the ghost-vector vertex in the
physical limit ($\L=0$ and \UV).
Finally we give a schematic proof of perturbative renormalizability.
In sect.~6 we show that the cutoff effective action depends, as the
classical Lagrangian, on the combination $w^a_\mu=u^a_\mu/g+\PD_\mu
\bc^a$. This fact simplify greatly the analysis of relevant field
monomials.
Sect.~8 contains some conclusions.

\section{Renormalization group flow and effective action}
In this section we introduce the cutoff effective action
and deduce the RG group flow equations. We follow the same
steps \cite{BDM,QED} as in the scalar and QED theory.
This allows us to set up the necessary notations.

In the $SU(2)$ gauge theory the classical Lagrangian in the
Feynman gauge is
\beq\nome{Scl}
S_{cl}[A_\mu,c,\bc]
=\int d^4x
\left\{
-\frac 1 4 \left( F_{\mu\nu}  \cdot F_{\mu\nu}\right)
-\half  \left(\PD_\mu A_\mu \right)^2
- \bc \cdot \PD_\mu D_\mu c
\right\}\,,
\eeq
where
$$
F^a_{\mu\nu}(x)=\PD_\mu A^a_\nu -\PD_\nu A^a_\mu
+ g \left( A_\mu \wedge A_\nu\right)^a \,,
\;\;\;\;\;\;
\left(A_\mu \wedge A_\nu\right)^a=
\epsilon^{abc} A^b_\mu A^c_\nu \,,
$$
$$
F_{\mu\nu} \cdot F_{\mu\nu} =F_{\mu\nu} ^a F_{\mu\nu} ^a\,,
\;\;\;\;\;\;
D_\mu c =\PD_\mu c + gA_\mu \wedge c\,.
$$
This action is invariant under the BRS transformations \cite{BRS}
$$
\delta A_\mu= \frac 1 g \eta D_\mu c
\,,
\;\;\;\;\;\;\;
\delta c= -\eta \half c \wedge c
\,,
\;\;\;\;\;\;\;
\delta \bc= - \frac 1 g \eta \PD_\mu A_\mu
\,,
$$
with $\eta$ a Grassmann parameter.
Introducing the sources $u_\mu$ and $v$ associated to the variations
of $A_\mu$ and $c$ one has the BRS action
\beq\nome{Stot}
S_{BRS}[A_\mu,c,\bc,u_\mu,v]
=
S_{cl} +
\int d^4x
\left\{ \frac 1 g u_\mu  \cdot D_\mu c -\half v \cdot c \wedge c
\right\}=S_2+S_{int}\,,
\eeq
with
\beq\nome{S2}
S_2=\int d^4 x \left\{
\half A_\mu\cdot\partial^2  A_\mu +
w_\mu \cdot \partial_\mu c
\,- \half
v\cdot c\wedge c \right\}
\eeq
and
\begin{eqnarray*}
&&
S_{int}= \int d^4x \left\{ig\;
(\partial_\nu A_\mu) \cdot A_\nu \wedge A_\mu
-g \; w_\mu \cdot c \wedge A_\mu
-\frac {g^2}{4}\;
(A_\mu\wedge A_\nu)\cdot (A_\mu\wedge A_\nu) \right\}
\,.
\end{eqnarray*}
These expressions are functions of the combination
$w_\mu=\frac 1 g u_\mu+\PD_\mu \bc$.
One expects that also the effective action depends on this
combination of fields.
In sect.~6 we will show that this property holds also for
the interacting part of the cutoff effective action.

\subsection{Cutoff effective action}
We now define for this theory the cutoff effective action discussed
in the introduction.
In order to compute the vertices one
needs a regularization procedure of the ultraviolet divergences.
We regularize these divergences by assuming that in the path integral
one integrates only the fields with frequencies smaller than a given
UV cutoff $\L_0$. This procedure is equivalent to assume that the free
propagators vanish for $p^2 > \L_0^2$.
The physical theory is obtained by considering the limit \UV.
In order to study the Wilson renormalization group flow \cite{W,P},
one introduces in the free propagators also an infrared cutoff $\L$.
The quadratic part of the action \re{S2} becomes, in momentum space,
\begin{eqnarray*}
S_2^{\L,\L_0}= & \int_p  \;
\left\{
-\half A_\mu(-p)\cdot A_\mu(p) \, p^2  \;+\; p^2 \bc(-p)\cdot c(p)
\right\}
\,K^{-1}_{\L\L_0}(p)
\\ &
-\frac {i}{g} \int_p  p_\mu u_\mu(-p)\cdot c(p)
\,- \frac 1 2
\int_p\int_q v(p)\cdot c(q)\wedge c(-p-q)
\,,
\end{eqnarray*}
where $K_{\L\L_0}(p)=1$ in the region
$\L^2 \, \ltap \,p^2\, \ltap \, \L_0^2$ and rapidly vanishing outside
and
$$
\int_p \equiv \int \frac {d^4p}{(2\pi)^4}\,.
$$
The introduction of a cutoff in the propagators breaks the gauge
invariance properties of the action.
Therefore at the UV scale the interaction
$S_{int}^{\L_0}$ must contain all
the nine monomials which are $SU(2)$ singlets and Lorentz scalar
with dimension not higher than four. The first seven monomials
\beq\nome{7s}
\eqalign{
&
A_\mu \cdot A_\mu \,,
\;\;\;\;\;\;\;
(\partial_\nu A_\mu) \cdot (\partial_\nu A_\mu) \,,
\;\;\;\;\;\;\;
(\partial_\mu A_\mu) \cdot (\partial_\nu A_\nu) \,,
\;\;\;\;\;\;\;
w_\mu \cdot \partial_\mu c \,,
\cr &
(A_\mu\wedge A_\nu) \cdot (\partial_\mu A_\nu) \,,
\;\;\;\;\;\;\;
w_\mu \cdot c \wedge A_\mu \,,
\;\;\;\;\;\;\;
(A_\mu \wedge A_\nu)\cdot (A_\mu \wedge A_\nu)
}
\eeq
are present in the classical action \re{Scl}.
The remaining two
\beq\nome{2r}
2(A_\mu \cdot A_\nu)\, (A_\mu \cdot A_\nu)
\,+\,
(A_\mu \cdot A_\mu)\, (A_\nu \cdot A_\nu) \,,
\;\;\;\;\;\;
v \cdot c \wedge c
\eeq
are not present in \re{Scl} and are generated by the interaction.

In general we consider two functionals $\Pi_{1;rel}$ and
$\Pi_{2;rel}$, containing the two groups of monomials in
\re{7s} and \re{2r} respectively.
The first one depends on the seven couplings $\s_i$
\beq\nome{Pis}
\eqalign{
& \Pi_{1;rel}[\Phi;\s_i]
=\int d^4x \biggl\{
\frac 1 2 A_\mu \cdot
\left[g_{\mu\nu}(\s_{m_A} -\s_\alpha \partial^2)
- \s_A (g_{\mu\nu} \partial^2-\partial_\mu \partial_\nu)\right] A_\nu+
i\s_{wc}\; w_\mu \cdot \partial_\mu c
\cr \;\;& +
i\s_{3A}\;(\partial_\nu A_\mu) \cdot A_\nu \wedge A_\mu+
\frac{\s_{4A}}{4}\;(A_\mu\wedge A_\nu)\cdot(A_\mu\wedge A_\nu)+
\s_{wcA} \;w_\mu\cdot c\wedge A_\mu \biggr\}\,.
}
\eeq
The second functional depends on two additional couplings.
Since they will play a particular r\^ole, we denote them with the
different letter $\r$. The functional $\Pi_{2;rel}$ is then
\beq\nome{Pir}
\Pi_{2;rel}[\Phi;\r_i]
=\int d^4x \left\{
\frac{\r_{vcc}}{2}\;
v\cdot c \wedge c +
\frac{\r_{4A}}{8}\;
[2\,(A_\mu\cdot A_\nu)\,(A_\mu\cdot A_\nu)\, +
(A_\mu\cdot A_\mu)\,(A_\nu\cdot A_\nu)]
\right\}
\eeq
and contains the contributions from field monomials
in \re{2r}, which do not appear in the classical action.
By $\Phi$ we indicate the classical fields and sources
$A_\mu$, $c$, $w_\mu$ and $v$.
In conclusion the interacting part of the UV action can be written as
\beq\nome{intac}
S_{int}^{\L_0}=\Pi_{1;rel}[\Phi;\s_i^B]
+\Pi_{2;rel}[\Phi;\r_i^B]
\,,
\eeq
where
$\s_i^B$ and $\r_i^B$ are the bare couplings.
The mass parameter $\s^B_{m_A}$ has positive dimension, while the other
parameters are dimensionless. $\s^B_A$ and $\s^B_{wc}$ are related to the
vector and ghost wave function renormalization,
$\s^B_\alpha$ to the gauge fixing parameter renormalization,
$\s^B_{wcA}$,
$\s^B_{3A}$ and
$\s^B_{4A}$
are related to the bare interaction couplings.
The additional coupling $\r^B_{4A}$ corresponds to the interaction among
four vectors with a group structure not present in the classical
action \re{Scl}.
Similarly $\r^B_{vcc}$ is an interaction term not present in $S_{cl}$.

The generating functional of the Green's functions is
$$
Z[j,u,v;\L]=\exp i W[j,u,v;\L] =\,
\int D \left[ \phi_A \right]\;
\exp i\left\{ S_2^{\L,\L_0}+ S_{int}^{\L_0} +
(j,\phi) \right\} \,,
$$
where we introduced the compact notation for the fields and
corresponding sources
$$
\phi_A=(A^a_\mu,\,c^a,\,\bc^a )\, ,
\;\;\;\;\;\;\;\;
j_A=(j^a_\mu,\, \bar \chi^a ,\, - \chi^a )\,,
$$
$$
(j,\phi) \equiv \int_p j_\mu(-p) \cdot A_\mu(p)
+ \bar \chi(-p) \cdot c(p)
+ \bc(-p) \cdot \chi(p)
\,.
$$
The free cutoff propagators are described by the matrix $D_{AB}$
defined by
$$
\int_p\half \phi_A(-p) D^{-1}_{AB}(p;\L)\phi_B(p)
\equiv
\int_p  \;
\left\{
-\half A_\mu(-p)\cdot A_\mu(p) \, p^2  \;+\; p^2 \bc(-p)\cdot c(p)
\right\}
\,K^{-1}_{\L\L_0}(p)\,.
$$
In $Z[j,u,v;\L]$  and $D_{AB}(p;\L)$ we have
explicitly written only the cutoff
$\L$ since we will consider in any case the limit \UV.
The physical functional $Z[j,u,v]$ is obtained by taking
the limits \IR and \UV.

Finally, the cutoff effective action is defined by taking the Legendre
transform
$$
\G[\phi,u,v;\L]=W[j,u,v;\L] - W[0,0,0;\L] - (j,\phi) \, ,
\;\;\;\;\,\;\;
\phi_A(p) = (2\pi)^4  \frac {\delta W[j,u,v;\L] }{\delta j_A(-p)} \, .
$$

\subsection{Exact renormalization group equation}
Usually the RG equations are obtained by requiring that the
physical quantities are independent of $\L$. In the present
formulation the same flow equation can be simply obtained by observing
that all the $\L$ dependence of $Z[j,u,v;\L]$ is contained in the cutoffs
in the propagators. Thus one easily derives \cite{P} the equation
$$
\dL Z[j,u,v;\L]=-i\frac{ (2\pi)^8}{2}\; \int_q\;\dL{D^{-1}_{BA}(q;\L)}
\frac {\delta^2 Z[j,u,v;\L] }{\delta j_B(q) \delta j_A(-q) } \, .
$$
In the same way one finds the corresponding equation for the cutoff
effective action. The equation is schematically given in fig.~1 and
is obtained by observing that the operator $\L\partial_\L$ acts on each
internal propagator. The first term involves only one vertex;
the second term involves two vertices in such a way to reproduce,
upon $q$-integration, a one-particle irreducible contribution. The
other terms will involve any possible number of vertices.
Of course one can deduce formally this equation. By following the same
steps of refs. \cite{BDM,QED} we obtain
\beq\nome{eveq}
\dL \Pi[\Phi;\L]
=
- \frac i 2 \int_q M_{BA}(q;\L)\; \bG_{AB}[-q,q;\Phi;\L]
\equiv I[\Phi;\L]\,,
\eeq
where
\beq\nome{gamma}
\G[\phi,u,v;\L]=S_2^{\L,\L_0}+\Pi[\Phi;\L]\,,
\eeq
$$
M_{BA}=\Delta_{BC}(q;\L) \dL{ D_{CD}^{-1}(q;\L)} \Delta_{DA}(q;\L)
$$
and $\Delta_{AB}(q;\L)$ is the full propagator.
The auxiliary functional $ \bG_{AB}[q,q';\Phi]\,$
is the inverse of
${\delta^2\G[\phi,u,v]}/ {\delta\phi_A(q)\delta\phi_B(q')}$.
To obtain this we isolate the contribution of the two point function
$$
(2\pi)^8\frac {\delta^2\G[\phi,u,v]} {\delta\phi_A(q)\delta\phi_B(q')}
=
(2\pi)^4\delta^4(q+q') \, \Delta^{-1}_{BA}(q;\L)+
\G_{BA}^{int}[q',q;\Phi]\,.
$$
The auxiliary functional $ \bG_{AB}[q,q';\Phi]\,$ is then given by
the integral equation
\beq \nome{bG}
\bG_{AB}[q,q';\Phi] = (-)^{\delta_B}\,\G_{AB}^{int}[q,q';\Phi]\,-
\int_{q''} \bG_{CB}[-q'',q';\Phi]
\Delta_{DC}(q'';\L)
\G_{AD}^{int}[q,q'';\Phi]
  \, ,
\eeq
where $\delta_B$ is the ghost number and
the indices $A$ and $B$ run over the indices of the fields
$A^a_\mu$, $\bc^a$ and $c^a$.
In terms of the proper vertices
of $\Pi[\Phi;\L]$ the evolution equations are
$$
\dL \Pi_{C_1\cdots C_n}(p_1, \cdots p_{n};\L)
=  -\frac i 2 \int_q M_{BA}(q;\L)\;
\bG_{AB,C_1\cdots C_n}(-q,q;p_1, \cdots p_{n};\L) \,,
$$
where $\bG_{AB,C_1\cdots C_n}(q,q';p_1, \cdots p_{n};\L)$
are the vertices of the auxiliary functional $\bG[\Phi;\L]$
obtained by differentiating with respect to the fields
$\Phi$.
The vertices of this auxiliary functional are obtained
in terms of the proper vertices by expanding \re{bG} and one finds
\beq
\eqalign{
&
\bG_{AB, C_1\cdots C_n} (q,q';p_1,\cdots p_{n};\L)=
 \G_{A B C_1\cdots C_n} (q,q',p_1,\cdots p_{n};\L)
\cr&
-{\sum}'
\G_{A C_{i_1}\cdots C_{i_{k}}C'} (q,p_{i_1},\cdots,p_{i_{k}},Q;\L)
\;\Delta_{C'C''}(Q;\L)
\bG_{C''B,C_{i_{k+1}}\cdots C_{i_{n}}}
(-Q,q';p_{i_{k+1}},\cdots p_{i_{n}};\L) \,,
}
\eeq
where $Q=q+p_{i_1}+\cdots p_{i_{k}}$,
and $\sum'$ is the sum over the combinations of photon and
ghost indices $(i_1 \cdots i_{n})$ taking properly into account the
symmetrization and anti-symmetrization.
Notice that this construction involves vertices with $\bc$ external fields,
which can be obtained from the relation
$\G_{\bc\cdots}(p,\cdots)=-ip_\mu\G_{w_\mu \cdots}(p,\cdots)$.

\section{Relevant parameters and physical conditions}
In order to define the boundary conditions for the evolution
equation of the cutoff effective action
we have to distinguish the relevant and irrelevant vertices.
The interaction part $\Pi[\Phi;\L]$ in \re{gamma} can be written as
$$
\Pi[\Phi;\L]=
 \Pi_{1;rel}[\Phi;\s_i(\L)]
+\Pi_{2;rel}[\Phi;\r_i(\L)]
+\G_{irrel}[\Phi;\L] \,,
$$
where the functional $\G_{irrel}$ contains all irrelevant vertices.
The relevant part of $\Pi[\Phi;\L]$ involves only field monomials with
dimension not larger than four. Therefore the structure of $\Pi_{1,2;rel}$
is the same as the one defined in \re{Pis} and \re{Pir} for the UV
action. Now the couplings $\s_i$ and $\r_i$ depend on $\L$. These nine
relevant couplings are defined as the value of some vertex functions
or their derivatives at a given normalization point. In order to
split the functional $\Pi[\Phi;\L]$ in its relevant and irrelevant
parts, it is then necessary to specify the procedure of extracting the
relevant parameters. This is done in full detail in the next
subsection.

\subsection{Relevant couplings and subtraction conditions}
In this subsection we define the couplings $\s_i(\L)$ and $\r_i(\L)$
as the values at the subtraction points of the vertices with
non-negative dimension. We work in the momentum space.
The subtraction point for the two point functions is assumed
at $p^2=\mu^2$,
while for the $N$ point vertex functions is assumed at the symmetric point
($NSP$) defined by
$$
\bar p_i \bar p_j=\frac{\mu^2}{N-1}(N\delta_{ij}-1)\,,
\;\;\;\;\;\;\;\;\;N=3,4,...\,.
$$
The relevant couplings are involved in the following six
$SU(2)$ singlets vertices with
$$
\dim \le 4\,,
$$
where $n_i$ is the number of fields of the type $i$.

\noindent
1) The vector propagator is the coefficient of
$\half A_\mu(-p)\cdot A_\nu(p)$ and has the form
\beq\nome{propp}
\G_{\mu\nu}(p;\L)=
-g_{\mu\nu}\, p^2 \,K_{\L\L_0}^{-1}(p) +
g_{\mu\nu}\Pi_L(p;\L) + t_{\mu\nu}(p)\,\Pi_T(p;\L)\,,
\eeq
with
$$
t_{\mu\nu}(p)\equiv g_{\mu\nu}\;-\;\frac {p_\mu p_\nu}{p^2} \,.
$$
The three relevant couplings are defined by
\beq\nome{sl}
\Pi_L(p;\L)=\s_{m_A}  (\L)+p^2 \s_\alpha(\L) + \Sigma_L(p;\L)\,,
\;\;\;\;\;
\Sigma_{L}(0;\L)=0\,,
\;\;\;\;\;
\frac{\partial \Sigma_{L}(p;\L)}{\partial p^2}|_{p^2=\mu^2}=0\,,
\eeq
and
\beq\nome{st}
\Pi_T(p;\L)=p^2 \s_A(\L) + \Sigma_T(p;\L)\,,
\;\;\;\;
\Sigma_{T}(0;\L)=0\,,
\;\;\;\;
\frac{\partial \Sigma_{T}(p;\L)}{\partial p^2}|_{p^2=\mu^2}=0\,.
\eeq
{}From these conditions we can factorize in the vertices $\Sigma_{L,T}$
a dimensional function of $p$. Thus $\Sigma_{L,T}$ are ``irrelevant''
and contribute to $\G_{irrel}[\L]$.

\par\noindent
2) The contribution to $\G[\L]$ from three vectors can be written as
$$
A_\mu(p)\cdot A_\nu(q)\wedge A_\r(k)
\; \G_{\mu\nu\r}^{(3A)}(p,q,k;\L)\,,
$$
\beq\nome{3A}
\G_{\mu\nu\r}^{(3A)}(p,q,k;\L)=
p_\nu g_{\mu\r} \G^{(3A)}(p,q,k;\L)
+
\tilde \G_{\mu\nu\r}^{(3A)}(p,q,k;\L)
\,.
\eeq
In the last term all three Lorentz indices are carried by external
momenta. Thus, after factorizing these momenta, one remains with a
function of dimension $-2$. Then the vertex
$\tilde \G_{\mu\nu\r}^{(3A)}(p,q,k;\L)$ is irrelevant and
contributes to $\G_{irrel}$.
The relevant coupling $\s_{3A}(\L)$ is defined by
$$
\G^{(3A)}(p,q,k;\L)= \s_{3A}(\L)+\S^{(3A)}(p,q,k;\L)\,,
\;\;\;\;\;\;\;
\S^{(3A)}(p,q,k;\L)|_{3SP}=0 \,,
$$
so that $\S^{(3A)}(p,q,k;\L)$ is irrelevant.

\par\noindent
3) The contribution to $\G[\L]$ from four vectors is given by two
different $SU(2)$ scalars. The first one involves the field combination
which appears in the classical action:
$$
\frac{1}{4} [A_\mu(p)\wedge A_\nu(q)]
\cdot
[A_\r(k)\wedge A_\s(h)]
\, \G_{1;\mu\nu\r\s}^{(4A)}(p,q,k,h;\L)\,,
$$
\beq\nome{4A1}
\G_{1;\mu\nu\r\s}^{(4A)}(p,q,k,h;\L)=
g_{\mu\r} g_{\nu\s}\, \G_1^{(4A)}(p,q,k,h;\L)
+
\tilde \G_{1;\mu\nu\r\s}^{(4A)}(p,q,k,h;\L)
\,.
\eeq
The other singlet structure is given by
$$
\frac 1 8 [A_\mu(p)\cdot A_\nu(q)]\; [A_\r(k)\cdot A_\s(h)]
\, \G_{2;\mu\nu\r\s}^{(4A)}(p,q,k,h;\L)\,,
$$
\beq\nome{4A2}
\G_{2;\mu\nu\r\s}^{(4A)}(p,q,k,h;\L)
=
(2g_{\mu\r} g_{\nu\s}\,+ g_{\mu\nu} g_{\r\s}\,)\;
\G_2^{(4A)}(p,q,k,h;\L)
+
\tilde \G_{2;\mu\nu\r\s}^{(4A)}(p,q,k,h;\L)
\,.
\eeq
For the two vertices $\tilde \G_{1,2;\mu\nu\r\s}^{(4A)}$
at least two Lorentz indices are carried by external
momenta. Thus these vertices are irrelevant.
The two relevant couplings $\s_{4A}(\L)$ and $\r_{4A}(\L)$ are defined by
$$
\G_1^{(4A)}(p,q,k,h;\L)= \s_{4A}(\L)+\S_1^{(4A)}(p,q,k,h;\L)\,,
\;\;\;\;\;
\S_1^{(4A)}(p,q,k,h;\L)|_{4SP}=0\,,
$$
$$
\G_2^{(4A)}(p,q,k,h;\L)= \r_{4A}(\L)+\S_2^{(4A)}(p,q,k,h;\L)\,,
\;\;\;\;
\S_2^{(4A)}(p,q,k,h;\L)|_{4SP}=0\,.
$$

\par\noindent
4) The interaction part of the ghost propagator and $u$-$c$ vertex
is given by the same function.
The ghost propagator is the coefficient of $\bc(-p)\cdot  c(p)$
\beq\nome{bcc}
\G^{(\bc c)}(p;\L)=
p^2K_{\L\L_0}^{-1}(p) + i p^2 \Pi^{(wc)}(p;\L)
\,.
\eeq
The coefficient of $u_\mu(-p)\cdot c(p)$ has the form
\beq\nome{uc}
\G_\mu^{(uc)}(p;\L)=
-\frac{ip_\mu}{g} +\frac{p_\mu}{g} \Pi^{(wc)}(p;\L) \,.
\eeq
The function $\Pi^{(wc)}$ contains the relevant coupling $\s_{wc}(\L)$
\beq\nome{wc}
\Pi^{(wc)}(p;\L)=
\s_{wc}(\L)+\S^{(wc)}(p;\L) \,,
\;\;\;\;\;\;
\partial_{p^2} \left( p^2\S^{(wc)}(p;\L) \right)|_{p^2=\mu^2}
=0\,.
\eeq

\par\noindent
5) The contribution to $\G[\L]$ from $w$-$c$-$A$ is given by the
$SU(2)$ scalar
$$
w_\mu(p)\cdot c(q)\wedge A_\nu(k)\;
\, \G_{\mu\nu}^{(wcA)}(p,q,k;\L)\,,
$$
\beq\nome{wca?}
\G_{\mu\nu}^{(wcA)}(p,q,k;\L)=
g_{\mu\nu}\,\G^{(wcA)}(p,q,k;\L)
+
\tilde \G_{\mu\nu}^{(wcA)}(p,q,k;\L)\,.
\eeq
The two Lorentz indices in $\tilde \G_{\mu\nu}^{(wcA)}(p,q,k;\L)$
are carried by external momenta. Thus this vertex is irrelevant.
The coupling $\s_{wcA}(\L)$ is defined by
\beq\nome{wcA}
\G^{(wcA)}(p,q,k;\L) = \s_{wcA}(\L) +
\S^{(wcA)}(p,q,k;\L)\,,
\;\;\;\;\;
\S^{(wcA)}(p,q,k;\L)|_{3SP}=0 \,.
\eeq
Due to the fact that $\G$ depends on $\bc$ and $u$ only through the
combination $w_\mu$ one has that the coefficient of $\bc$-$c$-$A$ is
\beq
\nome{bccA}
\G_{\nu}^{(\bc cA)}(p,q,k;\L)=-ip_\mu\,
\, \G_{\mu\nu}^{(wcA)}(p,q,k;\L)\,.
\eeq

\par\noindent
6) The only vertex involving the source $v$
which contains a relevant coupling is the coefficient of
$\half v(p)\cdot c(q) \wedge c(k)$. The coupling $\r_{vcc}$ is given by
$$
\G^{(vcc)}(p,q,k;\L)=-1+\r_{vcc}(\L)+ \S^{(vcc)}(p,q,k;\L)\,,
\;\;\;\;
\S^{(vcc)}(p,q,k;\L)|_{3SP}=0\,.
$$
All the remaining vertices are irrelevant, since they are coefficients of
monomials with dimension higher than four.

\subsection{Boundary conditions}
As discussed in the introduction for the irrelevant vertices
we assume the following boundary condition
\beq\nome{bcirr}
\G_{irr}[\Phi,\L_0] =0 \,,
\eeq
since, due to dimensional reasons, they vanish at the UV scale.
For $\L=\L_0$ the cutoff effective action becomes then local and
corresponds to the bare action \re{intac}, where
the nine bare parameters $\s_i^B$ and $\r_i^B$ are given by
$\s_i(\L_0)$ and $\r_i(\L_0)$. However, in our procedure these
couplings are fixed at $\L=0$ and their value at $\L=\L_0$ can be
perturbatively computed.

For the relevant part, we first give the boundary conditions for the
terms which appear also in the classical action.
For them we impose
$$
\Pi_{1;rel}[\Phi;\s_i(\L=0)] =S_{int}\,,
$$
which gives
\beq\nome{bcs}
\eqalign{
&
\s_{m_A}(0)=\s_{\alpha}(0)=\s_{A}(0)=\s_{wc}(0)=0\,,
\cr&
\s_{3A}(0)=ig
\,,\;\;\;
\s_{4A}(0)=-g^2
\,,\;\;\;
\s_{wcA}(0)=-g\,.
}
\eeq
The remaining two couplings $\r_{4A}(0)$ and $\r_{vcc}(0)$ are fixed by
the ST identities and given in terms of some irrelevant vertices as
follows.

The physical effective action, namely $\G[\Phi;\L=0]$, must satisfy the
ST identities (in the rest of this subsection it will be understood
that $\L=0$ and \UV).
Since, as discussed, the effective action depends on $\bc$ and $u_\mu$
fields only through the combination $w_\mu$, the Slavnov-Taylor identities
are
\beq\nome{ST}
\int_s \;
\left\{
\frac{\delta\G'}{\delta u_\mu^a(s)}
\frac{\delta\G'}{\delta A_\mu^a(-s)}
+
\frac{\delta\G'}{\delta v^a(s)}
\frac{\delta\G'}{\delta c^a(-s)}
\right\}
=0 \,,
\eeq
where  $\G'$ is obtained from
the effective action $\G$ by subtracting the gauge fixing contribution,
\ie the second term in \re{Scl}.

These relations, due to their non-linearity, couple the relevant and
irrelevant vertices.
Since the total action $S_{BRS}$ (without the gauge fixing term)
satisfies these identities,
only the relevant couplings $\r_{4A}(0)$ and $\r_{vcc}(0)$ are
coupled to irrelevant vertices.
We now determine the relations giving the couplings
$\r_{4A}(0)$ and $\r_{vcc}(0)$.

To find the relation for $\r_{vcc}(0)$ we use the simplest identity
involving the $v$-$c$-$c$ vertex. This is obtained by
differentiating \re{ST} with respect to the fields $c^a(p)\,$, $c^b(q)$ and
$\bc^c(k)$ and setting the fields to zero.
We obtain
\beq\nome{ST1}
\G_\mu^{(uc)}(p)\,
\G_{\mu}^{(\bc cA)}(k,q,p)
+
\G_\mu^{(uc)}(q)\,
\G_{\mu}^{(\bc cA)}(k,p,q)
+
\G^{(\bc c)}(k)\,
\G^{(vcc)}(k,p,q)=0\,.
\eeq
By taking $p$, $q$ and $k$ at the symmetric point and
using \re{bcc}, \re{uc} and \re{bccA} we obtain
\beq\nome{rovcc}
\r_{vcc}(0)=-\frac{2}{g}
\left[ \frac {k_\mu p_\nu}{p^2} \tilde \G_{\mu\nu}^{(wcA)}(k,q,p)
\right]_{3SP}
\,,
\eeq
where we used $\G^{(vcc)}(p,q,k)|_{3SP}=-1+\r_{vcc}(0)$
and $\G^{(wcA)}(p,q,k)|_{3SP}=\s_{wcA}(0)=-g$.

We discuss now the determination of value of $\r_{4A}(0)$.
{}From the ST identities for three and four vector vertices, one deduces
$$
\left[ p_\mu q_\nu k_\r h_\s
\G_{2;\mu\nu\r\s}^{(4A)}(p,q,k,h)\right]_{4SP}=0\,,
$$
which gives
\beq\nome{ro4A}
\r_{4A}(0) = - \left[ \frac{3}{p^4}
p_\mu q_\nu k_\r h_\s \, \tilde \G^{(4A)}_{2;\mu\nu\r\s}(p,q,k,h)
\right]_{4SP}
\,.
\eeq

\section{Integral equation}
Once the boundary conditions \re{bcirr}, \re{bcs}, \re{rovcc} and
\re{ro4A} are given, the RG equation \re{eveq} allows one to compute
all vertices, at least perturbatively. The nine relevant parameters
are fixed to satisfy the ST identities by definition and this avoids
the fine tuning problem. The couplings $\s_i(\L_0)$ and $\r_i(\L_0)$
can be computed as a function of $g$ and $\L_0/\mu$ and they will be
already fine tuned.

The boundary conditions we use are of the mixed type: the irrelevant
vertices are fixed at $\L=\L_0$, the relevant couplings at $\L=0$.
Moreover, two of the relevant couplings are given in terms of
irrelevant vertices at the physical point $\L=0$. This kind  of
boundary conditions does not allow to solve the RG equation as an
evolution equation, as in the usual case in which one fixes both
irrelevant and relevant vertices at $\L=\L_0$.
In our case the best way to formulate the problem is to convert the RG
equation \re{eveq} into a Volterra integral equation which embodies
the boundary conditions. As we shall show this formulation is also
quite suitable for generating the iterative solution, \ie loop
expansion, and for proving renormalizability. Moreover, it could be
used in principle also for nonperturbative studies.

We introduce the vertices of the functional $I[\Phi;\L]$, defined in
\re{eveq}
\beq\nome{Ivert}
I_{C_1\cdots C_n}(p_1,\cdots p_n;\L)=
-\frac i 2 \int_q M_{BA}(q;\L)\,
\bG_{AB,C_1 \cdots C_n}(-q,q;p_1,\cdots p_n;\L)\,.
\eeq
The functional $I[\Phi;\L]$ can be separated into its relevant and
irrelevant part
$$
I[\Phi;\L]=I_{rel}[\Phi;\L]+I_{irr}[\Phi;\L]\,.
$$
To extract the relevant parameters in $I_{rel}[\Phi;\L]$ we proceed
as in subsect. 3.1. We consider the six vertices in $I[\Phi;\L]$
with non-negative dimension and perform the Lorentz
decomposition and the Taylor type of expansion as done in
subsect.~3.1. For example from the following vertices of $I[\Phi;\L]$
\beq\nome{Ir}
\eqalign{
&
I_{\mu\nu}(p;\L)=
g_{\mu\nu}\,I_L(p;\L)+ t_{\mu\nu}\,I_T(p;\L)\,, \cr
&
I_{\mu\nu}^{(wcA)}(p,q,k;\L)=
g_{\mu\nu}\,I^{(wcA)}(p,q,k;\L)
+\tilde I_{\mu\nu}^{(wcA)}(p,q,k;\L)\,,
\cr&
I_{2;\mu\nu\r\s}^{(4A)}(p,q,k,h;\L)
=
(2g_{\mu\r} g_{\nu\s}\,+ g_{\mu\nu} g_{\r\s}\,)\;
I_2^{(4A)}(p,q,k,h;\L)
+
\tilde I_{2;\mu\nu\r\s}^{(4A)}(p,q,k,h;\L)\,,
}
\eeq
we extract the relevant parameters
\begin{eqnarray*}
&& I_{L}(p;\L)=I_{L}(0;\L)+p^2\partial_{\bp^2}I_{L}(\bp;\L)+\cdots
\equiv
S_{m_A}(\L)+p^2 S_\alpha(\L)+\cdots\,,
\\
&& I_{T}(p;\L)=p^2\partial_{\bp^2}I_{T}(\bp;\L)+\cdots
\equiv
p^2 S_A(\L)+\cdots\,,
\\
&& I^{(wcA)}(p,q,k;\L)=I^{(wcA)}(p,q,k;\L)|_{3SP}+\cdots
\equiv
S_{wcA}(\L)+\cdots\,,
\\
&& I_{2}^{(4A)}(p,q,k,h;\L)=I_2^{(4A)}(p,q,k,h;\L)|_{4SP}+\cdots
\equiv
R_{4A}(\L)+\cdots\,,
\end{eqnarray*}
where the dots stand for irrelevant parts.
The relevant part of the functional $I[\Phi;\L]$ can be again written
in terms of $\Pi_{1,2;rel}$
$$
I_{rel}[\Phi;\L]=\Pi_{1;rel}[\Phi;S_i(\L)]+\Pi_{2;rel}[\Phi;R_i(\L)]
\,.
$$
The integral equations for the relevant and irrelevant parts are then
\beq\nome{int1}
\Pi_{rel}[\Phi;\L]=\Pi_{rel}[\Phi;0]+\int_0^\L \frac{d\l}{\l}
I_{rel}[\Phi;\l]\,,
\eeq
\beq\nome{int2}
\Pi_{irr}[\Phi;\L]=-\int_\L^{\L_0} \frac{d\l}{\l} I_{irr}[\Phi;\l]\,.
\eeq
Finally, by using the form of $\Pi_{rel}[\Phi;0]$, we have
\beq\nome{inteq}
\Pi[\Phi;\L]= S_{int}[\Phi] + \Pi_{2;rel}[\Phi;\r_i(0)]
+  \int_0^\L \, \frac{d\l}{\l}I[\Phi;\l]
-  \int^\infty_0 \, \frac{d\l}{\l} \left\{I[\Phi;\l]-
I_{rel}[\Phi;\l] \right\}
\,,
\eeq
where $S_{int}[\Phi]$ is the interacting part of the BRS action
\re{Stot}. $\Pi_{2;rel}[\Phi;\r_i(0)]$ contains the two additional
relevant couplings fixed by the ST constraint and given by
\beq\nome{r0}
\eqalign{&
\r_{vcc}(0)
=\frac 2 g \int_0^\infty \frac{d\l}{\l} \left[
\frac {k_\mu p_\nu}{p^2} \tilde I^{(wcA)}_{\mu\nu}(k,q,p;\l)
\right]_{3SP}\,,
\cr &
\r_{4A}(0)
=3 \int_0^\infty \frac{d\l}{\l} \left[
\frac{p_\mu q_\nu k_\r h_\s}{p^4} \tilde
I^{(4A)}_{2;\mu\nu\r\s}(p,q,k,h;\l) \right]_{4SP}\,,
}
\eeq
where $\tilde I_{\mu\nu}^{(wcA)}$ and $\tilde I_{2;\mu\nu\r\s}^{(4A)}$
are given in \re{Ir}.

In the last $\l$ integration in \re{inteq} we have sent \UV.
The proof that the subtractions make all integrals finite
in the \UV limit is sketched in subsect.~5.4.
In the next section, where we compute at one loop some vertex functions,
we will see how the subtractions make all integrals finite in the \UV
limit. Only for $\L=0$ the total effective action
$\G[\Phi]=S_2+\Pi[\Phi;0]$ must satisfy the ST identities. This will be
analyzed in the next section at one loop level.

\section{Loop expansion}
The loop expansion is obtained by solving iteratively \re{inteq}.
The starting point is the zero loop order
$\Pi^{(0)}_{1;rel}[\Phi;\s_i(0)]=S_{int}[\Phi]\,.$
This corresponds to
give the relevant vertices in the tree approximation, \ie
\begin{eqnarray*}
&&
\s_{m_A}^{(0)}(\L)=\s_{\alpha}^{(0)}(\L)=
\s_{A}^{(0)}(\L)=\s_{wc}^{(0)}(\L)=0\,,
\\ &&
\s_{3A}^{(0)}(\L)=ig
\,,\;\;\;
\s_{4A}^{(0)}(\L)=-g^2
\,,\;\;\;
\s_{wcA}^{(0)}(\L)=-g\,,
\\ &&
\r_{4A}^{(0)}(\L)=\r_{vcc}^{(0)}(\L)=0\,.
\end{eqnarray*}
At zero loop we have also the auxiliary vertices in $\bG^{(0)}[\Phi;\L]$.
For example we have the vector-vector contributions to order $g^2$
given in fig.~2. They correspond to the usual Feynman diagrams at tree
level with cutoff propagators.

Assume we have computed the cutoff effective action
$\Pi^{(\ell)}[\Phi;\L]$ up to loop $\ell$.
At the next loop this functional can be computed iteratively by the
following steps.
First from eq.~\re{Ivert} we  compute the vertices
$I^{(\ell+1)}_{C_1\cdots C_n}(p_1,\cdots p_n;\l)$.
{}From eq.~\re{Ir} and eq.~\re{r0} we obtain
the vertices $\tilde I_{\mu\nu}^{(wcA)(\ell+1)}(\L)$
and $\tilde I_{2;\mu\nu\r\s}^{(4A)(\ell+1)}(\L)$
and the couplings
$\r_{vcc}^{(\ell+1)}(0)$ and $\r_{4A}^{(\ell+1)}(0)$.
We then obtain the functionals
$I^{(\ell+1)}[\Phi;\L]$, $\,I^{(\ell+1)}_{rel}[\Phi;\L]$ and
$\Pi^{(\ell+1)}_{2;rel}[\Phi;\r_i(0)]$.
Finally from eq.~\re{inteq} we compute
the functional $\Pi^{(\ell+1)}[\Phi;\L]$.

\subsection{One loop vertex functions}
In this subsection we compute to one loop the six vertices
with $\dim \le 4$ which contain the relevant couplings.
We also evaluate to one loop the couplings $\s_i^B$ and $\r_i^B$
of the bare action \re{intac} given by $\s_i(\L_0)$ and
$\r_i(\L_0)$.

\noindent
1) Vector propagator.

\noindent
We have three contributions to the integrand, corresponding to the three usual
Feynman amplitudes
\beq\eqalign{
& I_{\mu\nu}(p;\L)= \;
- \frac{i g^2}{3}
\dL \int_q \frac{K_{\L\L_0}(q)}{q^2}
t^{abba}_{\mu\alpha\alpha\nu}
\cr&
\;\;\; - i g^2
\dL \int_q \frac{K_{\L\L_0}(q,p+q)}{q^2 (p+q)^2}
\left[
V_{\mu\alpha\beta}(p,q,-p-q)
V_{\alpha\beta\nu}(-q,p+q,-p)
+
q_\mu (p+q)_\nu
\right]
\,,
}
\eeq
where $K_{\L\L_0}(q,p+q)=K_{\L\L_0}(q)K_{\L\L_0}(p+q)$ and
$$
V_{\mu\nu\r}(p,q,k)=
g_{\mu\nu}(p-q)_\r
+g_{\nu\r}(q-k)_\mu
+g_{\mu\r}(k-p)_\nu
\,,
$$
\begin{eqnarray*}
t^{a_1 \cdots a_4}_{\mu_1 \cdots \mu_4}
&=&
\left(\eps^{a_1a_2c}\eps^{ca_3a_4}-\eps^{a_1a_4c}\eps^{ca_2a_3}
\right)g_{\mu_1\mu_3}g_{\mu_2\mu_4} +
\left(\eps^{a_1a_3c}\eps^{ca_2a_4}-\eps^{a_1a_4c}\eps^{ca_3a_2}
\right)g_{\mu_1\mu_2}g_{\mu_3\mu_4} \\
&+&
\left(\eps^{a_1a_3c}\eps^{ca_4a_2}-\eps^{a_1a_2c}\eps^{ca_3a_4}
\right)g_{\mu_1\mu_4}g_{\mu_2\mu_3}\,,
\end{eqnarray*}
which are the three and four vector elementary vertices, respectively.
Notice that there is a factor $i$ coming from the $q$-integration.
Thus we get
\beq\nome{vectorI}
I_{L}(p;\L)=
-i g^2 \dL \left[
\int_q \frac{K_{\L\L_0}(q,p+q)}{q^2 (p+q)^2}
(2 q^2+10 pq+3p^2+8\frac{(pq)^2}{p^2})
-6 \int_q \frac{K_{\L\L_0}(q)}{q^2} \right] \,,
\eeq
\beq
I_{T}(p;\L)=
\frac{-ig^2}{3} \dL \left[
\int_q \frac{K_{\L\L_0}(q,p+q)}{q^2 (p+q)^2}
(8 q^2-24 pq+6 p^2-32 \frac{(pq)^2}{p^2})
\right] \,.
\eeq
The relevant couplings for large $\L$ are
\beq
\eqalign{
&
\s^{(1)}_{m_A}(\L) =
\int_0^{\L}\frac{d\l}{\l}I_L(0;\l)
= ig^2 \int_q \frac{K_{0\L}(q)}{q^4} (8\frac{(pq)^2}{p^2}-4 q^2)
= \frac{g^2}{8 \pi^2} \L^2 + {\cal O}(1) \,.
\cr&
\s^{(1)}_{\alpha}(\L)=
\int_0^{\L}\frac{d\l}{\l} \frac{\partial}{\partial \bp^2} I_L(\bp;\l)
={\cal O}(1) \,,
\cr&
\s^{(1)}_{A}(\L)=
\int_0^{\L}\frac{d\l}{\l} \frac{\partial}{\partial \bp^2} I_T(\bp;\l)
=-\frac{10}{3}\frac{g^2}{16 \pi^2} \log (\frac{\L^2}{\mu^2}) + {\cal O}(1) \,.
}
\eeq
{}From \re{int2} the irrelevant parts of the vector propagator are
$$
\Sigma_{L,T}(p;\L)=-\int_\L^{\L_0} \frac{d\l}{\l}
\left[ I_{L,T}(p;\l)-I_{L,T}(0;\l)-p^2
\frac{\partial}{\partial \bp^2} I_{L,T}(\bp;\l)
\right] \,.
$$
Because of the subtractions, we can take the \UV limit. The physical
value is obtained setting $\L=0$ and we have
$$
\Sigma_L(p;0)=
-i g^2 \int_q \left[
\frac{2 q^2+10 pq+3p^2+8\frac{(pq)^2}{p^2}}{q^2 (p+q)^2}
-\frac{4}{q^4}-p^2\frac{\partial}{\partial \bp^2}
\frac{2 q^2+10 \bp q+3\bp^2+8\frac{(\bp q)^2}{\bp^2}}{q^2 (\bp+q)^2}
\right] \,,
$$
$$
\Sigma_T(p;0)=
\frac{-ig^2}{3} \int_q \left[
\frac{8 q^2-24 pq+6 p^2-32 \frac{(pq)^2}{p^2}}{q^2 (p+q)^2}
-p^2\frac{\partial}{\partial \bp^2}
\frac{8 q^2-24 \bp q+6 \bp^2-32 \frac{(\bp q)^2}{\bp^2}}{q^2 (\bp+q)^2}
\right] \,.
$$
In subsect.~5.3 we will show that $\S_L(p;0)=0$, namely the vector propagator
is
transverse, as required by the ST identities.
For $\Sigma_T(p;0)$, for instance, we can write
$$
\frac{\partial}{\partial p^2}\Sigma_T(p;0)
=-\frac{10}{3}\frac{g^2}{16 \pi^2} \log (\frac{p^2}{\mu^2})\,.
$$

\noindent
2) Ghost propagator.

\noindent
The ghost propagator is given in terms of $\Pi^{(wc)}$.
We have one contribution
\beq\nome{Imu}
p_\mu I^{(wc)}(p;\L)=-2 g^2
\dL \left[ \int_q \frac{K_{\L\L_0}(q,p+q)}{q^2 (p+q)^2}
q_\mu \right]  \,.
\eeq
We obtain then the ghost wave function coupling
\beq
\s^{(1)}_{wc}(\L)=
\int_0^{\L}\frac{d\l}{\l} \frac{\partial}{\partial \bp^2}
\left(\bp^2 I^{(wc)}(\bp;\l)\right)
=-i \frac{g^2}{16 \pi^2} \log (\frac{\L^2}{\mu^2}) + {\cal O}(1) \,,
\eeq
for large $\L$. From \re{int2} the irrelevant part of the $w$-$c$ vertex
is, at $\L=0$
$$
p^2\S^{(wc)}(p;0)=-2 g^2
\int_q \left[ \frac{pq}{q^2 (p+q)^2}-p^2
\frac{\partial}{\partial \bp^2} \frac{\bp q}{q^2 (\bp+q)^2}
\right]  \,,
$$
where again we have removed the UV cutoff since the momentum
integration is convergent.

\noindent
3) $w$-$c$-$A$ vertex.

\noindent
In this case the integrand has two contributions
\begin{eqnarray}\nome{Imunu}
\lefteqn{
I^{(wcA)}_{\mu\nu}(p,k,-p-k;\L)= \dL \biggr\{
i g^3 \int_q \frac{K_{\L\L_0}(q,q+k,p-q)}
{q^2 (q+k)^2 (p-q)^2}
}
&& \nonumber \\ && \times
[g_{\mu\nu}(q^2-qk-2qp)-q_\mu k_\nu +
q_\nu (3k_\mu +p_\mu) -p_\nu k_\mu] \biggr\}
\,.
\end{eqnarray}
The coupling $\s_{wcA}(\L)$ for large $\L$ is
\beq
\s^{(1)}_{wcA}(\L)=
\int_0^{\L}\frac{d\l}{\l} I^{(wc A)}(p,k,q;\l)|_{3SP}
=\frac{g^3}{16 \pi^2} \log (\frac{\L^2}{\mu^2}) + {\cal O}(1) \,.
\eeq
This vertex allows one to obtain the coupling $\r_{vcc}(0)$.
Extracting from \re{Imunu} the vertex $\tilde I_{\mu\nu}^{(wca)}$
and inserting it in \re{r0} we find
\beq
\r^{(1)}_{vcc}(0)=
-i g^2 \int_q \frac{p k + 2q p + q k }
{q^2 (q+k)^2 (p-q)^2}\vert_{3SP}
\simeq\; 4.6878\;\frac{g^2}{16\pi^2}\,,
\eeq
where the value is found by numerical integration
over the Feynman parameters.

\noindent
4) $v$-$c$-$c$ vertex.

\noindent
In this case the integrand has one contribution and we have
\beq\nome{Iv2c}
I^{(vcc)}(p,k,-p-k;\L)=- i g^2 \dL \left[
\int_q \frac{K_{\L\L_0}(q,q+k,p-q)}
{q^2 (q+k)^2 (p-q)^2} \;
(pq-q^2) \right]
\,.
\eeq
The relevant coupling for large $\L$ is
\beq
\r^{(1)}_{vcc}(\L)=
\int_0^{\L}\frac{d\l}{\l} I^{(vcc)}(\bp;\l)
=\frac{g^2}{16 \pi^2} \log (\frac{\L^2}{\mu^2}) + {\cal O}(1) \,.
\eeq

\noindent
5) Three vector vertex.

\noindent
The integrand for the three vector vertex is too long
and we give only the result for the coupling for large $\L$
\beq
\s^{(1)}_{3A}(\L)=
i\frac 4 3 \frac{g^3}{16 \pi^2} \log (\frac{\L^2}{\mu^2}) + {\cal O}(1) \,.
\eeq

\noindent
6) Four vector vertex.

\noindent
Also for this vertex the integrand is too long and we give only the
values for the two couplings for large $\L$
$$
\s^{(1)}_{4A}(\L)=
\frac 2 3 \frac{g^4}{16 \pi^2} \log (\frac{\L^2}{\mu^2}) + {\cal O}(1) \,,
$$
$$
\r^{(1)}_{4A}(\L)=
-10 \frac{g^4}{16 \pi^2} \log (\frac{\L^2}{\mu^2}) + {\cal O}(1) \,.
$$
{}From this vertex we obtain the coupling $\r_{4A}(0)$ given in
\re{r0}. From a numerical integration over the Feynman parameters
we find
\beq
\r^{(1)}_{4A}(0)\simeq\; 1.2329\;\frac{g^2}{16\pi^2}\,.
\eeq

\subsection{One loop beta function}

We discuss in the present formulation the role of the subtraction
point $\mu$ and deduce the beta function (see also ref.~\cite{H} for the
scalar case).
In this subsection we take the physical values $\L=0$ and \UV.
Denote by $\G_{nA,mwc}(g,\mu)$
the vertices with $n$ vector fields and $m$ pairs $w_\mu\, c$, which
satisfy the physical conditions \re{bcs} with couplings $\s_i$  and $\r_i$
defined as in subsect.~3.1 at the scale $\mu$.
Therefore $\mu$ is the only dimensional parameter,
apart for the external momenta.
However physical measurements should not depend on the specific
value of $\mu$.
Consider the vertices $\G_{nA,mwc}(g',\mu')$ with coupling $g'$ at a
new scale $\mu'$.
The request that the two sets of vertices  $\G_{nA,mwc}(g,\mu)$
and $\G_{nA,mwc}(g',\mu')$ describe the same theory
implies that the corresponding effective actions
$\G[\Phi;g,\mu]$  and $\G[\Phi';g',\mu']$ are equal, where the
fields in $\Phi$ and $\Phi'$ are related by a rescaling,
$A'_\mu(p)=\sqrt{Z_A}A_\mu(p)$ and $w_\mu'(q)c'(p)=Z_c w_\mu(q) c(p)$.
This implies that the two sets of vertices are related by
$$
\G_{nA,mwc}(g',\mu')=Z_A^{-n/2} Z_c^{-m} \G_{nA,mwc}(g,\mu) \,.
$$
To obtain $Z_A$ we use \re{propp}-\re{st}, which give
$$
Z_A=1-\frac{\partial}{\partial p^2}\S_T(p;g,\mu)|_{p^2=\mu'^2}
\equiv 1-f_A(\mu';g,\mu) \,.
$$
Similarly from \re{wc} we find
$$
Z_c=1+i\frac{\partial}{\partial p^2}\left[p^2 \S^{(wc)}(p;g,\mu)
\right]|_{p^2=\mu'^2}
\equiv 1+f_c(\mu';g,\mu)  \,.
$$
We can now give the expression of the beta function.
By using the physical condition \re{wcA} for the $w$-$c$-$A$ vertex
and the relation
$\G^{(wcA)}_{\alpha\beta}(g',\mu')=Z_A^{-1/2}Z^{-1}_c
\G^{(wcA)}_{\alpha\beta}(g,\mu)$
we obtain the renormalization group relation
$$
g'= \frac{g-f_g(\mu';g,\mu)}
{\left( 1+f_c(\mu';g,\mu) \right)\left( 1-f_A(\mu';g,\mu) \right)^{1/2}}
\,,
$$
where $f_g(\mu';g,\mu)=\S^{(wca)}(p,q,k;g,\mu)|_{3SP'}$.
The beta function is obtained by considering an infinitesimal
scale change and is given by
$$
\beta(g)=\mu'\partder{\mu'}
\left\{-f_g(\mu';g,\mu)-g\,f_c(\mu';g,\mu)+\half g\, f_A(\mu';g,\mu)
\right\}|_{\mu'=\mu}\,.
$$
The three dimensionless quantities $f_i(\mu';g,\mu)$ are functions of $g$
and the ratio $\mu'/\mu$, thus the beta function depends only on $g$.
{}From the above results we find, at one loop order,
$$
f_A=-\frac{10}{3}\frac{g^2}{16\pi^2}\log(\frac{\mu'^2}{\mu^2})
\,,\;\;\;\;
f_c=\frac{g^2}{16\pi^2}\log(\frac{\mu'^2}{\mu^2})
\,,\;\;\;\;
f_g=\frac{g^3}{16\pi^2}\log(\frac{\mu'^2}{\mu^2})
\,.
$$
Thus
$$
\beta^{(1)}(g)=
-\frac {11}{3} \frac {1}{16\pi^2}g^3 \mu'\partder{\mu'} \ln\frac{\mu'^2}{\mu^2}
=-\frac {22}{3}\frac{g^3}{16\pi^2} \,,
$$
which is the usual one loop result.

Since there is only one coupling, the beta function can be defined
also from the three and four vector vertex. One finds
$$
\beta(g)=\mu'\partder{\mu'}
\left\{-if_{3A}(\mu';g,\mu)+\frac 3 2 g\, f_A(\mu';g,\mu)
\right\}|_{\mu'=\mu}
$$
and
$$
\beta(g)=\mu'\partder{\mu'}
\left\{-\frac{1}{2g} f_{4A}(\mu';g,\mu)+ g\, f_A(\mu';g,\mu)
\right\}|_{\mu'=\mu}\,,
$$
where $f_{3A}(\mu';g,\mu)=\G^{(3A)}(p_i;g,\mu)|_{3SP'}$ and
$f_{4A}(\mu';g,\mu)=\G^{(4A)}_1(p_i;g,\mu)|_{4SP'}$. At one loop they
are given by
$$
f_{3A}=i\frac{4}{3}\frac{g^3}{16\pi^2}\log(\frac{\mu'^2}{\mu^2})
\,,\;\;\;\;
f_{4A}=\frac{2}{3}\frac{g^4}{16\pi^2}\log(\frac{\mu'^2}{\mu^2})
\,.
$$
We have then that these definitions give the same one loop
value for the beta function.

\subsection{One loop ST identities}
In this subsection we show how two simple ST identities are satisfied
at one loop order in the physical limit. Thus in this subsection we set $\L=0$
and we will take \UV.
The first one is the vector propagator transversality and is obtained from
\re{ST} by derivating with respect to $c^a(p)$ and $A_\mu^b(q)$. We get
$$
\G_\mu^{(uc)}(p) \left[ p_\mu p_\nu + \G_{\mu\nu}(p)\right]=0 \,,
$$
which implies $\Sigma_L(p)=0$.
To show this at one loop, we consider first the unsubtracted integral
$$
\Sigma'_L(p)=-\int_0^{\L_0} \frac{d\l}{\l} I_L(p;\l) \,.
$$
{}From \re{vectorI}, after some algebra, we find
$$
\Sigma'_L(p)=-i \int_q \frac{K_{0\L_0}(q)}{q^2}
\biggr[\frac{ 4pq}{p^2} \left( K_{0\L_0}(q+p)-K_{0\L_0}(q-p) \right)
+6\left( K_{0\L_0}(q+p)-K_{0\L_0}(q) \right) \biggr] \,.
$$
Due to the difference of the two cutoff functions we have that $q^2$
is forced into the region $q^2 \sim \L_0^2$.
By taking for instance an exponential UV cutoff one has
$$
K_{0\L_0}(q+p)- K_{0\L_0}(q-p)=-4\frac{p\cdot q}{\L_0^2}
\left\{1-\frac{p^2}{\L_0^2}+\frac 2 {3}
\frac{(p\cdot q)^2}{\L_0^4} +\cdots\right\}
\, e^{-{q^2}/{\L_0^2}}
$$
and obtains
$$
\Sigma'_L(p) = a\L_0^2 + b p^2 + {\cal O} (\frac{p^2}{\L^2_0})\,,
$$
where $a$ and $b$ are numerical constants.
In this calculation the effect of the non invariant regularization
is clear.
The divergent integral with the cutoff functions gives a surface
term which destroys the transversality of the propagator.
However, the longitudinal contributions are of relevant type, thus they
are cancelled by imposing the boundary conditions and taking the limit \UV.
One finds
$$
\Sigma_L^{(1)}(p)=\Sigma'_L(p)-\Sigma'_L(0)
-p^2 \partial_{\bp^2} \Sigma'_L(\bp)|_{\bp^2=\mu^2}=
{\cal O} (\frac{p^2,\mu^2}{\L^2_0}) \to 0\,.
$$
Notice that for $\L\neq 0$ the longitudinal part of the photon propagator
is different from zero both in the relevant ($\s_{m_A}(\L)$ and
$\s_\alpha(\L)$) and irrelevant ($\S_L(p;\L)$) parts.

We now verify the identity \re{ST1}. At one loop order it becomes
\begin{eqnarray*}
&&
\frac 1 g \int_0^{\L_0} \frac{d\l}{\l}
\biggr[ p_\nu k_\mu \left( I_{\mu\nu}^{(wcA)}(k,h,p;\l)
-g_{\mu\nu}I_{rel}^{(wcA)}(\mu^2;\l) \right) +
\;\;\;\;h\leftrightarrow p \;\; \biggr]
\\ &&
-i \int_0^{\L_0} \frac{d\l}{\l}\biggr[ k p \left( I^{(wc)}(p;\l)-
I_{rel}^{(wc)}(\mu^2;\l)\right)+\;\;\; p\rightarrow h \;\;\;+
\;\;\;p \rightarrow k \biggr]
\\ && +
k^2 \int_0^{\L_0} \frac{d\l}{\l} \left( I^{(vcc)}(k,p,h;\l)-
I_{rel}^{(vcc)}(\mu^2;\l) \right)
-k^2 \r_{vcc}(0) \to 0 \;\;\;\;\; \mbox{for} \;\L_0\to \infty\,.
\end{eqnarray*}
As above we keep the UV cutoff $\L_0$, so that we can consider separately
the integrands $I$ and their subtractions $I_{rel}$.
{}From \re{Imu}, \re{Imunu} and \re{Iv2c}
the unsubtracted contribution can be cast in the form
$$
ig^2k_\mu\biggr\{\int_q k_\mu\frac{K_{0\L_0}(q,q+k)}{q^2(q+k)^2}
\left[K_{0\L_0}(q-h)-1\right]\;\;+\;\;
(k\rightarrow  h,\;h\rightarrow  k)\;\;+\;\;
(k\rightarrow  p,\; h\rightarrow  h)\biggr\}\,.
$$
As in the previous case, the factor $(K_{0\L_0}-1)$ forces the
integration momentum to be of the order $q^2 \sim \L_0^2$ so that
the integral vanishes for \UV.
It is easy to verify that also the remaining terms vanish for \UV.

\subsection{Perturbative renormalizability}
Perturbative renormalization is essentially based on power counting.
To prove that the theory is perturbatively renormalizable one has to
show  that the integral equations give a finite result in the limit \UV.
This can be done perturbatively by iterating eqs.~\re{int1} and \re{int2}.
In order to see that the integrations
over $\l$ are convergent, we have to estimate the behaviour of the integrands
for large $\l$.
The analysis can be simplified, following Polchinski, by introducing the norm
$$
|\G(n_A,n_{\bc c},n_u,n_v)|_\L\,
 \equiv \Maxlp |\G_{C_1\cdots C_n}(p_1,\cdots p_n;\L)|\,,
$$
which allows us to ignore the momentum dependence.
Since the $\L$-dependence is fixed only by the
number of fields, to simplify the notation, we have indicated
in the vertices only the numbers $n_A$ of vectors, $n_{\bc c}$
of ghost-antighost pairs, $n_u$ and $n_v$ of $u$ and $v$ sources.
We deduce perturbative renormalizability by proving that the
$\L$ dependence of this norm is given by power counting.
We show this by induction on the number of loops.
Namely, at loop $\ell$, we assume for large $\L$ the
following behaviour
\beq\nome{pwc}
|\G^{(\ell)}(n_A,n_{\bc c},n_u,n_v)|_\L\,
=\,{\cal O}(\L^{4-n_A-2n_{\bc c}-2n_u-2n_v})\,,
\eeq
which is satisfied at $\ell=0$.
We neglect for simplicity all possible $\ell$-dependent powers of
$\log\frac{\L}{\mu}$.

We now proceed by iteration and prove that the behaviours \re{pwc} are
reproduced at the loop $\ell+1$.
We follow the method of ref.~\cite{BDM,QED}.
First of all one notices that the norm of the auxiliary vertices have
the same behaviours as the corresponding vertices, as can be
seen from their definition. We have then
\beq
|\bG^{(\ell)}_{AB}(-q,q;n_A,n_{\bc c},n_u,n_v;\L)|_\L=
{\cal O}(\L^{2-n_A-2n_{\bc c}-2n_u-2n_v})\,.
\eeq
{}From this behaviour one obtains the bound for the integrands \re{Ivert}
of the vertices at loop $\ell+1$
\beq
|I^{(\ell+1)}(n_A,n_{\bc c},n_u,n_v)|_\L\,
=\,{\cal O}(\L^{4-n_A-2n_{\bc c}-2n_u-2n_v})\,,
\eeq
where one uses the fact that $M_{AB}$ gives a factor ${\L}^{-2}$.
Moreover, the derivative of the integrands $I^{(\ell+1)}$
with respect to the external momenta reduces the powers of $\L$
\beq\nome{der}
|\partial_p^m I^{(\ell+1)}(n_A,n_{\bc c},n_u,n_v)|_\L\,
=\,{\cal O}(\L^{4-n_A-2n_{\bc c}-2n_u-2n_v-m})\,.
\eeq
The proof of this is given for instance in ref.~\cite{BDM}.

Finally, from these behaviours to obtain the vertices at loop $\ell+1$.
For the relevant couplings the integrand in \re{int1}
grows with $\l$ and the result is dominated by the upper limit $\L$.
This reproduces immediately \re{pwc}.

For the irrelevant vertices, which are given by \re{int2}, the
$\l$-integration goes up to infinity.
We treat separately the irrelevant vertices with negative and
non-negative dimension. For the first ones
the integration over $\l$ is convergent and the integral
is dominated by the lower limit, thus reproducing
at loop $\ell+1$ the assumption \re{pwc}. For the second ones
due to the subtractions the integrand can be expressed as
derivative with respect to the external momenta for
which we can use \re{der}.
Consider for instance the case of $\S^{(\ell+1)}_L(p;\L)$
We have
\beq
|I^{(\ell+1)}_L(p;\l)-I^{(\ell+1)}_{rel\,L}(p;\l)|_\l\sim p^4|
\partial^4 I^{(\ell+1)}_L(p;\l)|_\l
=p^4{\cal O}(\l^{-2})
\eeq
for $p^2<\l^2$.
Then by inserting the behaviour \re{der} in \re{int2} we have that
the $\l$-integration is convergent and dominated by the lower limit $\L$.
Thus the assumption \re{pwc} is recovered.

\section{Functional identity}
In this paper we have assumed that the interaction part of the cutoff
effective action depends only on the combination of fields $w_\mu(p)$,
namely we have the following functional identity
\beq\nome{FI}
-igp_\mu\frac{\delta\Pi[\Phi;\L]}{\delta u_\mu^a(p)}=
\frac{\delta\Pi[\Phi;\L]}{\delta \bc^a(p)}\,.
\eeq
Since $w_\mu$ has mass dimension $2$ we have reduced the number of
vertices with non-negative dimension and consequently the number of
relevant couplings. Moreover, this identity simplifies the ST identities.

In this section we will prove that, by a proper choice of the boundary
conditions of the evolution equation, this identity is maintained at
all loops.

If we do not assume \re{FI} the fields $\bc$ and $u_\mu$ enter
independently in the effective action.
Then instead of the two monomials containing $w_\mu$ in \re{7s}
we would have four relevant couplings associated to the monomials
$$
\bc \cdot c \,, \;\;\;u_\mu \cdot c \,,\;\;\;
(\partial_\mu \bc) \cdot (c \wedge A_\mu) \,, \;\;\;
u_\mu \cdot (c \wedge A_\mu)\,.
$$
The four couplings associated with these monomials are defined by
\begin{eqnarray*}
&&\Pi^{(\bc c)}(p;\L)=
i p^2 (\s_{\bc c}(\L) +\S^{(\bc c)}(p;\L))\,,
\\
&&\Pi_\mu^{(uc)}(p;\L)=
p_\mu (\s_{uc}(\L)+\S^{(uc)}(p;\L)) \,,
\\
&&\G_{\mu}^{(\bc cA)}(p,q;\L)=p_\mu[\s_{\bc cA}+
\S^{(\bc cA)}(p,q;\L)]+
q_\mu \tilde\G^{(\bc cA)}(p,q;\L)\,,
\\
&&\G_{\mu\nu}^{(ucA)}(p,q;\L)=
g_{\mu\nu}[\s_{ucA}(\L) + \S^{(ucA)}(p,q;\L)]+
\tilde \G_{\mu\nu}^{(ucA)}(p,q;\L)\,,
\end{eqnarray*}
where the normalization conditions are
$$
\partial_{p^2} \left( p^2 \S^{(\bc c)}(p;\L) \right)|_{p^2=\mu^2}=
\partial_{p^2} \left( p^2 \S^{(uc)}(p;\L) \right)|_{p^2=\mu^2}=0\,,
$$
$$
\S^{(\bc cA)}(p,q;\L)|_{3SP}=
\S^{(ucA)}(p,q;\L)|_{3SP}=0 \,.
$$
The evolution equations and their integral solutions are completely analogous
to the previous case.

The field $\bc$ enters in four other monomials of dimension four
$$
(\partial_\mu c) \cdot (\bc \wedge A_\mu) \,, \;\;\;
(\bc \wedge c)\cdot (A_\mu \wedge A_\mu)\,, \;\;\;
(\bc \cdot A_\mu) (c \cdot A_\mu)\,, \;\;\;
(\bc \wedge c)\cdot (\bc \wedge c)\,.
$$
If the functional identity holds,
whenever there is a $\bc$ field in a vertex, we can extract a power
of its momentum and we are left with an irrelevant vertex.
We can therefore assume that there are no relevant couplings
associated with these monomials (notice that for this reason we have assumed
that there is only one relevant coupling in $\G_{\mu}^{(\bc cA)}$).
The corresponding vertices are then fixed to vanish at $\L=\L_0$.

Since the Lorentz decomposition and the normalization conditions for the
vertices $w$-$c$ and $w$-$c$-$A$ in \re{uc} and \re{wcA} are analogous to
the one for the vertices $u$-$c$ and $u$-$c$-$A$ given above, we select the
boundary conditions
$$
\s_{u c}(0)=\frac 1 g \s_{w c}(0)=0\,,\;\;\;\;\;
\s_{u cA}(0)=\frac 1 g \s_{w cA}(0)=-1\,.
$$
This corresponds to fix these two couplings as they appear in the total action
\re{Stot}.
We now show that it is possible to fix the boundary conditions on the
parameters $\s_{\bc c}$ and $\s_{\bc cA}$ in such a way that
the identity \re{FI} is valid at any loop and for any $\L$.
At the tree level this requirement gives
$$
\s_{\bc c}^{(0)}(0)=0\,,\;\;\;\;\;
\s_{\bc cA}^{(0)}(0)=ig\,.
$$
Let us suppose that the identity is true for the cutoff effective action
$\Pi^{(\ell)}$ at loop $\ell$.
The identity then holds for the auxiliary vertices $\bG^{(\ell)}$ and,
from \re{Ivert}, also for the integrands $I^{(\ell+1)}$.
Then the validity of \re{FI} at loop $\ell+1$ depends only on the boundary
conditions for $\s_{\bc c}^{(\ell+1)}$ and $\s_{\bc cA}^{(\ell+1)}$.
In the following, in order to simplify the notation, in
all the integrands and vertices we do not write the loop order which is
understood to be $\ell+1$.

\noindent
1) Ghost propagator.

The integral equations give ($\bp^2=\mu^2$)
$$
\Pi^{(\bc c)}(p;\L)=ip^2 \s^{(\ell+1)}_{\bc c}(0)+
ip^2\int_0^{\L_0}\,\frac{d\l}{\l}
\partial_{\bp^2} \bp^2 I^{(\bc c)}(\bp;\l)
-ip^2 \int_\L^{\L_0}\,\frac{d\l}{\l}
I^{(\bc c)}(p;\l)
$$
and
$$
p_\mu\Pi_\mu^{(uc)}(p;\L)=p^2\int_0^{\L_0}\,\frac{d\l}{\l}
\partial_{\bp^2} (\bp^2 I^{(u c) }(\bp;\l))
-p^2 \int_\L^{\L_0}\,\frac{d\l}{\l}
I^{(u c) }(p;\l) \,.
$$
{}From the inductive hypothesis
$$
g I^{(u c) }(p;\l)=I^{(\bc c) }(p;\l)\,,
$$
we have that the identity is satisfied at
any loop and for any $\L$, by imposing $\s_{\bc c}^{(\ell+1)}(0)=0$, which
implies to all loop order
$$
\s_{\bc c}(0)=0\,.
$$

\noindent
2) Vector-ghost vertex.

The integral equations give
$$
\G_\mu^{(\bc cA)}(p,q;\L)= p_\mu\s_{\bc cA}^{(\ell+1)}(0)
+p_\mu \int_0^{\L_0}\,\frac{d\l}{\l}I^{(\bc c A)}(p,q;\l)\vert_{3SP}
-\int_\L^{\L_0}\,\frac{d\l}{\l}
I_\mu^{(\bc c A)}(p,q;\l)
$$
and
$$
\G_{\nu\mu}^{(u cA)}(p,q;\L)= -g_{\mu\nu}
+g_{\mu\nu}\int_0^{\L_0}\,\frac{d\l}{\l}I^{(uc A)}(p,q;\l)\vert_{3SP}
-\int_\L^{\L_0}\,\frac{d\l}{\l}
I_{\nu\mu}^{(uc A)}(p,q;\l)
\,.
$$
The inductive hypothesis
$$
-ig p_\nu I_{\nu\mu}^{(u c A)}(p,q;\l)=I_\mu^{(\bc c A)}(p,q;\l)\,,
$$
at $3SP$ gives
$$
-ig \left[I^{(u c A)}(p,q;\l)+\frac{2}{3p^2} p_\mu Q_\nu
\tilde I_{\mu\nu}^{(ucA)}(p,q;\l)\right]_{3SP}
=I^{(\bc c A)}(p,q;\l)\vert_{3SP}\,,
$$
where $Q_\nu=2p_\nu+q_\nu$ is vector orthogonal to $q_\nu$ at $3SP$.
Using these two results one finds that in this case the identity \re{FI}
is valid for any $\L$ if we fix
$$
\s^{(\ell+1)}_{\bc c A}(0)=ig\left[\frac{2}{3p^2} Q_\alpha p_\beta
\int_0^{\L_0}\,\frac{d\l}{\l}
\tilde I^{(u c A)}_{\beta\alpha}(p,q;\l)\right]_{3SP}\,.
$$
To all loops we have then
$$
\s_{\bc c A}(0)=ig \left[1-\frac{2}{3p^2}  p_\beta
\tilde \G^{(u c A)}_{\beta\alpha}(p,q;0)Q_\alpha|_{3SP}\right]\,.
$$


Notice that this condition implies
$$
-ig\s_{u c A}(\L_0)=
\s_{\bc c A}(\L_0)\,,
$$
namely, in the usual language, no counterterms violating \re{FI} are
generated by the evolution equation.

\section{Conclusions}
In this paper we have analyzed a RG formulation of a non-Abelian
gauge theory which avoids the usual fine tuning problem
\cite{B,romani}. This problem arises when one treats the RG equations
as evolution equations and starts by giving the relevant couplings at
the UV scale $\L_0$ (bare parameters). These various parameters have
to be fine tuned to depend on a single parameter, the coupling $g$ at
the normalization point $\mu$,
in such a way that the ST identities are satisfied by the
computed physical effective action. Our procedure avoids the fine
tuning problem simply by fixing the relevant parameters at the
physical point $\L=0$. We have shown that the formulation so obtained is quite
practical and deduced the usual loop expansion of the renormalized
vertex functions in terms of the physical coupling $g$ at $\mu$.
The calculations do not seem more complex than the ones done in
dimensional regularization.
The cancellations of UV divergences in the Feynman graphs is provided
by subtractions which are systematically generated by implementing the
boundary conditions. These cancellations are very similar to the
ones obtained by the Bogoliubov $R$-operation \cite{Bog}.
We described very briefly the proof of perturbative renormalizability.
This proof is simply based on power counting and does not involve any
new feature with respect to the one for the scalar case and QED.

We have performed detailed calculations only to one loop.
We computed the six vertices with non-negative dimension
which contain the relevant couplings.
{}From them we obtained for large $\L$ the nine relevant couplings $\s_i(\L)$
and
$\r_i(\L)$, which are used to compute the usual one
loop beta function and are the coefficients of the UV bare action
\re{intac}.

One then has to show that the complete (physical) effective action
indeed satisfies ST identities.
In the QED case we have shown \cite{QED} to all perturbative orders
that the complete effective action computed by this method does indeed
satisfy the Ward identities.
This implies that, in the RG flow, all couplings
and vertices are constrained for every $\L$ by the symmetry.
In the YM case we have verified the ST identities only to one loop
and for two vertices. A general proof of ST identities,
based on the results of the recent work \cite{B} by Becchi, is under
study.

Our analysis has been perturbative. However, contrary to dimensional
regularization, the basic integral equations in \re{inteq} and \re{r0}
are in principle nonperturbative. The only possible perturbative
element at the basis of \re{inteq} is simply the separation of
$\G[\Phi;\L]$ in relevant and irrelevant parts, which is based on
na\^{\i}ve power counting. This should be correct for an
asympotically free theory. A next natural step in this formulation is
the analysis of anomalies and chiral gauge theories. Since we work in
four dimensions, the presence of $\g_5$ does not lead to any direct
complication. This analysis is left to a separate publication
\cite{anom}.

\vspace{3mm}\noindent{\large\bf Acknowledgements}

We have benefited greatly from discussions with C. Becchi, G. Bottazzi
and M. Tonin.

\eject
\newpage

\begin{thebibliography}{99}
\bibitem{eps}
         G. t'Hooft and M. Veltman, \np{44}{189}{72};
         K. Wilson, \pr{7}{2911}{73};
         G. Bollini and J. Giambiagi, \nc{12}{20}{72}, \apa{38}{211}{73};
	 L. Ashmore, \lnc{4}{289}{72};
	 E.M. Cicuta and E. Montaldi, \lnc{4}{392}{72};
	 B. Butera, E.M. Cicuta and E. Montaldi, \nc{19}{513}{74};
	 D.A. Akiempong and R. Delburgo, \nc{17}{578}{73},\nc{18}{94}{73},
	 \nc{19}{219}{73};
         T. Marinucci and M. Tonin, \nc{31}{381}{76};
	 G. t'Hooft, \np{61}{455}{73};
	 E. Speer, \jmp{15}{1}{74};
         P. Breitenlohner and D. Maison, \cmp{52}{11,39,55}{77}.
\bibitem{chi}
	G. Costa, T. Marinucci, J. Julve and M. Tonin,
		Il Nuovo Cimento 38 A (1977) 373;
	S. Aoyama and M. Tonin, \np{179}{293}{81};
	G. Bandelloni, C. Becchi, A. Blasi and R. Collina,
		\cmp{72}{239}{80};
	J.L. Trueman, \pl{88}{331}{79};
	S. Gottlieb and J.T. Donohue, \pr{20}{3378}{79};
	G. Bonneau, \np{177}{523}{81}.
\bibitem{W}
        K.G. Wilson, \prb{4}{3174,3184}{71};
        K.G. Wilson and J.G. Kogut, \prep{12}{75}{74}.
\bibitem{P}
        J. Polchinski, \np{231}{269}{84}.
\bibitem{G}
        G. Gallavotti, \rmp{57}{471}{85}.
\bibitem{B}
        C. Becchi, Lectures on the renormalization of gauge theories,
        in: Relativity, groups and topology II (Les Houches 1983)
        Eds. B.S. DeWitt and R. Stora (Elsevier Science Pub. 1984);
        C. Becchi, On the construction of renormalized quantum field
        theory using renormalization group techniques,
        in {\it Elementary particles, Field theory and Statistical
        mechanics}, Eds. M. Bonini, G. Marchesini and E. Onofri,
        Parma University 1993.
\bibitem{Warr}
	B.J. Warr, \ap{183}{1,89}{88}.
\bibitem{BDM}
        M. Bonini, M. D'Attanasio and G. Marchesini,
        \np{409}{441}{93}.
\bibitem{QED}
        M. Bonini, M. D'Attanasio and G. Marchesini,
	Ward identities and Wilson renormalization group for QED,
	Parma preprint, UPRF 93-382, Nucl. Phys. B to be published.
\bibitem{oth}
        G. Keller and C. Kopper, \pl{273}{323}{91};
        G. Keller, C. Kopper and M. Salmhofer, Helv. Phys. Acta 65 (1992) 32;
	C. Wetterich, \pl{301}{90}{93}, Effective average action for gauge
        theories and exact evolution equations, HD-THEP-93-40 preprint;
        U. Ellwanger, Heidelberg preprint HD-THEP-92-33;
	L. Girardello and A. Zaffaroni, Exact renormalization group
	equation and decoupling in quantum field theory,
	Milano-SISSA/EP preprint 1993.
\bibitem{romani}
        A. Borrelli, L. Maiani, G. C. Rossi, R. Sisto and M. Testa,
        \np{333}{335}{90}.
\bibitem{Riv}
        V. Rivasseau, From perturbative to constructive renormalization,
        Princeton University press, 1991;
        J. Magnen, V. Rivasseau  and R. S\'en\'eor, \cmp {155}{325}{93}.
\bibitem{BRS}
        C. Becchi, A. Rouet and R. Stora, \cmp{42}{127}{75},
        \ap{98}{287}{76}.
\bibitem{H}
         J. Hughes and J. Liu, \np{307}{183}{88}.
\bibitem{Bog}
        N.N. Bogoliubov and D. Shirkov, Introduction to the theory of
        quantized fields, Wiley Interscience, New York;
        for a recent review see the book of V. Rivasseau, already
	quoted in ref.~\cite{Riv}.
\bibitem{anom}
        M. Bonini, M. D'Attanasio and G. Marchesini,
	Axial anomalies in gauge theories within exact
	renormalization group method, Parma preprint, UPRF 94-392.
\end{thebibliography}
\eject
\newpage
\begin{figcap}

\item
The circles represent vertex functions with the IR cutoff $\L$.
Internal lines involve only the fields $\phi_A=(A_\mu,c,\bar c )$,
while the external lines could involve also the sources $u_\mu$ and $v$.
The crosses represent the derivative with respect to $\L$ of the internal
propagators. Integration over $q$ in the loop is understood.

\item
a) Graphical representation of the contribution to
$\bG^{(0)b_1b_2aa'}_{\nu_1\nu_2,\mu\mu'}(q_1,q_2;p,p';\L)$ obtained
by expanding \re{bG} to second order in $\G^{int}$.

b) Graphical representation of the contribution to
$\bG^{(0)b_1b_2aa'}_{\nu_1\nu_2,\mu\mu'}(q_1,q_2;p,p';\L)$ obtained
by expanding \re{bG} to first order in $\G^{int}$.

c) Graphical representation of the contribution to
$\bG^{(0)b_1b_2aa'}_{\mu\mu'}(q_1,q_2;p,p';\L)$ obtained
by expanding \re{bG} to second order in $\G^{int}$.


\end{figcap}

\end{document}

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1917 939 D
1913 937 D
1911 933 D
S
N
1951 901 M
1937 903 D
S
N
1951 900 M
1937 903 D
S
N
2175 1698 M
2174 1700 D
2173 1702 D
2172 1704 D
2171 1705 D
2169 1707 D
2168 1709 D
2167 1710 D
2166 1711 D
2165 1712 D
2164 1713 D
2163 1713 D
2162 1714 D
2160 1714 D
2159 1713 D
2158 1713 D
2157 1712 D
2156 1711 D
2155 1710 D
2154 1709 D
2152 1708 D
2151 1706 D
2150 1704 D
2149 1703 D
2148 1701 D
2147 1699 D
2146 1697 D
2145 1695 D
2144 1693 D
2142 1692 D
2141 1690 D
2140 1688 D
2139 1687 D
2138 1686 D
2137 1685 D
2136 1684 D
2134 1683 D
2133 1683 D
2132 1683 D
2131 1683 D
2130 1683 D
2129 1684 D
2128 1685 D
2126 1686 D
2125 1687 D
2124 1688 D
2123 1690 D
2122 1691 D
2121 1693 D
2120 1695 D
2119 1697 D
2118 1699 D
2116 1701 D
2115 1702 D
2114 1704 D
2113 1706 D
2112 1707 D
2111 1709 D
2110 1710 D
2109 1711 D
2107 1712 D
2106 1713 D
2105 1713 D
2104 1714 D
2103 1714 D
2102 1713 D
2101 1713 D
2099 1712 D
2098 1711 D
2097 1710 D
2096 1709 D
2095 1707 D
2094 1706 D
2093 1704 D
2092 1702 D
2090 1700 D
2089 1698 D
2088 1697 D
2087 1695 D
2086 1693 D
2085 1691 D
2084 1689 D
2083 1688 D
2081 1687 D
2080 1686 D
2079 1685 D
2078 1684 D
2077 1683 D
2076 1683 D
2075 1683 D
2073 1683 D
2072 1684 D
2071 1684 D
2070 1685 D
2069 1686 D
2068 1687 D
2067 1689 D
2066 1690 D
2064 1692 D
2063 1694 D
2062 1696 D
2061 1697 D
2060 1699 D
2059 1701 D
2058 1703 D
2057 1705 D
2056 1706 D
2054 1708 D
2053 1709 D
2052 1711 D
2051 1712 D
2050 1712 D
2049 1713 D
2048 1713 D
2046 1714 D
2045 1714 D
2044 1713 D
2043 1713 D
2042 1712 D
2041 1711 D
2040 1710 D
2038 1708 D
2037 1707 D
2036 1705 D
2035 1703 D
2034 1702 D
2033 1700 D
2032 1698 D
2031 1696 D
2029 1694 D
2028 1692 D
2027 1691 D
2026 1689 D
2025 1688 D
2024 1686 D
2023 1685 D
2022 1684 D
2021 1684 D
2019 1683 D
2018 1683 D
2017 1683 D
2016 1683 D
2015 1684 D
2014 1684 D
2013 1685 D
2011 1686 D
2010 1688 D
2009 1689 D
2008 1691 D
2007 1692 D
2006 1694 D
2005 1696 D
2004 1698 D
2002 1700 D
2001 1702 D
2000 1703 D
1999 1705 D
1998 1707 D
1997 1708 D
1996 1710 D
1995 1711 D
1993 1712 D
1992 1713 D
1991 1713 D
1990 1714 D
1989 1714 D
1988 1713 D
1987 1713 D
1985 1712 D
1984 1712 D
1983 1711 D
1982 1709 D
1981 1708 D
1980 1706 D
1979 1705 D
1978 1703 D
1976 1701 D
1975 1699 D
1974 1697 D
1973 1696 D
1972 1694 D
1971 1692 D
1970 1690 D
1969 1689 D
1967 1687 D
1966 1686 D
1965 1685 D
1964 1684 D
1963 1684 D
1962 1683 D
1961 1683 D
1959 1683 D
1958 1683 D
S
N
1958 1683 M
1957 1684 D
1956 1685 D
1955 1685 D
1954 1687 D
1953 1688 D
1952 1689 D
1950 1691 D
1949 1693 D
1948 1695 D
1947 1697 D
1946 1698 D
1945 1700 D
1944 1702 D
1943 1704 D
1941 1706 D
1940 1707 D
1939 1709 D
1938 1710 D
1937 1711 D
1936 1712 D
1935 1713 D
1934 1713 D
1932 1714 D
1931 1714 D
1930 1713 D
1929 1713 D
1928 1712 D
1927 1711 D
1926 1710 D
1925 1709 D
1923 1707 D
1922 1706 D
1921 1704 D
1920 1702 D
1919 1701 D
1918 1699 D
1917 1697 D
1915 1695 D
1914 1693 D
1913 1691 D
1912 1690 D
1911 1688 D
1910 1687 D
1909 1686 D
1908 1685 D
1907 1684 D
1905 1683 D
1904 1683 D
1903 1683 D
S
N
1903 1683 M
1902 1683 D
1901 1683 D
1900 1684 D
1898 1685 D
1897 1686 D
1896 1687 D
1895 1688 D
1894 1690 D
1893 1692 D
1892 1693 D
1891 1695 D
1890 1697 D
1888 1699 D
1887 1701 D
1886 1703 D
1885 1704 D
1884 1706 D
1883 1708 D
1882 1709 D
1881 1710 D
1879 1711 D
1878 1712 D
1877 1713 D
1876 1713 D
1875 1714 D
1874 1714 D
1873 1713 D
1871 1713 D
1870 1712 D
1869 1711 D
1868 1710 D
1867 1709 D
1866 1707 D
1865 1705 D
1864 1704 D
1862 1702 D
1861 1700 D
1860 1698 D
S
N
2179 1948 M
2139 1948 D
S
N
2179 1947 M
2139 1947 D
S
N
2173 1948 M
2177 1952 D
2179 1956 D
2179 1960 D
2177 1965 D
2173 1969 D
2168 1971 D
2164 1971 D
2158 1969 D
2154 1965 D
2153 1960 D
2153 1956 D
2154 1952 D
2158 1948 D
S
N
2179 1960 M
2177 1963 D
2173 1967 D
2168 1969 D
2164 1969 D
2158 1967 D
2154 1963 D
2153 1960 D
S
N
2139 1954 M
2139 1941 D
S
N
2159 1929 M
2160 1927 D
2164 1923 D
2138 1923 D
S
N
2163 1924 M
2138 1924 D
S
N
2138 1929 M
2138 1918 D
S
N
2153 1900 M
2154 1902 D
2156 1900 D
2154 1898 D
2151 1898 D
2147 1900 D
2145 1902 D
S
N
2179 1879 M
2153 1883 D
S
N
2179 1877 M
2168 1879 D
2158 1881 D
2153 1883 D
S
N
2179 1858 M
2171 1860 D
2164 1864 D
S
N
2179 1857 M
2173 1858 D
2169 1860 D
2164 1864 D
2160 1868 D
2156 1873 D
2154 1877 D
2153 1883 D
S
N
2179 1885 M
2179 1877 D
S
N
2159 1843 M
2160 1840 D
2164 1837 D
2138 1837 D
S
N
2163 1838 M
2138 1838 D
S
N
2138 1843 M
2138 1832 D
S
N
2153 1813 M
2154 1815 D
2156 1813 D
2154 1812 D
2151 1812 D
2147 1813 D
2145 1815 D
S
N
2192 1795 M
2153 1795 D
S
N
2192 1793 M
2153 1793 D
S
N
2173 1793 M
2177 1789 D
2179 1785 D
2179 1782 D
2177 1776 D
2173 1772 D
2168 1770 D
2164 1770 D
2158 1772 D
2154 1776 D
2153 1782 D
2153 1785 D
2154 1789 D
2158 1793 D
S
N
2179 1782 M
2177 1778 D
2173 1774 D
2168 1772 D
2164 1772 D
2158 1774 D
2154 1778 D
2153 1782 D
S
N
2192 1800 M
2192 1793 D
S
N
2159 1755 M
2160 1752 D
2164 1748 D
2138 1748 D
S
N
2163 1750 M
2138 1750 D
S
N
2138 1755 M
2138 1743 D
S
N
2179 1345 M
2139 1345 D
S
N
2179 1343 M
2139 1343 D
S
N
2173 1345 M
2177 1348 D
2179 1352 D
2179 1356 D
2177 1362 D
2173 1365 D
2168 1367 D
2164 1367 D
2158 1365 D
2154 1362 D
2153 1356 D
2153 1352 D
2154 1348 D
2158 1345 D
S
N
2179 1356 M
2177 1360 D
2173 1363 D
2168 1365 D
2164 1365 D
2158 1363 D
2154 1360 D
2153 1356 D
S
N
2139 1350 M
2139 1337 D
S
N
2159 1328 M
2158 1327 D
2156 1328 D
2158 1329 D
2159 1329 D
2161 1328 D
2163 1327 D
2164 1323 D
2164 1318 D
2163 1314 D
2161 1313 D
2159 1312 D
2156 1312 D
2154 1313 D
2151 1317 D
2149 1323 D
2148 1325 D
2145 1328 D
2141 1329 D
2138 1329 D
S
N
2164 1318 M
2163 1315 D
2161 1314 D
2159 1313 D
2156 1313 D
2154 1314 D
2151 1318 D
2149 1323 D
S
N
2140 1329 M
2141 1328 D
2141 1325 D
2139 1319 D
2139 1315 D
2140 1313 D
2141 1312 D
S
N
2141 1325 M
2138 1319 D
2138 1314 D
2139 1313 D
2141 1312 D
2144 1312 D
S
N
2153 1296 M
2154 1298 D
2156 1296 D
2154 1294 D
2151 1294 D
2147 1296 D
2145 1298 D
S
N
2179 1275 M
2153 1279 D
S
N
2179 1273 M
2168 1275 D
2158 1277 D
2153 1279 D
S
N
2179 1255 M
2171 1257 D
2164 1260 D
S
N
2179 1253 M
2173 1255 D
2169 1257 D
2164 1260 D
2160 1264 D
2156 1270 D
2154 1273 D
2153 1279 D
S
N
2179 1281 M
2179 1273 D
S
N
2159 1242 M
2158 1240 D
2156 1242 D
2158 1243 D
2159 1243 D
2161 1242 D
2163 1240 D
2164 1237 D
2164 1232 D
2163 1228 D
2161 1227 D
2159 1225 D
2156 1225 D
2154 1227 D
2151 1230 D
2149 1237 D
2148 1239 D
2145 1242 D
2141 1243 D
2138 1243 D
S
N
2164 1232 M
2163 1229 D
2161 1228 D
2159 1227 D
2156 1227 D
2154 1228 D
2151 1232 D
2149 1237 D
S
N
2140 1243 M
2141 1242 D
2141 1239 D
2139 1233 D
2139 1229 D
2140 1227 D
2141 1225 D
S
N
2141 1239 M
2138 1233 D
2138 1228 D
2139 1227 D
2141 1225 D
2144 1225 D
S
N
2153 1210 M
2154 1212 D
2156 1210 D
2154 1208 D
2151 1208 D
2147 1210 D
2145 1212 D
S
N
2192 1191 M
2153 1191 D
S
N
2192 1189 M
2153 1189 D
S
N
2173 1189 M
2177 1185 D
2179 1182 D
2179 1178 D
2177 1172 D
2173 1168 D
2168 1167 D
2164 1167 D
2158 1168 D
2154 1172 D
2153 1178 D
2153 1182 D
2154 1185 D
2158 1189 D
S
N
2179 1178 M
2177 1174 D
2173 1170 D
2168 1168 D
2164 1168 D
2158 1170 D
2154 1174 D
2153 1178 D
S
N
2192 1197 M
2192 1189 D
S
N
2159 1153 M
2158 1152 D
2156 1153 D
2158 1155 D
2159 1155 D
2161 1153 D
2163 1152 D
2164 1148 D
2164 1143 D
2163 1140 D
2161 1138 D
2159 1137 D
2156 1137 D
2154 1138 D
2151 1142 D
2149 1148 D
2148 1151 D
2145 1153 D
2141 1155 D
2138 1155 D
S
N
2164 1143 M
2163 1141 D
2161 1140 D
2159 1138 D
2156 1138 D
2154 1140 D
2151 1143 D
2149 1148 D
S
N
2140 1155 M
2141 1153 D
2141 1151 D
2139 1145 D
2139 1141 D
2140 1138 D
2141 1137 D
S
N
2141 1151 M
2138 1145 D
2138 1140 D
2139 1138 D
2141 1137 D
2144 1137 D
S
N
1860 1412 M
1861 1410 D
1862 1408 D
1864 1406 D
1865 1404 D
1866 1403 D
1867 1401 D
1868 1400 D
1869 1399 D
1870 1398 D
1871 1397 D
1873 1397 D
1874 1396 D
1875 1396 D
1876 1397 D
1877 1397 D
1878 1398 D
1879 1399 D
1881 1400 D
1882 1401 D
1883 1402 D
1884 1404 D
1885 1406 D
1886 1407 D
1887 1409 D
1888 1411 D
1890 1413 D
1891 1415 D
1892 1417 D
1893 1418 D
1894 1420 D
1895 1421 D
1896 1423 D
1897 1424 D
1898 1425 D
1900 1426 D
1901 1427 D
1902 1427 D
1903 1427 D
1904 1427 D
1905 1427 D
1907 1426 D
1908 1425 D
1909 1424 D
1910 1423 D
1911 1422 D
1912 1420 D
1913 1418 D
1914 1417 D
1915 1415 D
1917 1413 D
1918 1411 D
1919 1409 D
1920 1407 D
1921 1406 D
1922 1404 D
1923 1402 D
1925 1401 D
1926 1400 D
1927 1399 D
1928 1398 D
1929 1397 D
1930 1397 D
1931 1396 D
1932 1396 D
1934 1397 D
1935 1397 D
1936 1398 D
1937 1399 D
1938 1400 D
1939 1401 D
1940 1403 D
1941 1404 D
1943 1406 D
1944 1408 D
1945 1410 D
1946 1412 D
1947 1413 D
1948 1415 D
1949 1417 D
1950 1419 D
1952 1420 D
1953 1422 D
1954 1423 D
1955 1424 D
1956 1425 D
1957 1426 D
1958 1427 D
1959 1427 D
1961 1427 D
1962 1427 D
1963 1426 D
1964 1426 D
1965 1425 D
1966 1424 D
1967 1423 D
1969 1421 D
1970 1420 D
1971 1418 D
1972 1416 D
1973 1414 D
1974 1413 D
1975 1411 D
1976 1409 D
1978 1407 D
1979 1405 D
1980 1404 D
1981 1402 D
1982 1401 D
1983 1399 D
1984 1398 D
1985 1397 D
1987 1397 D
1988 1397 D
1989 1396 D
1990 1396 D
1991 1397 D
1992 1397 D
1993 1398 D
1995 1399 D
1996 1400 D
1997 1402 D
1998 1403 D
1999 1405 D
2000 1407 D
2001 1408 D
2002 1410 D
2004 1412 D
2005 1414 D
2006 1416 D
2007 1418 D
2008 1419 D
2009 1421 D
2010 1422 D
2011 1424 D
2013 1425 D
2014 1426 D
2015 1426 D
2016 1427 D
2017 1427 D
2018 1427 D
2019 1427 D
2021 1426 D
2022 1426 D
2023 1425 D
2024 1424 D
2025 1422 D
2026 1421 D
2027 1419 D
2028 1418 D
2029 1416 D
2031 1414 D
2032 1412 D
2033 1410 D
2034 1408 D
2035 1407 D
2036 1405 D
2037 1403 D
2038 1402 D
2040 1400 D
2041 1399 D
2042 1398 D
2043 1397 D
2044 1397 D
2045 1396 D
2046 1396 D
2048 1397 D
2049 1397 D
2050 1397 D
2051 1398 D
2052 1399 D
2053 1401 D
2054 1402 D
2056 1404 D
2057 1405 D
2058 1407 D
2059 1409 D
2060 1411 D
2061 1413 D
2062 1414 D
2063 1416 D
2064 1418 D
2066 1420 D
2067 1421 D
2068 1423 D
2069 1424 D
2070 1425 D
2071 1426 D
2072 1426 D
2073 1427 D
2075 1427 D
2076 1427 D
2077 1427 D
S
N
2077 1427 M
2078 1426 D
2079 1425 D
2080 1424 D
2081 1423 D
2083 1422 D
2084 1420 D
2085 1419 D
2086 1417 D
2087 1415 D
2088 1413 D
2089 1412 D
2090 1410 D
2092 1408 D
2093 1406 D
2094 1404 D
2095 1403 D
2096 1401 D
2097 1400 D
2098 1399 D
2099 1398 D
2101 1397 D
2102 1397 D
2103 1396 D
2104 1396 D
2105 1397 D
2106 1397 D
2107 1398 D
2109 1399 D
2110 1400 D
2111 1401 D
2112 1402 D
2113 1404 D
2114 1406 D
2115 1407 D
2116 1409 D
2118 1411 D
2119 1413 D
2120 1415 D
2121 1417 D
2122 1418 D
2123 1420 D
2124 1422 D
2125 1423 D
2126 1424 D
2128 1425 D
2129 1426 D
2130 1427 D
2131 1427 D
2132 1427 D
2133 1427 D
S
N
2133 1427 M
2134 1427 D
2136 1426 D
2137 1425 D
2138 1424 D
2139 1423 D
2140 1421 D
2141 1420 D
2142 1418 D
2144 1417 D
2145 1415 D
2146 1413 D
2147 1411 D
2148 1409 D
2149 1407 D
2150 1406 D
2151 1404 D
2152 1402 D
2154 1401 D
2155 1400 D
2156 1399 D
2157 1398 D
2158 1397 D
2159 1397 D
2160 1396 D
2162 1396 D
2163 1397 D
2164 1397 D
2165 1398 D
2166 1399 D
2167 1400 D
2168 1401 D
2169 1403 D
2171 1404 D
2172 1406 D
2173 1408 D
2174 1410 D
2175 1412 D
S
N
1804 1650 M
1764 1650 D
S
N
1804 1648 M
1764 1648 D
S
N
1792 1637 M
1777 1637 D
S
N
1804 1656 M
1804 1626 D
1792 1626 D
1804 1627 D
S
N
1785 1648 M
1785 1637 D
S
N
1764 1656 M
1764 1642 D
S
N
1804 1612 M
1802 1614 D
1800 1612 D
1802 1611 D
1804 1612 D
S
N
1790 1612 M
1764 1612 D
S
N
1790 1611 M
1764 1611 D
S
N
1790 1618 M
1790 1611 D
S
N
1764 1618 M
1764 1605 D
S
N
1790 1586 M
1789 1590 D
1787 1592 D
1783 1594 D
1779 1594 D
1775 1592 D
1774 1590 D
1772 1586 D
1772 1582 D
1774 1579 D
1775 1577 D
1779 1575 D
1783 1575 D
1787 1577 D
1789 1579 D
1790 1582 D
1790 1586 D
S
N
1789 1590 M
1785 1592 D
1777 1592 D
1774 1590 D
S
N
1774 1579 M
1777 1577 D
1785 1577 D
1789 1579 D
S
N
1787 1577 M
1789 1575 D
1790 1571 D
1789 1571 D
1789 1575 D
S
N
1775 1592 M
1774 1594 D
1770 1596 D
1768 1596 D
1764 1594 D
1762 1588 D
1762 1579 D
1760 1573 D
1759 1571 D
S
N
1768 1596 M
1766 1594 D
1764 1588 D
1764 1579 D
1762 1573 D
1759 1571 D
1757 1571 D
1753 1573 D
1751 1579 D
1751 1590 D
1753 1596 D
1757 1597 D
1759 1597 D
1762 1596 D
1764 1590 D
S
N
1796 1524 M
1798 1521 D
1804 1515 D
1764 1515 D
S
N
1802 1517 M
1764 1517 D
S
N
1764 1524 M
1764 1507 D
S
N
1787 1489 M
1785 1489 D
1785 1491 D
1787 1491 D
1789 1489 D
1790 1485 D
1790 1477 D
1789 1474 D
1787 1472 D
1783 1470 D
1770 1470 D
1766 1468 D
1764 1466 D
S
N
1787 1472 M
1770 1472 D
1766 1470 D
1764 1466 D
1764 1464 D
S
N
1783 1472 M
1781 1474 D
1779 1485 D
1777 1491 D
1774 1492 D
1770 1492 D
1766 1491 D
1764 1485 D
1764 1479 D
1766 1476 D
1770 1472 D
S
N
1779 1485 M
1777 1489 D
1774 1491 D
1770 1491 D
1766 1489 D
1764 1485 D
S
N
1262 1094 M
1267 1098 D
1272 1101 D
1275 1105 D
1277 1109 D
1277 1112 D
1276 1116 D
1272 1119 D
1267 1123 D
1262 1126 D
1257 1130 D
1252 1133 D
1248 1137 D
1246 1140 D
1246 1144 D
1248 1147 D
1252 1151 D
1256 1154 D
1262 1158 D
1267 1161 D
1272 1165 D
1275 1168 D
1277 1172 D
1277 1175 D
1276 1179 D
1272 1182 D
1268 1186 D
1262 1189 D
1257 1193 D
1252 1196 D
1249 1200 D
1247 1203 D
1246 1207 D
1248 1210 D
1251 1214 D
1256 1217 D
1261 1221 D
1267 1225 D
1271 1228 D
1275 1232 D
1277 1235 D
1278 1239 D
1276 1242 D
1273 1246 D
1268 1249 D
1263 1253 D
1257 1256 D
1253 1260 D
1249 1263 D
1247 1267 D
1246 1270 D
1248 1274 D
1251 1277 D
1255 1281 D
1261 1284 D
1266 1288 D
1271 1291 D
1275 1295 D
1277 1298 D
1278 1302 D
1276 1305 D
1273 1309 D
1268 1312 D
1263 1316 D
1258 1319 D
1253 1323 D
1249 1326 D
1247 1330 D
1246 1333 D
1248 1337 D
1251 1341 D
1255 1344 D
1260 1348 D
1266 1351 D
1271 1355 D
1275 1358 D
1277 1362 D
1278 1365 D
1276 1369 D
1273 1372 D
1269 1376 D
1264 1379 D
1258 1383 D
1253 1386 D
1249 1390 D
1247 1393 D
1246 1397 D
1247 1400 D
1250 1404 D
1255 1407 D
1260 1411 D
1265 1414 D
1270 1418 D
1274 1421 D
1277 1425 D
1278 1428 D
1276 1432 D
1273 1435 D
1269 1439 D
1264 1442 D
1259 1446 D
1253 1450 D
1249 1453 D
1247 1457 D
1246 1460 D
1247 1464 D
1250 1467 D
1254 1471 D
1260 1474 D
1265 1478 D
1270 1481 D
1274 1485 D
1277 1488 D
1278 1492 D
1277 1495 D
1274 1499 D
1269 1502 D
1264 1506 D
1259 1509 D
1254 1513 D
1250 1516 D
1247 1520 D
1246 1523 D
1247 1527 D
1250 1530 D
1254 1534 D
1259 1537 D
1265 1541 D
1270 1544 D
1274 1548 D
1277 1551 D
1278 1555 D
1277 1558 D
1274 1562 D
1270 1566 D
1265 1569 D
1259 1573 D
1254 1576 D
1250 1580 D
1247 1583 D
1246 1587 D
1247 1590 D
1250 1594 D
1254 1597 D
1259 1601 D
1264 1604 D
1269 1608 D
1274 1611 D
1277 1615 D
1278 1618 D
1277 1622 D
1274 1625 D
1270 1629 D
1265 1632 D
1260 1636 D
1254 1639 D
1250 1643 D
1247 1646 D
1246 1650 D
1247 1653 D
1249 1657 D
1253 1660 D
1259 1664 D
1264 1667 D
1269 1671 D
1273 1674 D
1276 1678 D
1278 1682 D
1277 1685 D
1274 1689 D
1270 1692 D
1265 1696 D
1260 1699 D
1255 1703 D
1250 1706 D
1247 1710 D
1246 1713 D
1247 1717 D
1249 1720 D
1253 1724 D
1258 1727 D
1264 1731 D
1269 1734 D
1273 1738 D
1276 1741 D
1278 1745 D
1277 1748 D
1275 1752 D
1271 1755 D
1266 1759 D
1260 1762 D
1255 1766 D
1251 1769 D
1248 1773 D
1246 1776 D
1247 1780 D
1249 1783 D
1253 1787 D
S
N
1253 1787 M
1258 1791 D
1263 1794 D
1268 1798 D
1273 1801 D
1276 1805 D
1278 1808 D
1277 1812 D
1275 1815 D
1271 1819 D
1266 1822 D
1261 1826 D
1255 1829 D
1251 1833 D
1248 1836 D
1246 1840 D
1247 1843 D
1249 1847 D
1253 1850 D
1257 1854 D
1263 1857 D
1268 1861 D
1273 1864 D
1276 1868 D
1278 1871 D
1277 1875 D
1275 1878 D
1271 1882 D
1267 1885 D
1261 1889 D
1256 1892 D
1251 1896 D
1248 1899 D
1246 1903 D
1247 1907 D
1249 1910 D
1252 1914 D
1257 1917 D
1262 1921 D
1268 1924 D
1272 1928 D
1276 1931 D
1277 1935 D
1277 1938 D
1275 1942 D
1272 1945 D
S
N
1272 1945 M
1267 1949 D
1262 1952 D
1256 1956 D
1252 1959 D
1248 1963 D
1246 1966 D
1246 1970 D
1248 1973 D
1252 1977 D
1257 1980 D
1262 1984 D
1267 1987 D
1272 1991 D
1276 1994 D
1277 1998 D
1277 2001 D
1275 2005 D
1272 2008 D
1267 2012 D
1262 2016 D
S
N
1339 2162 M
1300 2162 D
S
N
1339 2160 M
1300 2160 D
S
N
1333 2160 M
1337 2156 D
1339 2152 D
1339 2149 D
1337 2143 D
1333 2139 D
1328 2137 D
1324 2137 D
1318 2139 D
1315 2143 D
1313 2149 D
1313 2152 D
1315 2156 D
1318 2160 D
S
N
1339 2149 M
1337 2145 D
1333 2141 D
1328 2139 D
1324 2139 D
1318 2141 D
1315 2145 D
1313 2149 D
S
N
1339 2167 M
1339 2160 D
S
N
1300 2167 M
1300 2154 D
S
N
1313 2122 M
1315 2124 D
1317 2122 D
1315 2120 D
1311 2120 D
1307 2122 D
1305 2124 D
S
N
1339 2100 M
1300 2111 D
S
N
1339 2098 M
1300 2109 D
S
N
1333 2100 M
1322 2102 D
1317 2102 D
1313 2098 D
1313 2094 D
1315 2090 D
1318 2087 D
1324 2083 D
S
N
1339 2079 M
1318 2085 D
1315 2085 D
1313 2083 D
1313 2077 D
1317 2074 D
1320 2072 D
S
N
1339 2077 M
1318 2083 D
1315 2083 D
1313 2081 D
S
N
1313 2060 M
1315 2062 D
1317 2060 D
1315 2059 D
1311 2059 D
1307 2060 D
1305 2062 D
S
N
1335 2042 M
1333 2042 D
1333 2044 D
1335 2044 D
1337 2042 D
1339 2038 D
1339 2030 D
1337 2027 D
1335 2025 D
1332 2023 D
1318 2023 D
1315 2021 D
1313 2019 D
S
N
1335 2025 M
1318 2025 D
1315 2023 D
1313 2019 D
1313 2017 D
S
N
1332 2025 M
1330 2027 D
1328 2038 D
1326 2044 D
1322 2045 D
1318 2045 D
1315 2044 D
1313 2038 D
1313 2032 D
1315 2029 D
1318 2025 D
S
N
1328 2038 M
1326 2042 D
1322 2044 D
1318 2044 D
1315 2042 D
1313 2038 D
S
N
1339 1087 M
1300 1087 D
S
N
1339 1085 M
1300 1085 D
S
N
1333 1085 M
1337 1081 D
1339 1078 D
1339 1074 D
1337 1068 D
1333 1065 D
1328 1063 D
1324 1063 D
1318 1065 D
1315 1068 D
1313 1074 D
1313 1078 D
1315 1081 D
1318 1085 D
S
N
1339 1074 M
1337 1070 D
1333 1066 D
1328 1065 D
1324 1065 D
1318 1066 D
1315 1070 D
1313 1074 D
S
N
1339 1093 M
1339 1085 D
S
N
1300 1093 M
1300 1080 D
S
N
1352 1050 M
1339 1051 D
S
N
1352 1048 M
1339 1051 D
S
N
1313 1033 M
1315 1035 D
1317 1033 D
1315 1031 D
1311 1031 D
1307 1033 D
1305 1035 D
S
N
1339 1010 M
1300 1021 D
S
N
1339 1008 M
1300 1020 D
S
N
1333 1010 M
1322 1012 D
1317 1012 D
1313 1008 D
1313 1005 D
1315 1001 D
1318 997 D
1324 993 D
S
N
1339 990 M
1318 995 D
1315 995 D
1313 993 D
1313 988 D
1317 984 D
1320 982 D
S
N
1339 988 M
1318 993 D
1315 993 D
1313 991 D
S
N
1352 973 M
1339 975 D
S
N
1352 971 M
1339 975 D
S
N
1313 956 M
1315 958 D
1317 956 D
1315 954 D
1311 954 D
1307 956 D
1305 958 D
S
N
1335 937 M
1333 937 D
1333 939 D
1335 939 D
1337 937 D
1339 933 D
1339 926 D
1337 922 D
1335 920 D
1332 918 D
1318 918 D
1315 916 D
1313 915 D
S
N
1335 920 M
1318 920 D
1315 918 D
1313 915 D
1313 913 D
S
N
1332 920 M
1330 922 D
1328 933 D
1326 939 D
1322 941 D
1318 941 D
1315 939 D
1313 933 D
1313 928 D
1315 924 D
1318 920 D
S
N
1328 933 M
1326 937 D
1322 939 D
1318 939 D
1315 937 D
1313 933 D
S
N
1352 901 M
1339 903 D
S
N
1352 900 M
1339 903 D
S
N
1577 1708 M
1575 1710 D
1573 1711 D
1571 1712 D
1569 1713 D
1567 1714 D
1565 1714 D
1564 1715 D
1562 1715 D
1560 1716 D
1559 1716 D
1558 1716 D
1556 1715 D
1555 1715 D
1554 1714 D
1553 1713 D
1553 1712 D
1552 1711 D
1551 1709 D
1551 1708 D
1550 1706 D
1550 1704 D
1550 1702 D
1550 1700 D
1549 1698 D
1549 1695 D
1549 1693 D
1549 1691 D
1549 1689 D
1548 1687 D
1548 1685 D
1548 1683 D
1547 1681 D
1547 1679 D
1546 1678 D
1545 1677 D
1545 1675 D
1544 1675 D
1543 1674 D
1541 1673 D
1540 1673 D
1539 1673 D
1537 1673 D
1536 1674 D
1534 1674 D
1532 1675 D
1531 1676 D
1529 1677 D
1527 1678 D
1525 1679 D
1523 1680 D
1521 1681 D
1519 1682 D
1517 1683 D
1515 1684 D
1513 1685 D
1511 1686 D
1510 1687 D
1508 1687 D
1506 1688 D
1505 1688 D
1503 1688 D
1502 1688 D
1501 1687 D
1500 1687 D
1499 1686 D
1498 1685 D
1497 1684 D
1496 1682 D
1496 1681 D
1495 1679 D
1495 1677 D
1495 1675 D
1494 1673 D
1494 1671 D
1494 1669 D
1494 1667 D
1494 1665 D
1493 1662 D
1493 1660 D
1493 1658 D
1493 1656 D
1492 1654 D
1492 1653 D
1491 1651 D
1491 1650 D
1490 1648 D
1489 1647 D
1488 1646 D
1487 1646 D
1486 1645 D
1484 1645 D
1483 1645 D
1482 1646 D
1480 1646 D
1478 1646 D
1477 1647 D
1475 1648 D
1473 1649 D
1471 1650 D
1469 1651 D
1467 1652 D
1465 1653 D
1463 1654 D
1461 1655 D
1459 1657 D
1457 1657 D
1456 1658 D
1454 1659 D
1452 1659 D
1451 1660 D
1449 1660 D
1448 1660 D
1446 1660 D
1445 1659 D
1444 1659 D
1443 1658 D
1442 1657 D
1442 1656 D
1441 1654 D
1440 1653 D
1440 1651 D
1440 1649 D
1439 1647 D
1439 1645 D
1439 1643 D
1439 1641 D
1438 1638 D
1438 1636 D
1438 1634 D
1438 1632 D
1437 1630 D
1437 1628 D
1437 1626 D
1436 1624 D
1436 1623 D
1435 1621 D
1434 1620 D
1433 1619 D
1432 1618 D
1431 1618 D
1430 1617 D
1429 1617 D
1427 1617 D
1426 1618 D
1424 1618 D
1422 1619 D
1421 1619 D
1419 1620 D
1417 1621 D
1415 1622 D
1413 1623 D
1411 1625 D
1409 1626 D
1407 1627 D
1405 1628 D
1403 1629 D
1402 1630 D
1400 1630 D
1398 1631 D
1396 1632 D
1395 1632 D
1393 1632 D
1392 1632 D
1391 1632 D
1390 1631 D
1388 1631 D
1388 1630 D
1387 1629 D
1386 1627 D
1385 1626 D
1385 1624 D
1385 1622 D
1384 1620 D
1384 1619 D
1384 1616 D
1383 1614 D
1383 1612 D
1383 1610 D
1383 1608 D
1383 1605 D
1382 1603 D
1382 1601 D
1382 1599 D
1381 1598 D
1381 1596 D
1380 1594 D
1379 1593 D
1379 1592 D
1378 1591 D
1377 1590 D
1376 1590 D
1374 1590 D
S
N
1374 1590 M
1373 1589 D
1371 1590 D
1370 1590 D
1368 1590 D
1367 1591 D
1365 1592 D
1363 1593 D
1361 1594 D
1359 1595 D
1357 1596 D
1355 1597 D
1353 1598 D
1351 1599 D
1349 1600 D
1347 1601 D
1346 1602 D
1344 1603 D
1342 1603 D
1341 1604 D
1339 1604 D
1338 1604 D
1336 1604 D
1335 1604 D
1334 1603 D
1333 1602 D
1332 1602 D
1331 1600 D
1331 1599 D
1330 1597 D
1330 1596 D
1329 1594 D
1329 1592 D
1328 1590 D
1328 1588 D
1328 1586 D
1328 1583 D
1328 1581 D
1327 1579 D
1327 1577 D
1327 1575 D
1327 1573 D
1326 1571 D
1326 1569 D
1325 1568 D
1325 1566 D
1324 1565 D
1323 1564 D
1322 1563 D
1321 1562 D
1320 1562 D
S
N
1320 1562 M
1319 1562 D
1317 1562 D
1316 1562 D
1314 1562 D
1312 1563 D
1311 1563 D
1309 1564 D
1307 1565 D
1305 1566 D
1303 1567 D
1301 1568 D
1299 1569 D
1297 1570 D
1295 1572 D
1294 1573 D
1292 1574 D
1290 1574 D
1288 1575 D
1286 1576 D
1285 1576 D
1283 1576 D
1282 1576 D
1281 1576 D
1279 1576 D
1278 1575 D
1277 1574 D
1276 1573 D
1276 1572 D
1275 1571 D
1274 1569 D
1274 1567 D
1274 1566 D
1273 1564 D
1273 1562 D
1273 1559 D
1273 1557 D
1272 1555 D
S
N
1272 1555 M
1273 1553 D
1273 1551 D
1273 1548 D
1273 1546 D
1274 1544 D
1274 1542 D
1274 1541 D
1275 1539 D
1276 1538 D
1276 1537 D
1277 1536 D
1278 1535 D
1279 1534 D
1281 1534 D
1282 1534 D
1283 1534 D
1285 1534 D
1286 1534 D
1288 1535 D
1290 1536 D
1292 1536 D
1294 1537 D
1295 1538 D
1297 1539 D
1299 1541 D
1301 1542 D
1303 1543 D
1305 1544 D
1307 1545 D
1309 1546 D
1311 1547 D
1312 1547 D
1314 1548 D
1316 1548 D
1317 1548 D
1319 1548 D
1320 1548 D
1321 1548 D
1322 1547 D
1323 1546 D
1324 1545 D
1325 1544 D
1325 1542 D
1326 1541 D
1326 1539 D
1327 1537 D
1327 1535 D
1327 1533 D
1327 1531 D
1328 1529 D
1328 1526 D
1328 1524 D
1328 1522 D
1328 1520 D
1329 1518 D
1329 1516 D
1330 1514 D
1330 1512 D
1331 1511 D
1331 1510 D
1332 1508 D
1333 1507 D
1334 1507 D
1335 1506 D
1336 1506 D
1338 1506 D
1339 1506 D
1341 1506 D
1342 1507 D
1344 1507 D
1346 1508 D
1347 1509 D
1349 1510 D
1351 1511 D
1353 1512 D
1355 1513 D
1357 1514 D
1359 1515 D
1361 1516 D
1363 1517 D
1365 1518 D
1367 1519 D
1368 1520 D
1370 1520 D
1371 1520 D
1373 1520 D
1374 1520 D
1376 1520 D
1377 1520 D
1378 1519 D
1379 1518 D
1379 1517 D
1380 1516 D
1381 1514 D
1381 1512 D
1382 1511 D
1382 1509 D
1382 1507 D
1383 1505 D
1383 1502 D
1383 1500 D
1383 1498 D
1383 1496 D
1384 1494 D
1384 1491 D
1384 1489 D
1385 1488 D
1385 1486 D
1385 1484 D
1386 1483 D
1387 1481 D
1388 1480 D
1388 1479 D
1390 1479 D
1391 1478 D
1392 1478 D
1393 1478 D
1395 1478 D
1396 1478 D
1398 1479 D
1400 1479 D
1402 1480 D
1403 1481 D
1405 1482 D
1407 1483 D
1409 1484 D
1411 1485 D
1413 1487 D
1415 1488 D
1417 1489 D
1419 1490 D
1421 1491 D
1422 1491 D
1424 1492 D
1426 1492 D
1427 1493 D
1429 1493 D
1430 1492 D
1431 1492 D
1432 1492 D
1433 1491 D
1434 1490 D
1435 1489 D
1436 1487 D
1436 1486 D
1437 1484 D
1437 1482 D
1437 1480 D
1438 1478 D
1438 1476 D
1438 1474 D
1438 1472 D
1439 1469 D
1439 1467 D
1439 1465 D
1439 1463 D
1440 1461 D
1440 1459 D
1440 1457 D
1441 1456 D
1442 1454 D
1442 1453 D
1443 1452 D
1444 1451 D
1445 1451 D
1446 1450 D
1448 1450 D
1449 1450 D
1451 1450 D
1452 1450 D
1454 1451 D
1456 1452 D
1457 1453 D
1459 1453 D
1461 1455 D
1463 1456 D
1465 1457 D
1467 1458 D
1469 1459 D
1471 1460 D
1473 1461 D
1475 1462 D
1477 1463 D
1478 1463 D
1480 1464 D
1482 1464 D
1483 1465 D
1484 1465 D
1486 1464 D
1487 1464 D
1488 1463 D
1489 1463 D
S
N
1489 1463 M
1490 1462 D
1491 1460 D
1491 1459 D
1492 1457 D
1492 1456 D
1493 1454 D
1493 1452 D
1493 1450 D
1493 1448 D
1494 1445 D
1494 1443 D
1494 1441 D
1494 1439 D
1494 1437 D
1495 1434 D
1495 1433 D
1495 1431 D
1496 1429 D
1496 1427 D
1497 1426 D
1498 1425 D
1499 1424 D
1500 1423 D
1501 1423 D
1502 1422 D
1503 1422 D
1505 1422 D
1506 1422 D
1508 1423 D
1510 1423 D
1511 1424 D
1513 1425 D
1515 1426 D
1517 1427 D
1519 1428 D
1521 1429 D
1523 1430 D
1525 1431 D
1527 1432 D
1529 1433 D
1531 1434 D
1532 1435 D
1534 1436 D
1536 1436 D
1537 1437 D
1539 1437 D
1540 1437 D
1541 1436 D
1543 1436 D
1544 1435 D
S
N
1544 1435 M
1545 1434 D
1545 1433 D
1546 1432 D
1547 1431 D
1547 1429 D
1548 1427 D
1548 1425 D
1548 1423 D
1549 1421 D
1549 1419 D
1549 1417 D
1549 1415 D
1549 1412 D
1550 1410 D
1550 1408 D
1550 1406 D
1550 1404 D
1551 1402 D
1551 1401 D
1552 1399 D
1553 1398 D
1553 1397 D
1554 1396 D
1555 1395 D
1556 1394 D
1558 1394 D
1559 1394 D
1560 1394 D
1562 1394 D
1564 1395 D
1565 1396 D
1567 1396 D
1569 1397 D
1571 1398 D
1573 1399 D
1575 1400 D
1577 1401 D
S
N
1581 1948 M
1541 1948 D
S
N
1581 1947 M
1541 1947 D
S
N
1575 1948 M
1579 1952 D
1581 1956 D
1581 1960 D
1579 1965 D
1575 1969 D
1569 1971 D
1566 1971 D
1560 1969 D
1556 1965 D
1554 1960 D
1554 1956 D
1556 1952 D
1560 1948 D
S
N
1581 1960 M
1579 1963 D
1575 1967 D
1569 1969 D
1566 1969 D
1560 1967 D
1556 1963 D
1554 1960 D
S
N
1541 1954 M
1541 1941 D
S
N
1561 1929 M
1562 1927 D
1566 1923 D
1539 1923 D
S
N
1564 1924 M
1539 1924 D
S
N
1539 1929 M
1539 1918 D
S
N
1554 1900 M
1556 1902 D
1558 1900 D
1556 1898 D
1552 1898 D
1549 1900 D
1547 1902 D
S
N
1581 1879 M
1554 1883 D
S
N
1581 1877 M
1569 1879 D
1560 1881 D
1554 1883 D
S
N
1581 1858 M
1573 1860 D
1566 1864 D
S
N
1581 1857 M
1575 1858 D
1571 1860 D
1566 1864 D
1562 1868 D
1558 1873 D
1556 1877 D
1554 1883 D
S
N
1581 1885 M
1581 1877 D
S
N
1561 1843 M
1562 1840 D
1566 1837 D
1539 1837 D
S
N
1564 1838 M
1539 1838 D
S
N
1539 1843 M
1539 1832 D
S
N
1554 1813 M
1556 1815 D
1558 1813 D
1556 1812 D
1552 1812 D
1549 1813 D
1547 1815 D
S
N
1594 1795 M
1554 1795 D
S
N
1594 1793 M
1554 1793 D
S
N
1575 1793 M
1579 1789 D
1581 1785 D
1581 1782 D
1579 1776 D
1575 1772 D
1569 1770 D
1566 1770 D
1560 1772 D
1556 1776 D
1554 1782 D
1554 1785 D
1556 1789 D
1560 1793 D
S
N
1581 1782 M
1579 1778 D
1575 1774 D
1569 1772 D
1566 1772 D
1560 1774 D
1556 1778 D
1554 1782 D
S
N
1594 1800 M
1594 1793 D
S
N
1561 1755 M
1562 1752 D
1566 1748 D
1539 1748 D
S
N
1564 1750 M
1539 1750 D
S
N
1539 1755 M
1539 1743 D
S
N
1581 1345 M
1541 1345 D
S
N
1581 1343 M
1541 1343 D
S
N
1575 1345 M
1579 1348 D
1581 1352 D
1581 1356 D
1579 1362 D
1575 1365 D
1569 1367 D
1566 1367 D
1560 1365 D
1556 1362 D
1554 1356 D
1554 1352 D
1556 1348 D
1560 1345 D
S
N
1581 1356 M
1579 1360 D
1575 1363 D
1569 1365 D
1566 1365 D
1560 1363 D
1556 1360 D
1554 1356 D
S
N
1541 1350 M
1541 1337 D
S
N
1561 1328 M
1559 1327 D
1558 1328 D
1559 1329 D
1561 1329 D
1563 1328 D
1564 1327 D
1566 1323 D
1566 1318 D
1564 1314 D
1563 1313 D
1561 1312 D
1558 1312 D
1556 1313 D
1553 1317 D
1551 1323 D
1549 1325 D
1547 1328 D
1543 1329 D
1539 1329 D
S
N
1566 1318 M
1564 1315 D
1563 1314 D
1561 1313 D
1558 1313 D
1556 1314 D
1553 1318 D
1551 1323 D
S
N
1542 1329 M
1543 1328 D
1543 1325 D
1541 1319 D
1541 1315 D
1542 1313 D
1543 1312 D
S
N
1543 1325 M
1539 1319 D
1539 1314 D
1541 1313 D
1543 1312 D
1546 1312 D
S
N
1554 1296 M
1556 1298 D
1558 1296 D
1556 1294 D
1552 1294 D
1549 1296 D
1547 1298 D
S
N
1581 1275 M
1554 1279 D
S
N
1581 1273 M
1569 1275 D
1560 1277 D
1554 1279 D
S
N
1581 1255 M
1573 1257 D
1566 1260 D
S
N
1581 1253 M
1575 1255 D
1571 1257 D
1566 1260 D
1562 1264 D
1558 1270 D
1556 1273 D
1554 1279 D
S
N
1581 1281 M
1581 1273 D
S
N
1561 1242 M
1559 1240 D
1558 1242 D
1559 1243 D
1561 1243 D
1563 1242 D
1564 1240 D
1566 1237 D
1566 1232 D
1564 1228 D
1563 1227 D
1561 1225 D
1558 1225 D
1556 1227 D
1553 1230 D
1551 1237 D
1549 1239 D
1547 1242 D
1543 1243 D
1539 1243 D
S
N
1566 1232 M
1564 1229 D
1563 1228 D
1561 1227 D
1558 1227 D
1556 1228 D
1553 1232 D
1551 1237 D
S
N
1542 1243 M
1543 1242 D
1543 1239 D
1541 1233 D
1541 1229 D
1542 1227 D
1543 1225 D
S
N
1543 1239 M
1539 1233 D
1539 1228 D
1541 1227 D
1543 1225 D
1546 1225 D
S
N
1554 1210 M
1556 1212 D
1558 1210 D
1556 1208 D
1552 1208 D
1549 1210 D
1547 1212 D
S
N
1594 1191 M
1554 1191 D
S
N
1594 1189 M
1554 1189 D
S
N
1575 1189 M
1579 1185 D
1581 1182 D
1581 1178 D
1579 1172 D
1575 1168 D
1569 1167 D
1566 1167 D
1560 1168 D
1556 1172 D
1554 1178 D
1554 1182 D
1556 1185 D
1560 1189 D
S
N
1581 1178 M
1579 1174 D
1575 1170 D
1569 1168 D
1566 1168 D
1560 1170 D
1556 1174 D
1554 1178 D
S
N
1594 1197 M
1594 1189 D
S
N
1561 1153 M
1559 1152 D
1558 1153 D
1559 1155 D
1561 1155 D
1563 1153 D
1564 1152 D
1566 1148 D
1566 1143 D
1564 1140 D
1563 1138 D
1561 1137 D
1558 1137 D
1556 1138 D
1553 1142 D
1551 1148 D
1549 1151 D
1547 1153 D
1543 1155 D
1539 1155 D
S
N
1566 1143 M
1564 1141 D
1563 1140 D
1561 1138 D
1558 1138 D
1556 1140 D
1553 1143 D
1551 1148 D
S
N
1542 1155 M
1543 1153 D
1543 1151 D
1541 1145 D
1541 1141 D
1542 1138 D
1543 1137 D
S
N
1543 1151 M
1539 1145 D
1539 1140 D
1541 1138 D
1543 1137 D
1546 1137 D
S
N
1205 1650 M
1166 1650 D
S
N
1205 1648 M
1166 1648 D
S
N
1194 1637 M
1179 1637 D
S
N
1205 1656 M
1205 1626 D
1194 1626 D
1205 1627 D
S
N
1187 1648 M
1187 1637 D
S
N
1166 1656 M
1166 1642 D
S
N
1205 1612 M
1203 1614 D
1202 1612 D
1203 1611 D
1205 1612 D
S
N
1192 1612 M
1166 1612 D
S
N
1192 1611 M
1166 1611 D
S
N
1192 1618 M
1192 1611 D
S
N
1166 1618 M
1166 1605 D
S
N
1192 1586 M
1190 1590 D
1188 1592 D
1185 1594 D
1181 1594 D
1177 1592 D
1175 1590 D
1173 1586 D
1173 1582 D
1175 1579 D
1177 1577 D
1181 1575 D
1185 1575 D
1188 1577 D
1190 1579 D
1192 1582 D
1192 1586 D
S
N
1190 1590 M
1187 1592 D
1179 1592 D
1175 1590 D
S
N
1175 1579 M
1179 1577 D
1187 1577 D
1190 1579 D
S
N
1188 1577 M
1190 1575 D
1192 1571 D
1190 1571 D
1190 1575 D
S
N
1177 1592 M
1175 1594 D
1172 1596 D
1170 1596 D
1166 1594 D
1164 1588 D
1164 1579 D
1162 1573 D
1160 1571 D
S
N
1170 1596 M
1168 1594 D
1166 1588 D
1166 1579 D
1164 1573 D
1160 1571 D
1158 1571 D
1155 1573 D
1153 1579 D
1153 1590 D
1155 1596 D
1158 1597 D
1160 1597 D
1164 1596 D
1166 1590 D
S
N
1198 1524 M
1200 1521 D
1205 1515 D
1166 1515 D
S
N
1203 1517 M
1166 1517 D
S
N
1166 1524 M
1166 1507 D
S
N
1205 1489 M
1166 1489 D
S
N
1205 1487 M
1166 1487 D
S
N
1187 1487 M
1190 1483 D
1192 1479 D
1192 1476 D
1190 1470 D
1187 1466 D
1181 1464 D
1177 1464 D
1172 1466 D
1168 1470 D
1166 1476 D
1166 1479 D
1168 1483 D
1172 1487 D
S
N
1192 1476 M
1190 1472 D
1187 1468 D
1181 1466 D
1177 1466 D
1172 1468 D
1168 1472 D
1166 1476 D
S
N
1205 1494 M
1205 1487 D
S
N
622 1708 M
624 1710 D
625 1711 D
627 1712 D
629 1713 D
631 1714 D
632 1715 D
634 1716 D
635 1717 D
636 1718 D
636 1719 D
637 1720 D
637 1722 D
637 1723 D
637 1724 D
637 1725 D
636 1726 D
635 1727 D
634 1728 D
633 1729 D
631 1731 D
630 1732 D
628 1733 D
626 1734 D
624 1735 D
622 1736 D
620 1737 D
618 1738 D
617 1739 D
615 1740 D
613 1741 D
612 1743 D
610 1744 D
609 1745 D
608 1746 D
607 1747 D
606 1748 D
606 1749 D
606 1750 D
606 1751 D
606 1753 D
607 1754 D
608 1755 D
609 1756 D
610 1757 D
611 1758 D
613 1759 D
615 1760 D
616 1761 D
618 1762 D
620 1764 D
622 1765 D
624 1766 D
626 1767 D
628 1768 D
629 1769 D
631 1770 D
633 1771 D
634 1772 D
635 1773 D
636 1775 D
637 1776 D
637 1777 D
637 1778 D
637 1779 D
637 1780 D
637 1781 D
636 1782 D
635 1783 D
634 1784 D
632 1786 D
631 1787 D
629 1788 D
627 1789 D
626 1790 D
624 1791 D
622 1792 D
620 1793 D
618 1794 D
616 1795 D
614 1796 D
613 1798 D
611 1799 D
610 1800 D
609 1801 D
608 1802 D
607 1803 D
606 1804 D
606 1805 D
606 1806 D
606 1808 D
607 1809 D
607 1810 D
608 1811 D
609 1812 D
610 1813 D
612 1814 D
613 1815 D
615 1816 D
617 1817 D
619 1819 D
621 1820 D
623 1821 D
625 1822 D
626 1823 D
628 1824 D
630 1825 D
631 1826 D
633 1827 D
634 1828 D
635 1829 D
636 1831 D
637 1832 D
637 1833 D
637 1834 D
637 1835 D
637 1836 D
636 1837 D
635 1838 D
634 1840 D
633 1841 D
632 1842 D
630 1843 D
629 1844 D
627 1845 D
625 1846 D
623 1847 D
621 1848 D
619 1849 D
617 1850 D
616 1851 D
614 1853 D
612 1854 D
611 1855 D
609 1856 D
608 1857 D
607 1858 D
607 1859 D
606 1860 D
606 1861 D
606 1863 D
606 1864 D
607 1865 D
607 1866 D
608 1867 D
609 1868 D
611 1869 D
612 1870 D
614 1871 D
616 1872 D
617 1874 D
619 1875 D
621 1876 D
623 1877 D
625 1878 D
627 1879 D
629 1880 D
630 1881 D
632 1882 D
633 1883 D
634 1884 D
635 1886 D
636 1887 D
637 1888 D
637 1889 D
637 1890 D
637 1891 D
637 1892 D
636 1893 D
635 1895 D
634 1896 D
633 1897 D
631 1898 D
630 1899 D
628 1900 D
626 1901 D
625 1902 D
623 1903 D
621 1904 D
619 1905 D
617 1907 D
615 1908 D
613 1909 D
612 1910 D
610 1911 D
609 1912 D
608 1913 D
607 1914 D
607 1915 D
606 1916 D
606 1918 D
606 1919 D
606 1920 D
S
N
606 1920 M
607 1921 D
608 1922 D
609 1923 D
610 1924 D
611 1925 D
613 1926 D
614 1927 D
616 1929 D
618 1930 D
620 1931 D
622 1932 D
624 1933 D
626 1934 D
627 1935 D
629 1936 D
631 1937 D
632 1938 D
634 1940 D
635 1941 D
636 1942 D
637 1943 D
637 1944 D
637 1945 D
637 1946 D
637 1947 D
637 1948 D
636 1949 D
635 1951 D
634 1952 D
633 1953 D
631 1954 D
629 1955 D
628 1956 D
626 1957 D
624 1958 D
622 1959 D
620 1960 D
618 1962 D
616 1963 D
615 1964 D
613 1965 D
611 1966 D
610 1967 D
609 1968 D
608 1969 D
607 1970 D
606 1971 D
606 1973 D
606 1974 D
606 1975 D
S
N
606 1975 M
606 1976 D
607 1977 D
608 1978 D
609 1979 D
610 1980 D
612 1981 D
613 1982 D
615 1984 D
617 1985 D
618 1986 D
620 1987 D
622 1988 D
624 1989 D
626 1990 D
628 1991 D
630 1992 D
631 1993 D
633 1995 D
634 1996 D
635 1997 D
636 1998 D
637 1999 D
637 2000 D
637 2001 D
637 2002 D
637 2003 D
636 2005 D
636 2006 D
635 2007 D
634 2008 D
632 2009 D
631 2010 D
629 2011 D
627 2012 D
625 2013 D
624 2014 D
622 2016 D
S
N
699 2162 M
659 2162 D
S
N
699 2160 M
659 2160 D
S
N
693 2160 M
697 2156 D
699 2152 D
699 2149 D
697 2143 D
693 2139 D
688 2137 D
684 2137 D
678 2139 D
674 2143 D
673 2149 D
673 2152 D
674 2156 D
678 2160 D
S
N
699 2149 M
697 2145 D
693 2141 D
688 2139 D
684 2139 D
678 2141 D
674 2145 D
673 2149 D
S
N
699 2167 M
699 2160 D
S
N
659 2167 M
659 2154 D
S
N
673 2122 M
674 2124 D
676 2122 D
674 2120 D
671 2120 D
667 2122 D
665 2124 D
S
N
699 2100 M
659 2111 D
S
N
699 2098 M
659 2109 D
S
N
693 2100 M
682 2102 D
676 2102 D
673 2098 D
673 2094 D
674 2090 D
678 2087 D
684 2083 D
S
N
699 2079 M
678 2085 D
674 2085 D
673 2083 D
673 2077 D
676 2074 D
680 2072 D
S
N
699 2077 M
678 2083 D
674 2083 D
673 2081 D
S
N
673 2060 M
674 2062 D
676 2060 D
674 2059 D
671 2059 D
667 2060 D
665 2062 D
S
N
695 2042 M
693 2042 D
693 2044 D
695 2044 D
697 2042 D
699 2038 D
699 2030 D
697 2027 D
695 2025 D
691 2023 D
678 2023 D
674 2021 D
673 2019 D
S
N
695 2025 M
678 2025 D
674 2023 D
673 2019 D
673 2017 D
S
N
691 2025 M
689 2027 D
688 2038 D
686 2044 D
682 2045 D
678 2045 D
674 2044 D
673 2038 D
673 2032 D
674 2029 D
678 2025 D
S
N
688 2038 M
686 2042 D
682 2044 D
678 2044 D
674 2042 D
673 2038 D
S
N
699 1087 M
659 1087 D
S
N
699 1085 M
659 1085 D
S
N
693 1085 M
697 1081 D
699 1078 D
699 1074 D
697 1068 D
693 1065 D
688 1063 D
684 1063 D
678 1065 D
674 1068 D
673 1074 D
673 1078 D
674 1081 D
678 1085 D
S
N
699 1074 M
697 1070 D
693 1066 D
688 1065 D
684 1065 D
678 1066 D
674 1070 D
673 1074 D
S
N
699 1093 M
699 1085 D
S
N
659 1093 M
659 1080 D
S
N
712 1050 M
699 1051 D
S
N
712 1048 M
699 1051 D
S
N
673 1033 M
674 1035 D
676 1033 D
674 1031 D
671 1031 D
667 1033 D
665 1035 D
S
N
699 1010 M
659 1021 D
S
N
699 1008 M
659 1020 D
S
N
693 1010 M
682 1012 D
676 1012 D
673 1008 D
673 1005 D
674 1001 D
678 997 D
684 993 D
S
N
699 990 M
678 995 D
674 995 D
673 993 D
673 988 D
676 984 D
680 982 D
S
N
699 988 M
678 993 D
674 993 D
673 991 D
S
N
712 973 M
699 975 D
S
N
712 971 M
699 975 D
S
N
673 956 M
674 958 D
676 956 D
674 954 D
671 954 D
667 956 D
665 958 D
S
N
695 937 M
693 937 D
693 939 D
695 939 D
697 937 D
699 933 D
699 926 D
697 922 D
695 920 D
691 918 D
678 918 D
674 916 D
673 915 D
S
N
695 920 M
678 920 D
674 918 D
673 915 D
673 913 D
S
N
691 920 M
689 922 D
688 933 D
686 939 D
682 941 D
678 941 D
674 939 D
673 933 D
673 928 D
674 924 D
678 920 D
S
N
688 933 M
686 937 D
682 939 D
678 939 D
674 937 D
673 933 D
S
N
712 901 M
699 903 D
S
N
712 900 M
699 903 D
S
N
622 1094 M
624 1096 D
625 1097 D
627 1098 D
629 1099 D
631 1100 D
632 1101 D
634 1102 D
635 1103 D
636 1104 D
636 1105 D
637 1106 D
637 1108 D
637 1109 D
637 1110 D
637 1111 D
636 1112 D
635 1113 D
634 1114 D
633 1115 D
631 1117 D
630 1118 D
628 1119 D
626 1120 D
624 1121 D
622 1122 D
620 1123 D
618 1124 D
617 1125 D
615 1126 D
613 1127 D
612 1128 D
610 1130 D
609 1131 D
608 1132 D
607 1133 D
606 1134 D
606 1135 D
606 1136 D
606 1137 D
606 1139 D
607 1140 D
608 1141 D
609 1142 D
610 1143 D
611 1144 D
613 1145 D
615 1146 D
616 1147 D
618 1148 D
620 1149 D
622 1151 D
624 1152 D
626 1153 D
628 1154 D
629 1155 D
631 1156 D
633 1157 D
634 1158 D
635 1159 D
636 1160 D
637 1161 D
637 1163 D
637 1164 D
637 1165 D
637 1166 D
637 1167 D
636 1168 D
635 1169 D
634 1170 D
632 1172 D
631 1173 D
629 1174 D
627 1175 D
626 1176 D
624 1177 D
622 1178 D
620 1179 D
618 1180 D
616 1181 D
614 1182 D
613 1184 D
611 1185 D
610 1186 D
609 1187 D
608 1188 D
607 1189 D
606 1190 D
606 1191 D
606 1192 D
606 1193 D
607 1195 D
607 1196 D
608 1197 D
609 1198 D
610 1199 D
612 1200 D
613 1201 D
615 1202 D
617 1203 D
619 1204 D
621 1206 D
623 1207 D
625 1208 D
626 1209 D
628 1210 D
630 1211 D
631 1212 D
633 1213 D
634 1214 D
635 1215 D
636 1217 D
637 1218 D
637 1219 D
637 1220 D
637 1221 D
637 1222 D
636 1223 D
635 1224 D
634 1225 D
633 1226 D
632 1228 D
630 1229 D
629 1230 D
627 1231 D
625 1232 D
623 1233 D
621 1234 D
619 1235 D
617 1236 D
616 1237 D
614 1239 D
612 1240 D
611 1241 D
609 1242 D
608 1243 D
607 1244 D
607 1245 D
606 1246 D
606 1247 D
606 1249 D
606 1250 D
607 1251 D
607 1252 D
608 1253 D
609 1254 D
611 1255 D
612 1256 D
614 1257 D
616 1258 D
617 1260 D
619 1261 D
621 1262 D
623 1263 D
625 1264 D
627 1265 D
629 1266 D
630 1267 D
632 1268 D
633 1269 D
634 1270 D
635 1272 D
636 1273 D
637 1274 D
637 1275 D
637 1276 D
637 1277 D
637 1278 D
636 1279 D
635 1281 D
634 1282 D
633 1283 D
631 1284 D
630 1285 D
628 1286 D
626 1287 D
625 1288 D
623 1289 D
621 1290 D
619 1291 D
617 1293 D
615 1294 D
613 1295 D
612 1296 D
610 1297 D
609 1298 D
608 1299 D
607 1300 D
607 1301 D
606 1302 D
606 1304 D
606 1305 D
606 1306 D
S
N
606 1306 M
607 1307 D
608 1308 D
609 1309 D
610 1310 D
611 1311 D
613 1312 D
614 1313 D
616 1315 D
618 1316 D
620 1317 D
622 1318 D
624 1319 D
626 1320 D
627 1321 D
629 1322 D
631 1323 D
632 1324 D
634 1326 D
635 1327 D
636 1328 D
637 1329 D
637 1330 D
637 1331 D
637 1332 D
637 1333 D
637 1334 D
636 1335 D
635 1337 D
634 1338 D
633 1339 D
631 1340 D
629 1341 D
628 1342 D
626 1343 D
624 1344 D
622 1345 D
620 1346 D
618 1348 D
616 1349 D
615 1350 D
613 1351 D
611 1352 D
610 1353 D
609 1354 D
608 1355 D
607 1356 D
606 1357 D
606 1359 D
606 1360 D
606 1361 D
S
N
606 1361 M
606 1362 D
607 1363 D
608 1364 D
609 1365 D
610 1366 D
612 1367 D
613 1368 D
615 1370 D
617 1371 D
618 1372 D
620 1373 D
622 1374 D
624 1375 D
626 1376 D
628 1377 D
630 1378 D
631 1379 D
633 1380 D
634 1382 D
635 1383 D
636 1384 D
637 1385 D
637 1386 D
637 1387 D
637 1388 D
637 1389 D
636 1391 D
636 1392 D
635 1393 D
634 1394 D
632 1395 D
631 1396 D
629 1397 D
627 1398 D
625 1399 D
624 1400 D
622 1401 D
S
N
937 1708 M
622 1708 D
S
N
805 1698 M
803 1699 D
796 1700 D
788 1701 D
778 1703 D
769 1705 D
761 1706 D
755 1708 D
753 1708 D
755 1709 D
761 1711 D
769 1712 D
778 1714 D
788 1716 D
796 1717 D
803 1718 D
805 1719 D
S
N
622 1708 M
622 1401 D
S
N
632 1581 M
632 1577 D
630 1571 D
629 1562 D
627 1553 D
625 1544 D
623 1536 D
622 1531 D
622 1529 D
621 1532 D
620 1538 D
618 1546 D
616 1555 D
614 1564 D
613 1572 D
612 1578 D
611 1581 D
S
N
940 1948 M
901 1948 D
S
N
940 1947 M
901 1947 D
S
N
935 1948 M
938 1952 D
940 1956 D
940 1960 D
938 1965 D
935 1969 D
929 1971 D
925 1971 D
920 1969 D
916 1965 D
914 1960 D
914 1956 D
916 1952 D
920 1948 D
S
N
940 1960 M
938 1963 D
935 1967 D
929 1969 D
925 1969 D
920 1967 D
916 1963 D
914 1960 D
S
N
901 1954 M
901 1941 D
S
N
920 1929 M
922 1927 D
925 1923 D
899 1923 D
S
N
924 1924 M
899 1924 D
S
N
899 1929 M
899 1918 D
S
N
914 1900 M
916 1902 D
918 1900 D
916 1898 D
912 1898 D
908 1900 D
907 1902 D
S
N
940 1879 M
914 1883 D
S
N
940 1877 M
929 1879 D
920 1881 D
914 1883 D
S
N
940 1858 M
933 1860 D
925 1864 D
S
N
940 1857 M
935 1858 D
931 1860 D
925 1864 D
922 1868 D
918 1873 D
916 1877 D
914 1883 D
S
N
940 1885 M
940 1877 D
S
N
920 1843 M
922 1840 D
925 1837 D
899 1837 D
S
N
924 1838 M
899 1838 D
S
N
899 1843 M
899 1832 D
S
N
914 1813 M
916 1815 D
918 1813 D
916 1812 D
912 1812 D
908 1813 D
907 1815 D
S
N
953 1795 M
914 1795 D
S
N
953 1793 M
914 1793 D
S
N
935 1793 M
938 1789 D
940 1785 D
940 1782 D
938 1776 D
935 1772 D
929 1770 D
925 1770 D
920 1772 D
916 1776 D
914 1782 D
914 1785 D
916 1789 D
920 1793 D
S
N
940 1782 M
938 1778 D
935 1774 D
929 1772 D
925 1772 D
920 1774 D
916 1778 D
914 1782 D
S
N
953 1800 M
953 1793 D
S
N
920 1755 M
922 1752 D
925 1748 D
899 1748 D
S
N
924 1750 M
899 1750 D
S
N
899 1755 M
899 1743 D
S
N
940 1345 M
901 1345 D
S
N
940 1343 M
901 1343 D
S
N
935 1345 M
938 1348 D
940 1352 D
940 1356 D
938 1362 D
935 1365 D
929 1367 D
925 1367 D
920 1365 D
916 1362 D
914 1356 D
914 1352 D
916 1348 D
920 1345 D
S
N
940 1356 M
938 1360 D
935 1363 D
929 1365 D
925 1365 D
920 1363 D
916 1360 D
914 1356 D
S
N
901 1350 M
901 1337 D
S
N
920 1328 M
919 1327 D
918 1328 D
919 1329 D
920 1329 D
923 1328 D
924 1327 D
925 1323 D
925 1318 D
924 1314 D
923 1313 D
920 1312 D
918 1312 D
915 1313 D
913 1317 D
910 1323 D
909 1325 D
907 1328 D
903 1329 D
899 1329 D
S
N
925 1318 M
924 1315 D
923 1314 D
920 1313 D
918 1313 D
915 1314 D
913 1318 D
910 1323 D
S
N
902 1329 M
903 1328 D
903 1325 D
900 1319 D
900 1315 D
902 1313 D
903 1312 D
S
N
903 1325 M
899 1319 D
899 1314 D
900 1313 D
903 1312 D
905 1312 D
S
N
914 1296 M
916 1298 D
918 1296 D
916 1294 D
912 1294 D
908 1296 D
907 1298 D
S
N
940 1275 M
914 1279 D
S
N
940 1273 M
929 1275 D
920 1277 D
914 1279 D
S
N
940 1255 M
933 1257 D
925 1260 D
S
N
940 1253 M
935 1255 D
931 1257 D
925 1260 D
922 1264 D
918 1270 D
916 1273 D
914 1279 D
S
N
940 1281 M
940 1273 D
S
N
920 1242 M
919 1240 D
918 1242 D
919 1243 D
920 1243 D
923 1242 D
924 1240 D
925 1237 D
925 1232 D
924 1228 D
923 1227 D
920 1225 D
918 1225 D
915 1227 D
913 1230 D
910 1237 D
909 1239 D
907 1242 D
903 1243 D
899 1243 D
S
N
925 1232 M
924 1229 D
923 1228 D
920 1227 D
918 1227 D
915 1228 D
913 1232 D
910 1237 D
S
N
901 1243 M
903 1242 D
903 1239 D
900 1233 D
900 1229 D
901 1227 D
903 1225 D
S
N
903 1239 M
899 1233 D
899 1228 D
900 1227 D
903 1225 D
905 1225 D
S
N
914 1210 M
916 1212 D
918 1210 D
916 1208 D
912 1208 D
908 1210 D
906 1212 D
S
N
953 1191 M
914 1191 D
S
N
953 1189 M
914 1189 D
S
N
935 1189 M
938 1185 D
940 1182 D
940 1178 D
938 1172 D
935 1168 D
929 1167 D
925 1167 D
920 1168 D
916 1172 D
914 1178 D
914 1182 D
916 1185 D
920 1189 D
S
N
940 1178 M
938 1174 D
935 1170 D
929 1168 D
925 1168 D
920 1170 D
916 1174 D
914 1178 D
S
N
953 1197 M
953 1189 D
S
N
920 1153 M
919 1152 D
918 1153 D
919 1155 D
920 1155 D
923 1153 D
924 1152 D
925 1148 D
925 1143 D
924 1140 D
923 1138 D
920 1137 D
918 1137 D
915 1138 D
913 1142 D
910 1148 D
909 1151 D
906 1153 D
903 1155 D
899 1155 D
S
N
925 1143 M
924 1141 D
923 1140 D
920 1138 D
918 1138 D
915 1140 D
913 1143 D
910 1148 D
S
N
901 1155 M
903 1153 D
903 1151 D
900 1145 D
900 1141 D
901 1138 D
903 1137 D
S
N
903 1151 M
899 1145 D
899 1140 D
900 1138 D
903 1137 D
905 1137 D
S
N
622 1401 M
937 1401 D
S
N
753 1412 M
756 1411 D
762 1410 D
770 1409 D
780 1407 D
789 1405 D
797 1404 D
803 1402 D
805 1401 D
803 1401 D
797 1399 D
789 1398 D
780 1396 D
770 1394 D
762 1393 D
756 1392 D
753 1391 D
S
N
565 1650 M
526 1650 D
S
N
565 1648 M
526 1648 D
S
N
554 1637 M
539 1637 D
S
N
565 1656 M
565 1626 D
554 1626 D
565 1627 D
S
N
546 1648 M
546 1637 D
S
N
526 1656 M
526 1642 D
S
N
565 1612 M
563 1614 D
561 1612 D
563 1611 D
565 1612 D
S
N
552 1612 M
526 1612 D
S
N
552 1611 M
526 1611 D
S
N
552 1618 M
552 1611 D
S
N
526 1618 M
526 1605 D
S
N
552 1586 M
550 1590 D
548 1592 D
544 1594 D
541 1594 D
537 1592 D
535 1590 D
533 1586 D
533 1582 D
535 1579 D
537 1577 D
541 1575 D
544 1575 D
548 1577 D
550 1579 D
552 1582 D
552 1586 D
S
N
550 1590 M
546 1592 D
539 1592 D
535 1590 D
S
N
535 1579 M
539 1577 D
546 1577 D
550 1579 D
S
N
548 1577 M
550 1575 D
552 1571 D
550 1571 D
550 1575 D
S
N
537 1592 M
535 1594 D
531 1596 D
529 1596 D
526 1594 D
524 1588 D
524 1579 D
522 1573 D
520 1571 D
S
N
529 1596 M
528 1594 D
526 1588 D
526 1579 D
524 1573 D
520 1571 D
518 1571 D
514 1573 D
513 1579 D
513 1590 D
514 1596 D
518 1597 D
520 1597 D
524 1596 D
526 1590 D
S
N
558 1524 M
559 1521 D
565 1515 D
526 1515 D
S
N
563 1517 M
526 1517 D
S
N
526 1524 M
526 1507 D
S
N
546 1470 M
544 1472 D
543 1470 D
544 1468 D
546 1468 D
550 1472 D
552 1476 D
552 1481 D
550 1487 D
546 1491 D
541 1492 D
537 1492 D
531 1491 D
528 1487 D
526 1481 D
526 1477 D
528 1472 D
531 1468 D
S
N
552 1481 M
550 1485 D
546 1489 D
541 1491 D
537 1491 D
531 1489 D
528 1485 D
526 1481 D
S
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