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46 pages, to appear in Commun.Math.Phys.
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Renormalization, Yang-Mills Theory, Flow Equations
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\title{ Renormalization Proof for Spontaneously broken Yang-Mills
Theory with Flow Equations}



\author{Christoph Kopper\\
Centre de Physique Th{\'e}orique de l'Ecole Polytechnique\\
F-91128 Palaiseau, France
\and 
Volkhard F. M{\"u}ller\\
Fachbereich Physik, Universit{\"a}t Kaiserslautern\\
D-67653 Kaiserslautern, Germany }  


\date{ }


\maketitle

\begin{abstract}
Abstract: In this paper we present a renormalizability proof
for spontaneously broken $SU(2)$ gauge theory. It is based on Flow
Equations, i.e. on the Wilson renormalization group adapted to 
perturbation theory. The power counting part of the proof,
which is conceptually and technically simple,
follows the same lines as that for any other renormalizable theory.  
The main difficulty stems from the
fact that the regularization violates gauge invariance. We prove that
there exists a class of renormalization conditions such that the 
renormalized Green functions satisfy the Slavnov-Taylor identities
of $SU(2)$ Yang-Mills theory on which the gauge invariance of the
renormalized theory is based.


\end{abstract}
\newpage

\noindent
\section{Introduction }
In the early seventies Wilson and his collaborators published their
ideas on the renormalization group and effective Lagrangians [WiKo],
which have stimulated the progress of quantum field theory
and statistical mechanics ever since. In 1984 Polchinski [Pol] showed
that these ideas are suited as a basis for perturbative
renormalization theory.\footnote{Wilson himself remarked  
already in the late sixties 
that this should be possible,  as we learned from E. Br{\'e}zin.}
He proved Euclidean massive $\Phi^4_4$
to be renormalizable without introducing 
Feynman diagrams, thus sidestepping the associated complicated
analysis of their divergence/convergence properties. 
Instead, the problem  is solved by bounding inductively 
the solutions of a system of first order differential
equations, the Flow Equations (FE), which are a reduction of the Wilson
FE to their perturbative content.\\
Over the past decade Polchinski's argument has been considerably 
simplified technically, extended to physical renormalization conditions
and has been rendered rigorous [KKSa]. Beyond it has been applied, again in
mathematical rigour,  to nearly all situations of physical interest: 
The $\Phi^4_4$  proof itself already also holds for any other 
massive theory with global
symmetries only and renormalizable power counting, like
e.g. the Yukawa-models, O(N)-models etc. It could then be
extended to Euclidean massless $\Phi^4_4$ [KK1] and $QED_4$ [KK2] 
and also to theories in Minkowski-space [KKSc]. The FE method also
served to extract properties of, or bounds on Green functions
which were harder - if at all - to get by other methods.
We mention composite operator renormalization together with 
(generalized) Zimmermann identities [KK3], Wilson's operator product
expansion [KK4], Symanzik improvement in the convergence of the regularized 
theory [Ke1, Wie], de Calan-Rivasseau large order bounds on perturbation
theory [Ke2], bounds on the singularities of Green functions at
exceptional momenta [KK1], analyticity properties of Green functions
in Minkowski space [KKSc] and decoupling theorems [Kim].  
A recent review (in German) on previous work on FEs can be found in [Kop].
We should also mention that the interest in FEs over the last
decade goes far beyond mathematical physics and has led to many
interesting results, ideas and calculations in theoretical physics.
To give few examples we mention that critical exponents
for $\Phi^4_4$-type theories have been calculated in
[TeWe]. Truncated FE have also been applied  to the bound state
problem in [Ell], to Yang-Mills theory in [EHW], in particular to the study of
vacuum condensates in [ReWe].
 
Among the entries in our list on solved renormalization problems
there is still one missing, which is of fundamental importance in
physics, namely nonabelian gauge theory.  The present paper 
is intended to close this gap by treating spontaneously broken
SU(2)-Yang-Mills theory, which corresponds to the weak sector
of the standard model.\footnote{ for vanishing Weinberg angle.
This is however not of decisive importance for the line of the argument.
It matters insofar as the explicit description and treatment of the whole
SU(2)$\times$U(1)-theory would require much more space.} 
Another interesting problem, which should be studied, is QCD where the
problem of gauge invariance is intertwined with the infrared problem.
Since the latter has already been extensively studied 
we chose the
spontaneously broken theory which is infrared finite and thus simpler.
On the other hand  the Slavnov-Taylor identities (STI) or Ward
identities of the spontaneously broken symmetry are more 
complicated to analyse.\footnote{ We mention also that FE and STI 
for pure Yang-Mills theory in the limit case without UV
cutoff have been considered in [BAM2].}

 
The (ultraviolet) power  counting part of the FE
renormalization proof is (up to notational and other minor changes) 
the same and simple for all the above mentioned theories, which renders the
method attractive.  Gauge theories, however, present a
difficulty coming from the wellknown fact that gauge symmetry is
broken by cutoffs in momentum space, and it is just the flow of such a
cutoff which produces the FE. What we have to show is that gauge
invariance is restored when the cutoffs are taken away. On the level
of the Green functions  (which are not gauge invariant) 
this  means  that we have to verify the STI
 of the theory. They  then allow to argue that 
physical quantities such as the S-matrix are gauge-invariant[ZiJ].
On analysing the FE for a gauge theory one realizes that 
the restoration of the STI depends on the choice of the
renormalization conditions chosen and cannot be true in general.
More precisely, since gauge invariance is violated in the regularized
theory, the renormalization group flow will generally produce
nonvanishing contributions to all those relevant parameters of the
theory, which are forbidden by gauge invariance, e.g. a noninvariant
gauge field self\-coupling of the form $(\vec{A}^2)^2$. 
The question is then: Can we use the freedom in adjusting the
renormalization conditions such that the STI are nevertheless 
restored in the end? To answer this question a first observation,
already encountered when treating QED, is crucial: The violation of
the STI in the regularized theory can be expressed through Green functions
carrying an operator insertion, which depends on the regulators. FE theory
for such insertions tells us that these Green functions will vanish once the
cutoffs are removed, if we achieve renormalization conditions on
the theory such that the inserted Green functions (uniquely calculated
from those) have vanishing renormalization conditions
for all relevant terms, i.e. up to the dimension of the insertion 
(which is  5 in our case). Comparing the number of relevant terms for the
SU(2) theory - 37 (see App.A)-  and for the insertion 
- 53 (see App.C)-, we realize that it is not
possible to make vanish 53 terms on adjusting 37 free parameters,
unless there are linear interdependences. It is again the FE (in its
global integrated form) which helps us to make transparent these
interdependences. The problem of how to find one's way through the
STI and adjusting the renormalization conditions appropriately 
is somewhat complicated through
spontaneous symmetry breaking, since the latter mixes Green functions
of different dimension. 

One may of course ask the question whether such a proof of the
renormalizability of Yang-Mills theory is still necessary in view of
the fact that the problem has been settled in the seventies  
by the pioneering work  of 't Hooft and Veltman and successors.
Without going into details or giving references on work which
has made entrance into nearly all textbooks on quantum field theory or
particle physics we would still like to mention that there rests a bit
of uneasiness on the mathematical physicists' side on the form in
which the subject has settled in the course of time. This is because 
the standard way in which the argument is presented nowadays is based
on two main ingredients: the existence of an invariant regularization
scheme, i.e. dimensional regularization, and algebraic manipulations
on generating functionals, which can be given rigorous meaning for
regularized path integral formulations. To date nobody has achieved 
a (rigorous) definition of dimensionally regularized path integrals  
so that there remains a gap in the reasoning which could only be
closed if the analysis of the STI were directly performed on
individual Feynman graphs, a presumably awkward procedure.  
These arguments do not apply to the lattice regularization
\footnote{the above mentioned algebraic analysis is however based on the
continuum formulation.},
which allows for a (particularly transparent) path integral
formulation while respecting gauge invariance. It violates Euclidean
or Lorentz symmetry however. We emphasize the work of Rei\ss\ as a largely
coherent and 
rigorous analysis of the perturbative renormalization problem of 
(QCD type) gauge theories on the lattice [Rei].
His work is based on an adaptation of BPHZ renormalization to the
lattice, where quite a number of new problems appear.\\[.1cm]  
    
As a guide to the logical structure of the paper we now expose the
main line of arguments. Our starting point is a  massive UV
regularized theory. The generating functional $L^{\La,\Lao}$ of the connected
amputated Green functions (CAG) with momenta in the
interval $[\La,\Lao]$ satisfies a flow equation (35) with respect to
$\La$, which when reduced to its perturbative content (37) permits to
bound inductively the $l$-loop $n$-point functions $\cLlln$
in such a way (39, 43) that their existence for $\Lao \to \infty$
becomes obvious. This is true for all theories renormalizable by power
counting under the condition that all relevant terms, i.e. local terms
of mass dimension $\le 4$ are fixed by ($\Lao$-independent)
renormalization conditions (r.c.). In gauge theories the number of
such terms is generally much bigger than the number of free parameters
of the theory. For our model the respective numbers are 37 (listed in App.A)
and 8 (cf. (121)). So most of the r.c. cannot be freely chosen for a
gauge theory. A priori it does not seem possible to guess which
r.c. are the right ones. 

Thus we analyse the action $L^{0,\Lao}$ 
for general r.c. and expose the violation of the STI 
as a functional associated with an operator
insertion,  which turns out to be of dimension 5.
We denote it as  $L_1=L^{0,\Lao}_1$ (75).  
This is achieved on using an UV regularized version (62, 66) of the
BRS transformation (13, 14, 18).
General results from FE theory tell us that $L^{0,\Lao}_1$ will vanish
for $\Lao \to \infty$ if all its relevant terms, i.e. the local parts
of dimension $\le 5$, are fixed to be $0$ by the r.c. and if the
irrelevant terms in $L^{\Lao,\Lao}_1$ vanish sufficiently rapidly 
for $\Lao \to \infty$ (110). The $53$ renormalization parts for 
$L^{0,\Lao}_1$ (see App.C) are functions of the $37$ r.c. for $L^{0,\Lao}$ 
and $7$ free parameters in the BRS transformation (see App.B). 
Thus if the model can be renormalized respecting the STI there must be
linear interdependences among the $53$ relations. These are not
explicit in the theory $L^{0,\Lao}$, since $L^{0,\Lao}$ contains
irrelevant terms of arbitrary dimension which are not known explicitly.
We therefore derive the violated Slavnov-Taylor identities (VSTI) 
also in terms of the bare functionals $L^{\Lao,\Lao}$ and $L_1^{\Lao,\Lao}$
(98, 99), using again the FE for that purpose. The FE may also be used
(104, 113-120) to relate $L_1^{0,\Lao}$ and $L_1^{\Lao,\Lao}$ with
each other (111, 112) so that - respecting the inductive procedure, i.e.
climbing up in the loop order $l$, and for given $l$ in the number of
external legs $n$ - 
we may hope to satisfy the STI (for $\Lao \to \infty$) as well 
by imposing the relevant terms in $L_1^{\Lao,\Lao}$  to vanish 
(instead of those in $L_1^{0,\Lao}$). Since $L^{\Lao,\Lao}$  does not
contain unknown\footnote{ $L^{\Lao,\Lao}$  will include some
  well-behaved irrelevant terms (107, 108) linked to the particular
  nature of the cutoff (30) chosen.} irrelevant terms an explicit
analysis of the bare STI is possible, and we can make vanish $53$
terms order by order in $l$ on appropriately fixing $L^{\Lao,\Lao}$       
and the free BRS constants. However starting at the wrong end -
i.e. fixing counter terms instead of r.c. - we cannot prove
renormalizability.
Thus the task is threefold~:\\
i) Reveal a number of free renormalization constants 
corresponding to the free parameters of the theory (121).\\
ii) Satisfy a subset of the STI for the relevant parts by
choosing appropriate r.c. for $L^{0,\Lao}$ (125, 127).
This subset has to be chosen sufficiently large to get hold on the
finiteness problem, with the help of the FE and afterwards also
of the STI themselves.\\  
iii) Satisfy the remaining STI for the relevant parts by choosing
the appropriate $l$-loop terms in  $L^{\Lao,\Lao}$ (122, 123, 124).
It is possible indeed to show that all remaining STI ((128, 129, 130)
 and those
mentioned after (131)) can be satisfied. These are far more than the
constants fixed in iii).
All this has to be done respecting the order of the inductive
procedure.\\
If it were not possible to make ends meet (i.e. if either the subset  
in ii) is too  small to prove finiteness, or the one in iii) is too small
in order to  satisfy all STI) we would face what is called an anomaly.

Our procedure is complicated by a technical point. The analysis of the
relevant part of the STI at $\La=0$ is much more complicated for $L^{0,\Lao}$
 than for $\Ga^{0,\Lao}$, the generating functional of the 
one-particle irreducible functions. For $L^{0,\Lao}$ many more
 terms of the same loop order may appear in a single STI. 
Passing to one-particle irreducible objects achieves to a
considerable degree the disentangling of the $l$-loop renormalization
parts in the inhomogeneous linear equations of App.C. So App.C has indeed  
been written for the $\Ga^{0,\Lao}$- and not for the
$L^{0,\Lao}$-functional. The price to pay is that we have to provide
for the necessary machinery for the $\Ga$-functional (flow equations
(87), STI (82)) too, using the Legendre transform (78, 79).   
This should not obscure the fact that all results of this paper are to
be obtained from $L^{0,\Lao}$.\\[.1cm]
 
This paper is organized as follows. In chapter 2 we introduce the 
classical action of the model and fix notations.
In chapter 3  we introduce the concepts from FE theory and recall the
statements on renormalizability we need. As regards the general
aspects on bounding inductively solutions of the FE we tend to be 
short as long as the reasoning follows the lines of previous papers.
In chapter 4 we derive the VSTI 
for the regularized
theory in various forms, comment on the adaptation of the
renormalization results to the vertex functions, 
analyse the above mentioned operator
insertion and show how to make vanish its relevant parts step by step
on disposing of the freedom in choosing the renormalization conditions.
This is the key part of the paper.
With the aid of the results from chapter 3 it permits to prove that
the STI are restored and thus solves the
renormalization problem for spontaneously broken SU(2) Yang-Mills theory.


\section{Classical theory and Tree approximation}
\setcounter{equation}{0}
We collect some basic properties of the classical Euclidean SU(2)-
Yang-Mills-Higgs model in four dimensional Euclidean spacetime,
mainly to introduce the notation and the conventions. We 
largely follow the textbook of Faddeev and Slavnov [FaSl].

The action considered involves the real Yang-Mills field $\{A^a_{\mu}\}_{a =
1,2,3}$ and the 
complex scalar doublet $\{\phi_{\alpha}\}_{\alpha = 1,2}$. 
All bosonic fields appearing in this paper may be viewed as smooth
functions of (sufficiently) rapid fall-off. Details do not matter in
view of the fact that we do not perform any nonperturbative analysis of
path integrals. The action has the form
\begin{equation}
S_{\rm inv} = \int dx \left\{ \frac14 F^a_{\mu\nu} F^a_{\mu\nu} +%
\frac12(\nabla_{\mu}\phi)^{\ast}
\nabla_{\mu}\phi + \lambda(\phi^{\ast}\phi-\rho^2)^2\right\},
\label{2.1}
\end{equation}
with the curvature tensor
\begin{equation}
F^a_{\mu\nu}(x) = \partial_{\mu}A^a_{\nu}(x) - \partial_{\nu}A^a_{\mu}(x)
 +
g\epsilon^{abc}
A^b_{\mu}(x) A^c_{\nu}(x)
\label{2.2}
\end{equation}
and the covariant derivative
\begin{equation}
\nabla_{\mu} = \partial_{\mu} + g \frac{1}{2i} \sigma^aA^a_{\mu}(x)
\label{2.3}
\end{equation}
acting on the SU(2)-spinor $\phi$. The parameters $g, \lambda, \rho$ are 
real positive,
$\epsilon^{abc}$ is totally skew symmetric, $\epsilon^{123} = +1$, and
$\{\sigma^a\}_{a = 1,2,3}$
are the standard Pauli matrices. For simplicity the wave function
normalizations of the
fields are chosen equal to one. The action (\ref{2.1}) is invariant under 
local gauge
transformations of the fields
\eq
\frac{1}{2i} \sigma^a A^a_{\mu}(x) \longrightarrow u(x) \frac{1}{2i}
\sigma^a A^a_{\mu}(x) u^{\ast}(x)
+ g^{-1} u(x) \partial_{\mu}u^{\ast}(x)\,, \quad
\phi(x) \longrightarrow u(x) \phi(x)
 \label{2.4b}
\eqe
with $u:  \bbbr^4  \to $ SU(2)$\,$  smooth.
A stable ground state of the action (\ref{2.1}) implies spontaneous symmetry
breaking, taken into
account by reparametrizing the complex scalar doublet as
\begin{equation}
\phi(x) = \left( \begin{array}{c}
B^2(x) + i B^1(x) \\ \rho + h(x) - i B^3(x) \end{array} \right)\,\,,
\label{2.5}
\end{equation}
where $\{B^a(x)\}_{a=1,2,3}$ is a real triplet and $h(x)$ the real Higgs
field. Moreover, in
place of the parameters $\rho, \lambda$ we introduce the masses
\begin{equation}
m = \frac12 g \rho, \quad M = (8 \lambda \rho^2)^{\frac12}.
\label{2.6}
\end{equation}

Aiming at a quantized theory we choose the 't Hooft gauge fixing
\begin{equation}
S_{\rm g.f.} = \int dx \,\frac{1}{2\alpha} (\partial_{\mu}A^a_{\mu} - \alpha m
 B^a)^2,
\label{2.7}
\end{equation}
with $\alpha \in  \bbbr_+$, implemented by anticommuting Faddeev-Popov ghost
and antighost fields
$\{c^a\}_{a=1,2,3}$ and $\{ \bar{c}^a\}_{a=1,2,3}\,$, respectively, via
\eq
S_{\rm gh} = - \int dx \bar{c}^a \big\{ (-\partial_{\mu}\partial_{\mu} +
\alpha m^2)
\delta^{ab} + \frac12 \alpha gm \, h \delta^{ab}  
+\frac12 \alpha gm \epsilon^{acb} B^c - g \partial_{\mu}
\epsilon^{acb} A^c_{\mu}
\big\} c^b.
\label{2.8}
\eqe
Hence, the total "classical action" is
\alpheqn
\begin{equation}
S_{\rm BRS} = S_{\rm inv} + S_{\rm g.f.} + S_{\rm gh} ,
\label{2.9a}
\end{equation}
which we decompose as
\begin{equation}
S_{\rm BRS} = \int dx \left\{ {\cal L}_{\rm quad} (x) + {\cal L}_{\rm
int}(x) \right\}
\label{2.9b}
\end{equation}
\reseteqn
into its quadratic part, with $\Delta \equiv \partial_{\mu} \partial_{\mu}$,
\begin{eqnarray}
{\cal L}_{\rm quad} &=& \frac14 ( \partial_{\mu}A^a_{\nu} -%
\partial_{\nu}A^a_{\mu} )^2
+ \frac{1}{2\alpha} (\partial_{\mu}A^a_{\mu})^2 + \frac12%
m^2A^a_{\mu}A^a_{\mu} \nonumber \\
& & + \frac12 h (- \Delta + M^2)h + \frac12 B^a (- \Delta + \alpha m^2)B^a
%
\nonumber \\
& & - \bar{c}^a (- \Delta + \alpha m^2) c^a\, ,
\label{2.10}
\end{eqnarray}
and into its interaction part
\begin{eqnarray}
{\cal L}_{\rm int} &=& g \epsilon^{abc} (\partial_{\mu}A^a_{\nu})A^b_{\mu}
A^c_{\nu} +
\frac14 g^2(\epsilon^{abc} A^b_{\mu}A^c_{\nu})^2 \nonumber \\
& & + \frac12 g \left\{ (\partial_{\mu}h)A^a_{\mu}B^a - h
A^a_{\mu}\partial_{\mu}B^a -
\epsilon^{abc}A^a_{\mu}(\partial_{\mu}B^b) B^c \right\} \nonumber \\
& & + \frac18 g A^a_{\mu}A^a_{\mu} \left\{ 4 mh + g(h^2+B^aB^a) \right\}
\nonumber \\
& & + \frac14 g \frac{M^2}{m}\, h (h^2 + B^aB^a) + \frac{1}{32} g^2 \left(
\frac Mm \right)^2
(h^2+B^aB^a)^2 \nonumber \\
& & - \frac12 \alpha gm \bar{c}^a \left\{ h \delta^{ab} + \epsilon^{acb}B^c
\right\} c^b
\nonumber \\
& & - g \epsilon^{acb} (\partial_{\mu} \bar{c}^a) A^c_{\mu}c^b.
\label{2.11}
\end{eqnarray}
In (\ref{2.10}) we recognize that all fields are massive and that no
coupling term
$A^a_{\mu}\partial_{\mu} B^a$ appears.
The propagators of the Yang-Mills field $A_{\mu}^a$,  of the Higgs
field $h$,  and of the ghost field $c^a$ and  the Goldstone field $B^a$, 
are thus (respectively)
\eq
C_{\mu \nu}^{ab}(p)\,=\, {\de^{ab} \over  p^2+m^2} \{\de_{\mu \nu}
-(1-\al){p_{\mu}p_{\nu}\over  p^2+\al m^2}\}\,,\
C(p)\,=\, {1 \over  p^2+M^2}\,,\
S^{ab}(p)\,=\, {\de^{ab} \over  p^2+\al m^2}\,\,.
\label{2.12}
\eqe
The classical action $S_{\rm BRS}$ in (\ref{2.9b}) has the following
properties
\begin{enumerate}
\item[i)] Euclidean invariance: $S_{\rm BRS}$ is an O(4)-scalar.
\item[ii)] Rigid SO(3)-isosymmetry: The fields $\{A^a_{\mu}\}, \{B^a\},
\{c^a\}, \{ \bar{c}^a\}$
are isovectors and $h$ is an isoscalar; 
$S_{\rm BRS}$ is invariant under global SO(3)-transformations.
\item[iii)] BRS-invariance:
Introducing the classical composite fields
\alpheqn
\begin{eqnarray}
& &\psi^a_{\mu}(x) = \left\{ \partial_{\mu} \delta^{ab} + g%
\epsilon^{arb}A^r_{\mu}(x)\right \}
c^b(x),\\
\label{2.13a}
& &\psi(x) = - \frac12 g B^a(x) c^a(x), \\
\label{2.13b}
& &\psi^a(x) = \left\{ (m + \frac12 g\,h(x)) \delta^{ab} + \frac12 g%
\epsilon^{arb} B^r(x) \right\}
c^b(x), \\
\label{2.13c}
& &\Omega^a(x) = \frac12 g\epsilon^{apq}c^p(x)c^q(x)\,,
\label{2.13d}
\end{eqnarray}
\reseteqn
the BRS-transformations of the fields are defined as
\alpheqn
\begin{equation}
A^a_{\mu}(x) \longrightarrow A^a_{\mu}(x) - \psi^a_{\mu}(x)\epsilon,
\label{2.14a}
\end{equation}
\begin{equation}
h(x) \longrightarrow h(x) - \psi(x) \epsilon,
\label{2.14b}
\end{equation}
\begin{equation}
B^a(x) \longrightarrow B^a(x) - \psi^a(x) \epsilon,
\label{2.14c}
\end{equation}
\begin{equation}
c^a(x) \longrightarrow c^a(x) - \Omega^a(x) \epsilon,
\label{2.14d}
\end{equation}
\begin{equation}
\bar{c}^a(x) \longrightarrow \bar{c}^a(x) - \frac{1}{\alpha}%
(\partial_{\nu}A^a_{\nu}(x)
- \alpha m B^a(x)) \epsilon\,.
\label{2.14e}
\end{equation}
\reseteqn
\end{enumerate}
In these transformations $\epsilon$ is a spacetime independent Grassmann
element that 
commutes with the fields $\{A^a_{\mu}, h, B^a\}$ but anticommutes with the
(anti-)ghosts
$\{c^a, \bar{c}^a\}$. To show the BRS-invariance of the total classical 
action (9) one
first observes that the composite classical fields (13) are themselves 
invariant under
the BRS-transformations (14). Moreover, we can write (\ref{2.8}) in the form
\begin{equation}
S_{\rm gh} = - \int dx \bar{c}^a \{-\partial_{\mu}\psi^a_{\mu} + \alpha m
 \psi^a \}.
\label{2.15}
\end{equation}
Using these properties the BRS-invariance of (9) is straightforward
(if somewhat tedious) to  verify.

It is convenient to add to the classical action (9) source terms both for
 the
fields and the composite fields (13), defining
\eq
S_c = S_{\rm BRS} + \int dx \{ \gamma^a_{\mu} \psi^a_{\mu} + \gamma \psi +
\gamma^a \psi^a
+ \omega^a\Omega^a \} 
- \int dx \{ j^a_{\mu} A^a_{\mu} + sh + b^aB^a + \bar{\eta}^ac^a +
\bar{c}^a\eta^a \}.
\label{2.16}
\eqe
The sources $\gamma^a_{\mu}, \gamma, \gamma^a$ have dimension 2, ghost
number -1 and are
Grassmann elements, whereas $\omega^a$ has dimension 2 and ghost number -2;
the sources $\eta^a$
and $\bar{\eta}^a$ have ghost number +1 and -1, respectively, and are 
Grassmann elements.
The BRS-transformation (14) of $S_c$ can be written as
\begin{equation}
S_c \longrightarrow S_c + {\cal D} S_c \epsilon
\label{2.17}
\end{equation}
employing the BRS-operator ${\cal D}$, defined by
\begin{equation}
{\cal D} = \int dx \left\{ j^a_{\mu} \frac{\delta}{\delta\gamma^a_{\mu}} +
s\,\frac{\delta} 
{\delta\gamma} + b^a \frac{\delta}{\delta\gamma^a} + \bar{\eta}^a
\frac{\delta}{\delta\omega^a}
+ \eta^a \left(\frac{1}{\alpha} \partial_{\nu} \frac{\delta}{\delta
j^a_{\nu}} - m \frac{\delta}{\delta b^a} \right) \right\}\ .
\label{2.18}
\end{equation}
(Observe that $\epsilon$ anticommutes with $\eta, \bar{\eta}$, too.)

For some purposes it will turn out convenient to regard the
fields and functionals thereof in momentum space. Our conventions are
\eq
\phi(x)\,=\,\int_p e ^{ipx}\,\hat{\phi}(p)\,,\,\,
\,\,\int_p\,=\,\int {d^4p \over (2\pi)^4} \,,
\label{2.19}
\eqe 
where mostly we will omit the hat on $\phi(p)$.
From (\ref{2.19}) we obtain 
\[
{\de \over \de \phi(x)} \,=\, 
\int d^4p \,\,e ^{-ipx} {\de \over \de   \hat{\phi}(p)}
\,=\, (2\pi)^4 \int_p e ^{-ipx} {\de \over \de \hat{\phi}(p)}\ .
\]
For functionals with operator insertions like e.g.
\eq
S_{\ga(x)}~:={\de S_c \over \de \ga(x) }\,\, 
\mbox{ we define }
S_{\ga(p)}~:= \int d^4x\,\, e ^{ipx} \, S_{ \ga(x) } 
\label{2.20}
\eqe
(again in abusively shortened notation).
For later use it will be convenient to introduce a shortened
collective notation for the fields, sources and propagators. 
As for the latter, we will sometimes denote all propagators (\ref{2.12})
collectively by $C$. Furthermore we write 
\eq   
\mbox{for the
bosonic fields }\vp_{\tau}=(A_{\mu}^a,\ h,\ B^a)\, \mbox{ with corresponding
  sources }\, J_{\tau}=(j_{\mu}^a,\ s,\ b^a\,)\,,
\label{nota1}
\eqe
\eq 
\mbox{for all fields }
\,\Phi= (\vp_{\tau},\ c^a,\ \bc^a ) \ \mbox{ and  for their sources}
\,\,K= (J_{\tau},\ \etab^a,\ \eta^a)\,,
\label{nota2}
\eqe 
\eq
\mbox{ and for the insertion sources }\,
\xi = (\ga_{\mu}^a,\ \ga,\ \ga^a,\ \omega^a)
\mbox{ and }\ 
\ga_{\tau}= (\ga_{\mu}^a,\ \ga,\ \ga^a)\,.
\label{nota3}
\eqe 
The quantization of the classical theory amounts to constructing 
a well-defined version of the formal
functional integral respresentation for the generating functional W of the
connected Green
functions such that these functions satisfy the system of STI. Considering
the formal expression for the modified generating functional
\begin{equation}
\exp \frac{1}{\hbar} W = {\cal N} \int [d A\, dh\, dB\, dc\, d\bar{c}]
\exp \{ -\frac{1}{\hbar} S_c \}
\label{2.21}
\end{equation}
we observe that the quadratic part (\ref{2.10}) appearing in $S_c$ 
constitutes a well-defined Gaussian measure\footnote{Once 
we have introduced the regularization (\ref{regu})
the support of the measure consists of sufficiently well-behaved functions.}.
In a formal loop expansion of the remaining part of the exponent the
emerging order $\hbar^0$, i.e. the tree approximation, is well-defined and
satisfies
\begin{equation}
{\cal D}W|_{h^0} = 0\,,
\label{2.22}
\end{equation}
which follows from (\ref{2.17}) when using the 
invariance of the (formal) measure in (\ref{2.21})
under BRS transformations. In the sequel we will inductively 
tackle all orders $\hbar^l, \; l \in \bbbn$, of
the loop expansion. 



\section{Flow Equations: Renormalizability without
Slavnov-Taylor Identities}
\subsection{The Flow Equations for the SU(2) Yang-Mills Higgs model}
The FE of Wilson's renormalization group is obtained as a differential
equation w.r.t. the flow parameter $\La$, which  is the energy scale 
down  to which the degrees of freedom have been integrated out, starting
from the UV region. We will consider the generating
functional of the connected amputated Green functions (CAG) which we denote as
\eq 
L^{\La,\Lao}(\vp_{\tau},c,\bc)
\label{3.1}
\eqe 
with the following explanations:
We have introduced an UV regularization\footnote{Furthermore we 
should restrict the theory  to a
finite volume $V$ as long as field independent vacuum terms are
generated by the flow, which diverge in infinite volume by translation
invariance.  We do not make this explicit here and refer 
the interested reader to previous work [KKSa, KK3].} $\Lao\,$
to have a well-defined starting point, so that  
\eq
0\leq \La \leq \Lao\,<\, \infty\,\,.
\label{3.2}
\eqe
The functional $\,L^{\La,\Lao}(\vp,c,\bc)$ is to be viewed as a formal power
series in $\hbar$, since we are studying the perturbative
renormalization problem in the loop expansion.
To be more precise on its definition we write it as 
\eq
L^{\La,\Lao}\,=\,\sum_{|n|=3}^{\infty}L^{\La,\Lao}_{l=0,n}
\,+\,
\sum_{l=1}^{\infty} \hbar^l\,\sum_{|n|=1}^{\infty} L^{\La,\Lao}_{l,n}\,\,.
\label{3.3}
\eqe 
Here the multiindex $n$ denotes the  number of field variables 
of each species appearing:
\eq
n=\,\{n_A,\,n_h,\,n_B,\,n_{\bc},\,n_c\,\},\quad
|n|:=n_A+\,n_h+\,n_B+\,n_{\bc}+\,n_c\ .
\label{n}
\eqe
So for $|n|=4$ we are e.g. regarding a four point function. 
(\ref{3.3}) implies that, by definition, at 0 loop order  $L^{\La,\Lao}$
contains no contribution from the one- or two-point functions.
With this restriction it is the generating functional of the CAG
of the following theory: \\
i) The propagators are those from (\ref{2.12})
including the regulating factor
\eq
\si_{\La,\Lao}(p^2)\,=\,{\si_{\Lao}(p^2)
\,-\,\si_{\La}(p^2) \over \si_{\Lao}(0)}\,
\ \mbox{ with }\ 
\si_{\La}(p^2)\,=\,
e^{-{1 \over \La^6}[(p^2+m^2)(p^2+\al m^2)(p^2+M^2)]}\ .
\label{regu}
\eqe
In the sequel this choice of the cutoff function turns out to be 
technically convenient\footnote{There is of course a lot of
  arbitrariness in this choice. What is needed is a sufficiently
  well-behaved function tending to $1$ for $\La \to 0$, $\Lao \to
  \infty$, which is essentially supported for momenta between $\La$
and $\Lao$.
The verification of the restoration 
of the STI in Ch.4
would be somewhat easier using a suitable regulating function with compact
support of the type $\si_{\La}(p)\,=\,K({p^2+m^2 \over \La ^2})$,
where $K(x)=1, x\le 1$, $K(x)=0, x\ge 2$, $K$ monotonic and smooth.
But the choice (\ref{regu}) allows to perform the analytic continuation to
Minkowski space as shown in [KKSc], and it has the advantage that 
$(\si_{\La}(p))^{-1}$ is well-defined. Avoiding its appearance is
possible, but sometimes needs detours.}.
Besides being explicit it permits 
to verify easily the following bounds on the regularized propagators
$C^{\La,\Lao}(p)~:=\,C(p)\,\si_{\La,\Lao}(p^2)$
\eq
|(\prod_{i=1}^{|w|} {\pa \over \pa p_{\mu_i}}){\pa \over \pa \La}
C^{\La,\Lao}(p)| \leq\,
\left \{ \begin{array}{r@{\quad,\quad}l} 
C & \quad \mbox{for}
\quad 0 \leq \La \leq m     \\   
\La^{-3-|w| }\,{\cal P}(|p|/\La)\,\si_{\La}(p^2)
 & \quad \mbox{for} \quad m\leq \La \leq \Lao  
\end{array} \right\}  \,.
\label{probo}
\eqe
Here and in the following  ${\cal P}$ denotes (each time it appears
possibly a new) polynomial with nonnegative coefficients. These as well
as the constant $C$ depend on $\al,\,m,\,M,\, |w|$,
but not on $p,\,\La,\,\Lao$.\\ 
ii) The vertices are to be taken from   
our starting bare action (interaction Lagrangian inclu\-ding counter terms)
\eq
L^0\,:=\,L^{\Lao,\Lao}\,.
\label{3.4}
\eqe
In the case of an invariant regularization we would
choose here $S_{\rm BRS}$ from (\ref{2.9b}), modified by including counter
terms of any order $\hbar^l$, $l\ge 1$, of the same structure and by  
excluding the 0-loop quadratic part. In our case such a restricted
choice would not allow to prove restoration of the STI. Therefore 
 we will allow at first for  all counter terms permitted by the unbroken global
symmetries of the theory, i.e. O(4) and SO(3)$_{\rm iso}$.
 
These terms will then become unique functions of the renormalization
conditions chosen. There are 37 such local terms of dimension $\leq
4$, corresponding to those listed in  Appendix A. At the tree level $l=0$ we 
shall always consider the terms with $|n|+|w|
\le 4$ to be given by (\ref{2.11}).
We denote by 
\eq
(2 \pi)^{4(|n|-1)} \de^{n}_{\Phi(p)}L^{\La,\Lao}_l|_{\Phi \equiv 0}
\,=\,
\de(p_1+\ldots+p_{|n|})\, {\cLlln}(p_1,\ldots,p_{|n|-1})
\label{cag}
\eqe
the $n$-point CAG of loop order $l$ involving
the indicated number of ($A_{\mu},\,h,\,B,\,\bc,\,c$) fields.
We will also write $\vec{p}$ for  $(p_1,\,\ldots,\,p_{|n|-1})$ 
in the following.
We stay somewhat unprecise about the momentum assignment to the fields since
this would unnecessarily blow up the notation. We also omit
vector and isovector indices. Finally we will also use the shorthand
\eq
\pa^w:= \prod_{i=1}^{|n|-1}\prod_{\mu=1}^{4}
({\pa \over \pa p_{\mu_i}})^{w_{i,\mu}}\ \mbox{ with }\
w=(w_{i,1},\ldots,w_{|n|-1,4}),\  |w|=\sum w_{i,\mu}\,.
\eqe 
The Flow equations (FE) have been derived quite generally several times,
so we tend to be short. The Wilson FE written for  
$\Lll$ takes the form\footnote{$\Ill$ is the vacuum functional which
  strictly speaking exists only in finite volume. Since it plays 
hardly any role in the following, we do not discuss this issue here
and refer to [KKSa, KK3] for further comments.}
\eq
e ^{-{1 \over \hbar} (\Lll\,+\,\Ill})\,= \,
e ^{ \hbar \De(\La,\Lao)}\,e ^{-{1 \over \hbar} L^{0}}\ .
\label{wils}
\eqe
Here $\De(\La,\Lao)$ is the functional Laplace operator
which in our theory takes the form
\eq
\De(\La,\Lao)\!=\! 
\frac12 \langle {\de \over \de A_{\mu}^a},
\,\Cll_{\mu \nu}{\de \over \de A_{\nu}^a} \rangle\!+\,
\frac12 \langle {\de \over \de h},\,\Cll{\de \over \de h} \rangle\! +\, 
\frac12 \langle {\de \over \de B^a},
\,\Sll{\de \over \de B^a} \rangle\! +\,
\langle {\de \over \de c^a},
\,\Sll{\de \over \de \bc^a} \rangle\, .
\label{De}
\eqe
Using our shorthand notation we obtain the FE for the CAG $\,\cLll_{l,n}$
from (\ref{wils}) on deriving w.r.t. $\La$, expanding $L$ as in 
(\ref{3.3}) and using (\ref{cag})
\eq
\pa_{\La} \pa^w \,\cLlln (\vec{p}) \,=\,\!\!\!
\sum_{n',|n'|=|n|+2}\!\! c_{n'}\int_k (\pa_{\La}\Cll(k))\,\pa ^w 
\cLll_{l-1,n'}(\vec{p},k,-k)
\label{fequ}
\eqe
\[
-\!\!\!\!\!\!\!\!\!\!\!\sum_{l_1+l_2=l,\,w_1+w_2+w_3=w\atop
n_1,n_2,|n_1|+|n_2|=|n|+2} \!
\Biggl[ c_{n_1,n_2}\, \pa^{w_1} \cLll_{l_1,n_1}(p_1,\ldots,p_{|n_1|-1})
(\pa^{w_3}\pa_{\La}\Cll(p'))\,\,
\pa^{w_2} \cLll_{l_2,n_2}(-p',\ldots,p_{|n|-1})\Biggr]_{s,a}\!\!.
\]
The constants $c_{n'},\ c_{n_1,n_2}$ are combinatorial.
The field assignment of the propagators $\Cll$ is not written, it is
implicit in the multiindices $n',\ n_1, \ n_2$ related to $n$. On the
r.h.s. the integrated momentum $k$ refers to that of the fields from
$n'-n$, and $-p'= p_1+\ldots+p_{|n_1|-1}$. Furthermore the subscripts
$s,a$ indicate (anti)symmetrization according to the statistics of the 
various fields, since we assume the $\cLll_{l,n}$ to be
(anti)symmetrized from the beginning.







\subsection{Renormalizability}
The system of differential FE (\ref{fequ}) can be integrated
inductively, using mixed boundary conditions (b.c.)~:\\
$A_1$) At $\La=\Lao$ the $n$ point functions with
$|n|+|w|>4$, i.e. the irrelevant ones, are supposed to be smooth
functions of $\vec{p},\ \Lao$ obeying the bounds
\eq
|\partial^w \cLlol_{l,n}(\vec{p})| \,\leq\,
\Lao^{4-|n|-|w|}\,{\cal P}_1(log{\Lao \over m})\,
{\cal P}_2(\frac{|\vec{p}|}{\Lao})\;,
\quad |n|+|w|\,\geq\,5 \;.
\label{rand}
\eqe
The standard case are b.c., where the r.h.s. of  
(\ref{rand}) vanishes. We need to be slightly more general to compensate for
effects of the cutoff function $\si_{0,\Lao}$, see Ch.4, 
(\ref{irr}, \ref{wert}).\\
$A_2$) At $\La=0$ the CAG with $|n|+|w| \le 4$, i.e. the relevant ones,
are fixed, order by order
in $\hbar$ at the renormalization point, which
we choose at $\vec{p}=0$ for simplicity. The renormalization conditions  
(r.c.) may be chosen weakly $\Lao$-dependent, we restrict to smooth uniformly
bounded functions of $\Lao$ converging for $\Lao \to \infty$.
Of course we always restrict to b.c. respecting the global 
(Euclidean and Iso-)symmetries.\\
With the FE we can inductively obtain the following bounds
on the CAG $\,\cLll_{l,n}$~:\\[.1cm]
{\bf Proposition 1~:} 
\eq
|\pa^w \cLlln(\vec{p})| \le\,
(\La+m)^{4-|n|-|w| }\,{{\cal P}_1}(log {\La + m \over m})\,
{{\cal P}_2}({|\vec{p}| \over \La+m})\,.
\label{propo1}
\eqe
%\left \{ \begin{array}{r@{\quad,\quad}l} 
%{{\cal P}_1}(\vec{p}/m) & \quad \mbox{for}
%\quad 0 \leq \La \leq m     \\   
%\La^{4-|n|-|w| }\,{{\cal P}_2}(log (\La/m))\,
%{{\cal P}_3}(\vec{p}/\La)\,
%& \quad \mbox{for} \quad m\leq \La \leq \Lao  
%\end{array} \right\}  \,.
The polynomials ${{\cal P}_1},\ {{\cal P}_2}$ have nonnegative
coefficients depending on $l,\ n,\ w,\ \al,\ m,\ M$, but not on
$\vec{p}, \ \La,\ \Lao$. \\[.1cm]  
We do not present a proof of the proposition since 
the line of thought is the same as in the references [KKSa, KK3, Kop] 
and restrict to few comments. 
It proceeds by induction upwards in the number of loops and for given
loop order upwards in $|n|$ (in contrast to the procedure employed
when expanding in a coupling constant~: There one proceeds downwards in
$|n|$. For given $l,\,n$ we proceed downwards
in $|w|$, starting from some 
arbitrary\footnote{The minimal value of $3$ is needed, because for the relevant
terms the passage from the fixed momentum, at which the renormalization
conditions are imposed, to any momentum is achieved by the
Schl{\"o}milch or integrated
Taylor formula [KKSa,Pol]. For the two point function there thus appear up
to three derivatives. If one also wants to prove smoothness one has 
to admit for arbitrarily high $|w_{max}|$.} $|w_{max}|\ge 3$.
Thus we have to start at loop order $l=0$ and from $|n|= 3$,
since $\Lll_{l=0}$ does not contain contributions for $|n|\le 2$.
(\ref{fequ}) immediately gives 
\[
\cLll_{0,n}(\vec{p})\,=\,\cLlol_{0,n}(\vec{p})\,,\quad|n|= 3\,,
\]
since the r.h.s. vanishes. Thus the bound is satisfied.
For $|n|= 4,\ l=0$ we may also fix the b.c. at $\La=\Lao$,
{\it if we want to read them off the action (\ref{2.11})}, 
since here the second term on the r.h.s. of (\ref{fequ}) 
contributes and leads to a 
one particle reducible difference between  $\cLll_{0,n}$ 
and $\cLlol_{0,n}$. 
This digression of the rules $A_1$), $A_2$) is a pure matter of
convenience however. The inductive proof then proceeds by inserting 
the induction hypothesis on the r.h.s. of the FE 
(which has already been bounded) and performing the 
momentum and $\La$-integrals, starting from the respective b.c. and using 
the bound (\ref{probo}). An important point to note is the following~: 
Which bounds for the $\cLll$ can be obtained, depends only on the 
b.c. imposed and  on the propagators (and dimensionality). Note finally 
that for the  purpose of renormalizability only the bound on 
$\cLol$ in the limit $\Lao \to \infty$ is needed. The rest is of
technical nature.
In the next chapter we want to make use of the following also somewhat
technical\\[.1cm] 
{\bf Corollary~:} 
For given $l_0 >\,0$ and $n_0,\ w_0$ with $|n_0|+|w_0| \le 4$ 
we assume that the b.c. on the CAG 
$\,\pa^w \cLll_{l,n}$, ($|w| \le |w_{max}|\,$),
 have been imposed in agreement with 
$A_1$), $A_2$) for $l<l_0$ and arbitrary $n,\ w$~; and for 
$l=l_0$ and $|n| < |n_0|$ and $|w| \le |w_0|$.
Suppose that we fix the b.c. for $\pa ^{w_0}\cLll_{l_0,n_0}$ 
'on the wrong side', i.e. at $\Lao$, 
such that it obeys the bound
\eq
|\pa^{w_0}\cLlol_{l_0,n_0}(0)| \leq \Lao^{4-|n_0|-|w_0| }\,
{\cal P}(log (\Lao/m))\,.
\label{coro}
\eqe
Then we also have 
\eq
|\pa^{w_0}\cLll_{l_0,n_0}(0)| \leq \Lao^{4-|n_0|-|w_0| }\,
{\cal P}(log (\Lao/m))\,.
\label{coro1}
\eqe
{\it Proof~:} Due to our assumptions, the r.h.s. of the FE (\ref{fequ})  
is bounded by (\ref{propo1}), since the bounds on all terms preceding 
$(l_0,n_0,w_0)$ in the induction remain unchanged
apart from those with $|w|>|w_0|$. Those are not needed however
because we only make a statement at the renormalization point
$\vec{p}=0$ and thus do not require a bound on the Taylor remainder. 
The deterioration of the bound then stems from both  the b.c. contribution 
(\ref{coro}) {\it and} from the fact that the r.h.s. of the FE
has to be integrated from $\Lao$ to $\La$ (instead of integrating from
$0$ to $\La$), i.e. from the wrong side.  This gives the bound 
\[
|\pa^{w_0}\cLll_{l_0,n_0}(0)| \leq \Lao^{4-|n_0|-|w_0| }\,{\cal P}_1(log
(\Lao/m))\,+\, 
|\int_{\La}^{\Lao} d\La'\,\La'^{4-|n_0|-|w_0|-1}\,{\cal P}_2(log
(\La'/m))|
\]
\[
 \leq \Lao^{4-|n_0|-|w_0| }\,{\cal P}_3(log (\Lao/m))\,.
\]
Note that the bound does not improve, if we set the b.c. for
$\pa^{w_0}\cLlol_{l_0,n_0}(0)$ equal to zero.  \qed
We remark that statements similar to that of the Corollary 
could also be extended
to general external momenta, they are not needed however. 
In response to the remarks made before one may ask oneself whether the previous
bounds (\ref{propo1}) may be improved, if the b.c. are in some sense smaller.
This is indeed the case. Regard e.g. the CAG 
containing an odd number of scalar
fields, i.e. $n_h+n_B \in 2\bbbn-1$. Then the following improved
bounds hold~:  
\eq
|\pa^w \cLlln(\vec{p})| \le\,
(\La+m)^{3-|n|-|w| }\,{{\cal P}_1}(log {\La+m \over m})\,
{{\cal P}_2}({|\vec{p}| \over \La +m})\,.
\label{propo2}
\eqe
The main reason why we may expect an improvement of power counting for those
terms in our theory is that, as can be seen in App.A$\,$, at $l=0$ the terms 
in question are all proportional to a mass factor.
Since we will not need such sharpened statements we do not give a
proof of (\ref{propo2}) here.
As usual the bound on the Green functions should be complemented by a
convergence statement, since (\ref{propo1}, \ref{propo2}) would still
admit bounded but oscillating solutions\footnote{a possibility
generally only envisaged by mathematical physicists since such
oscillations are counterintuitive to any experience from
calculations}. 
Convergence follows from
\\[.1cm]  
{\bf Proposition 2~:} 
\eq
|\pa_{\Lao}\pa^w \cLlln(\vec{p})| \le\,
{1\over \Lao^2}(\La+m)^{5-|n|-|w| }\,{{\cal P}_1}(log (\Lao/m))\,
{{\cal P}_2}({|\vec{p}| \over \La +m})\,.
\label{propo3}
\eqe
%\left \{ \begin{array}{r@{\quad,\quad}l} 
%{1\over \Lao^2}\,{{\cal P}_1}(log(\Lao /m))
%{{\cal P}_2}(\vec{p}/m) & \quad \mbox{for}
%\quad 0 \leq \La \leq m     \\   
%{1\over \Lao^2}\La^{5-|n|-|w| }\,{{\cal P}_3}(log (\Lao/m))\,
%{{\cal P}_4}(\vec{p}/\La)\,
%& \quad \mbox{for} \quad m\leq \La \leq \Lao  
%\end{array} \right\}\ .  
As before the nonnegative coefficients in the (new) 
polynomials ${\cal P}_i$ may depend on $l,\ n,\ w,\ \al$, $m$, $M$, but not on
$\vec{p}, \ \La,\ \Lao$. For the proof, which follows the same
inductive scheme, we refer again to the earlier  
references [KKSa, KK3, Kop].


\subsection{Bounds on Green functions with Operator Insertions}
The problem of renormalizing Green functions with operator insertions
has been studied quite generally in [KK3, KK4]. Again we state the
propositions needed for SU(2) Yang-Mills theory without proofs,
restricting to remarks on the (minor) modifications needed.
We have to deal with two kinds of operator insertions here.
The first are the BRS insertions (13a)-(\ref{2.13d}). 
These are defined 
as operator insertions of dimension 2, ghost number one for
(13a)-(\ref{2.13b}) and ghost number 2 for
(\ref{2.13d}), which transform as vector-isovector, scalar-isoscalar, 
scalar-isovector and  scalar-isovector respectively. By the general
renormalization theory we thus have to allow for all counter terms 
of dimension $\le 2$ and of the same symmetry properties. In the bare
action the insertions take the form    
\alpheqn
\begin{eqnarray}
& &\psi^a_{\mu}(x) = R_1^0 \, \partial_{\mu} c^a(x) + R_2^0\, g\,
\epsilon^{arb}A^r_{\mu}(x)\,c^b(x),\\
\label{3.12a}
& &\psi(x) = - R_3^0 \,\frac12 g\, B^a(x) c^a(x), \\
\label{3.12b}
& &\psi^a(x) = R_4^0 \, m \,c^a(x) 
+ R_5^0 \,\frac12 g\,h(x)\,c^a(x)  + R_6^0 \, \frac12 g\,
\epsilon^{arb} B^r(x) \,c^b(x), \\
\label{3.12c}
& &\Omega^a(x) =  R_7^0 \,\frac12 g\,\epsilon^{apq}c^p(x)c^q(x)\,,
\label{3.12d}
\end{eqnarray}
\reseteqn
where we demand
\eq
 R_i^0 \,=\,1\,+\,O(\hbar)\ ,
\label{R}
\eqe
i.e. the counter terms are again viewed as formal power
series in $\hbar$, and we of course assume the insertions to agree
with (13a-13d) at the tree level. 

The following remark might be helpful, as regards the transformation 
(\ref{2.14e}) of the anti\-ghost~: We do not introduce constants 
$R_8^0,\,\ldots,\,R_{11}^0$, corresponding to the terms of dimension
$\le 2$ with the same symmetry properties (besides the ones in (\ref{2.14e})
these are $h\,B^a$ and $\vep^{abc}c^b\bc^c$). The claim implicit
(not only here, but throughout the literature) and verified 
in Ch.4
is then that it is possible to obtain a finite renormalized 
theory\footnote{This is related to the fact that (14e) is linear in $\Phi$.}  
satisfying the STI, by fixing these
constants at $\La=\Lao$, i.e. on the wrong side~; in fact setting
$R_8^0,\,R_9^0\,=1$, $R_{10}^0,\,R_{11}^0\,=\,0$. In the more general case 
one would have to admit arbitrary values for these four constants
and to introduce another
source for the respective composite operator. The (violated) STI (see below  
(\ref{vicag}, \ref{gasti}, \ref{4.2.8})) would then take a more
symmetric form, the terms involving $A_{\mu}^a, B^a$ being replaced by another
one of the form $\langle c^a,\, D\,L_{\overline{\ga}^a} \rangle$.

The insertions may be  generated 
by the respective sources as in (\ref{2.16}), we set
\eq
\Lllo_{\xi}\,=\,
\int dx \,\{ \gamma^a_{\mu}(x) \psi^a_{\mu}(x) + \gamma(x) \psi(x) +
\gamma^a (x)\psi^a (x)+ \omega^a(x)\Omega^a(x) \} \,,
\label{laxi}
\eqe
and also
\eq
\tLllo\,=\,\Lllo\,+\,\Lllo_{\xi}\,\,. 
\label{zwei}
\eqe
We again 
get a Wilson FE (cf. (\ref{wils})) for $\tLll$
generating the CAG with operator insertions\footnote{We will only 
regard insertions with nonvanishing ghost number. Therefore
the vacuum functional
$\tilde{I}$ equals $I$, since there are no vacuum diagrams with
nonvanishing ghost number, due to ghost number conservation under the
flow. Thus we will always write $I$  subsequently.}  
\eq
e ^{-{1 \over \hbar} (\tLll\,+\,\tilde{I}^{\La,\Lao})}\,= \,
e ^{ \hbar \De(\La,\Lao)}\,\,e ^{-{1 \over \hbar} \tLllo}\ .
\label{wilsin}
\eqe
Restricting our attention to CAG with {\it one} insertion, e.g.
\eq
\Lll_{\ga(x)}:= {\de \tLll \over \de \ga(x)}|_{\xi=0} 
\label{1ein}
\eqe
(similarly for the other insertions) we obtain 
by deriving (\ref{wilsin}) w.r.t. $\La$ a {\it linear}
FE for $\Lll_{\ga(x)}$. Writing similarly as in(\ref{cag})
\eq
(2 \pi)^{4(|n|-1)} \de^{n}_{\Phi(p)}L^{\La,\Lao}_{\ga(q);l}|_{\Phi\equiv 0}
\,=\,
\de(q+p_1+\ldots+p_{|n|})\, {\cLlgln}(p_1,\ldots,p_{|n|-1})
\label{cag1}
\eqe
we obtain the differential FE for CAG with one insertion
\eq
\pa_{\La} \pa^w \,\cLlgln (\vec{p}) \,=\,\!\!\!
\sum_{n',|n'|=|n|+2}\!\! c_{n'}\int_k (\pa_{\La}\Cll(k))\,\pa ^w 
\cLll_{\ga(q);l-1,n'}(\vec{p},k,-k)\label{fequ1}
\eqe
\[
-
\!\!\!\!\!\!\!\!\!\sum_{l_1+l_2=l,\,w_1+w_2+w_3=w\atop
n_1,n_2,|n_1|+|n_2|=|n|+2} \!
\Biggl[ c_{n_1,n_2}\, \pa^{w_1} \cLll_{\ga(q);l_1,n_1}(p_1,\ldots,p_{|n_1|-1})
(\pa^{w_3}\pa_{\La}\Cll(p'))\,
\pa^{w_2} \cLll_{l_2,n_2}(-p',\ldots,p_{|n|-1})\Biggr]_{s,a}
\]
the notation being that of (\ref{fequ}). 
Since ghost and antighost in
(\ref{De}) do not appear symmetrically, the 
$\bc$ ($c$)-derivative appears once
in $n_1$ ($n_2$) and once in $n_2$ ($n_1$). In the following we denote
for shortness by $\xi(q)$ any of the sources $\ga_{\mu}^a(q)$, 
$\ga(q)$, $\ga^a(q)$, $\omega^a(q)$. Obviously each of the insertions
leads to a FE as (\ref{fequ1}).
In the derivation
of (\ref{fequ1}) no use is made of the specific kind of insertion considered.
Thus even more generally we replace $\xi(q)$ by $\chi(q)$ when
talking of an insertion of dimension $D$ (instead of 2).
This is because we also want
to cover the CAG with one insertion of  dimension
5 describing the BRS violating terms
of the regularized theory. This insertion is analysed in Ch.4.1. 
The particular kind of insertion chosen only comes into play when
considering the b.c., which are fixed as follows~:\\
$B_1$) 
At $\La=\Lao$ the $n$ point functions $\pa^w \cLlcln$ with
$|n|+|w|>D$, i.e. the irrelevant ones, are supposed to obey the bounds
(cf. $A_1$, (\ref{rand}))
\eq
|\partial^w \cLlol_{\chi(q);l,n}(\vec{p})| \,\leq\,
\Lao^{D-|n|-|w|}\,{\cal P}_1(log{\Lao \over m})\,
{\cal P}_2(\frac{|\vec{p}|}{\Lao})\;,
\quad |n|+|w|\,>\,D\ . 
\label{randb}
\eqe
$B_2$) At $\La=0$ the CAG with $|n|+|w| \le D$, i.e. the relevant ones,
are fixed, order by order
 in $\hbar$ at the renormalization point
$\vec{p}=0$, with the same restrictions as in $A_2$).\\ 
Again (\ref{fequ1}) lends itself to an inductive scheme
through which we may prove the renormalizability of the CAG
with insertion.
For the $\cLlxln$ there are seven free r.c. which fix the seven
parameters  $R_i^0$ from (\ref{R}). For the CAG $\cLlcln$ with insertion 
$L^{\Lao,\Lao}_1$ from (\ref{llo1}) we have to fix 53 r.c. 
corresponding to the list in App.C. Under these conditions our
inductive scheme may now
also be employed to prove  boundedness and convergence of inserted
Green functions.\\[.1cm]
{\bf Proposition 3:} 
\eq
|\pa^w \cLlcln(\vec{p})| \le\,
(\La+m)^{D-|n|-|w| }\,{{\cal P}_1}(log {\La +m\over m})\,
{{\cal P}_2}({|\vec{p}| \over \La +m})\,,
\label{propo41}
\eqe
\eq
|\pa_{\Lao}\pa^w \cLlcln(\vec{p})| \le\,
{(\La+m)^{D+1-|n|-|w|} \over \Lao^2}\,{{\cal P}_1}(log (\Lao/m))\,
{{\cal P}_2}({|\vec{p}| \over \La +m})\,.
\label{propo42}
\eqe
Whereas the bounds from Proposition 3 are sufficient for our purposes
as regards the functions $\cLlxln$, we need a stronger result for the
BRS violating insertions $\cLlcln$, 
which we can achieve on imposing further restrictions on
the b.c. It is important in this respect that the FE for the inserted
CAG is linear. This implies e.g. that multiplying all CAG with a 
$\La$-
independent factor gives a new solution. If 
we want to show that the CAG $\cLlcln$ from Ch.4.1 vanish in the limit
$\Lao \to \infty$, the strategy is thus to reveal a negative power of
$\Lao$, which can be factorized from the CAG $\cLlcln$. 
It is quite conceivably a sufficient condition for achieving this,
to require that all r.c. be bounded by  a negative power of
$\Lao$. The main issue of Ch.4 will be to prove that there exist 
r.c. on the CAG such that the inserted CAG describing BRS violation
obey such suppressed r.c.
Once this is accomplished we can rely on the
following proposition for the restoration of BRS invariance~:\\[.1cm]  
{\bf Proposition 4:} Replace the  statements from $B_2)$ on the
renormalization conditions by\\
$B_3)$ At $\La=0$ the $\cLol_{\chi(q);l,n}$ with $|n|+|w| \le D$
are fixed at order  $\hbar^l$ and $\vec{p}=0$ 
to be smooth functions of $\Lao$  bounded by 
\eq {1 \over \Lao} \,\,{{\cal P}}(log (\Lao/m))\ .
\label{vpropo5}
\eqe
Then we have the bound
\eq
|\pa^w \cLlcln(\vec{p})| \le\,
{1 \over \Lao}\,
(\La+m)^{D+1-|n|-|w| }\,{{\cal P}_1}(log (\Lao/m))\,
{{\cal P}_2}({|\vec{p}| \over \La +m})\,.
\label{propo5}
\eqe

Again we do not give a proof, but refer to our previous remarks, to
[KK3] and in particular to Prop.7 in the paper on QED [KK2],
where similar results were obtained in the more complicated situation
of a massless theory.
Proposition 4  obviously shows that the CAG  $\cLlcln$ vanish for $\Lao \to
\infty$. We remark that in Ch.4 we will arrange for r.c. such that
the bound (\ref{vpropo5}) can be set to 0. This does not improve  
(\ref{propo5}), because of the nonvanishing b.c. for the irrelevant
terms (see B1), (\ref{randb}) above).




\section{Restoration of the Slavnov-Taylor Identities}
\subsection{Violated Slavnov-Taylor Identities
for Connected and Proper Green functions}
Once the physical free parameters of the theory, i.e.
$g,\,\la,\, m $ and the gauge fixing parameter $\al$
\footnote{on which physical quantities should not depend} 
 have been fixed,
the Yang-Mills-Higgs theory  should be uniquely determined up to 
normalizations of the fields. The standard tool to enforce this
uniqueness are the Slavnov-Taylor-identities. 
Whereas their role is twofold in renormalization procedures based on
invariant regularization schemes - apart from assuring uniqueness and
physical gauge invariance, they also serve as a technical  tool to
show inductively that the theory can be renormalized without
introducing counter terms not present in the bare action -     
we only have to ensure their validity for the first purpose. At an
intermediate stage they
are  inevitably violated by the regularization in momentum space, as
gauge invariance is. We want to show that they hold after removing 
the regularization, if we choose the renormalization conditions properly.
Our starting point is the generating functional of the regularized
Green functions at the physical value $\La=0$ of the flow parameter.
Remembering (\ref{nota1},\ref{nota2}) we write
\eq
\langle \Phi ,\,K \rangle \,=\,
\int dx\, \{ \sum_{\tau} \vp_{\tau}(x)  J_{\tau}(x)\,+\,
\bc^a(x) \eta ^a(x)\,+\,\etab^a(x) c^a(x)\}\ .
\label{phik}
\eqe
The Gaussian measure $d\mu_{\Lao}(\Phi)$
corresponding to the quadratic form ${1 \over \hbar}\,Q^{\Lao}$ with
\eq
Q^{\Lao} =\frac12 \langle
 A_{\mu}^a,\, (\Col\!)_{\mu\nu}^{-1} \,A_{\nu}^a\,\rangle +
\frac12 \langle h ,\,(\Col)^{-1}h\,\rangle +
\frac12 \langle B^a ,\,(\Sol)^{-1}B^a\,\rangle -
\langle \bc^a ,\,(\Sol)^{-1}c^a\,\rangle 
\label{Q}
\eqe
is given by its characteristic functional
\eq
\int\, d\mu_{\Lao}(\Phi) \, e^{ {1\over\hbar} \langle \Phi,\,K \rangle}
\,=\, e^{  {1\over\hbar} P(K)}
\label{peka}
\eqe 
with
\eq
P(K)\,=\,
\frac12 \langle j_{\mu}^a,\, \Col_{\mu\nu} \,j_{\nu}^a\rangle\,+\,
\frac12 \langle s ,\,\Col \,s\rangle\,+\,
\frac12 \langle b^a ,\,\Sol b^a\rangle\,-\,
 \langle \etab^a ,\,\Sol \eta^a\rangle\,.
\label{pekad}
\eqe
The generating functional of the regularized Green functions may now
be written as
\eq
Z^{0,\Lao}(K)\,=\,\int \, d\mu_{\Lao}(\Phi)\,  
e^{- {1\over\hbar} \Lllo \,+\,{1\over\hbar}\langle \Phi ,\,K \rangle}\ .
\label{zet}
\eqe 
Defining {\it regularized} BRS variations of the fields through
\eq
\de_{BRS}\, \vp_{\tau}(x)\,=\,-(\si_{0,\Lao}\psi_{\tau})(x)\, \vep
\,,\quad
\de_{BRS}\, c^a(x)\,=\,-(\si_{0,\Lao}\Omega ^a)(x)\, \vep\,,
\label{brsre}
\eqe
\[
\de_{BRS}\, \bc^a(x)\,=\,-[\si_{0,\Lao}({1\over\al}
 \pa_{\nu}A_{\nu}^a - m\,B^a)](x)\,
\vep\,\,,
\]
the BRS transform of the Gaussian measure is given by 
\eq
 d\mu_{\Lao}(\Phi)\,\mapsto \,
 d\mu_{\Lao}(\Phi)\biggl\{1\,+\,{1 \over \hbar}\sum_{\tau}
\langle \vp_{\tau} ,\,(\Col_{\tau})^{-1} \si_{0,\Lao} \psi_{\tau} \rangle
\,\vep\,-\,
{1 \over \hbar}\langle \bc^a,\,(\Sol)^{-1} \si_{0,\Lao} \Omega ^a \rangle
\,\vep
\label{mubrs}
\eqe
\[
\,+\,{1 \over \hbar}\langle {1\over\al}
 \pa_{\nu}A_{\nu}^a - m\,B^a,\, \si_{0,\Lao} (\Sol)^{-1} c^a \rangle
\,\vep\biggr\}\,=\,
d\mu_{\Lao}(\Phi)\biggl\{1\,-\,{1 \over \hbar}\,\de_{BRS}\,
Q^{\Lao}\, \biggr\}\ .
\]
The BRS-variation of the measure has mass dimension 5, since $\si_{0,\Lao}$
just cancels its inverse appearing in the inverted propagators in 
(\ref{mubrs}). This is convenient, and it is the basic  reason why 
we regularized the BRS-transformation. Requiring the invariance of 
the functional integral
in (\ref{zet}) under (regularized) BRS-transformations 
of the field variables 
\footnote{These transformations of variables and consequently
(\ref{vsti0}) can be given rigorous
meaning for the regularized Gaussian integrals. Arguing formally (\ref{vsti0})
amounts to the somewhat sloppy statement that the Jacobian of the 
BRS-transformation equals 1 which  in turn has  rigorous meaning 
for the lattice regularization, see e.g. [Rei].},  
(\ref{brsre}) provides us with the Violated Slavnov-Taylor identities
(VSTI)~:
\eq
0\,\stackrel {!} {= }\,\int \, d\mu_{\Lao}(\Phi) \, 
e^{- {1\over\hbar} \Lllo \,+\,{1\over\hbar} \langle \Phi ,\,K \rangle}\,
 \bigl\{ \de_{BRS}\, \langle \Phi ,\,K \rangle\,-\,
\de_{BRS}\,(Q^{\Lao}\,+\,\Lllo)\bigr\}\ .
\label{vsti0}
\eqe 
The BRS variations in (\ref{vsti0}) can be generated using an
appropriate operator insertion:\\
i) First we define the modified generating functional using 
(\ref{zwei})
\eq
\tilde{Z}^{0,\Lao}(K,\xi)\,=\,\int \, d\mu_{\Lao}(\Phi) \, 
e^{- {1\over\hbar} \tLllo \,+\,{1\over\hbar}
 \langle \Phi ,\,K \rangle}\,
\label{tzet}
\eqe
together with the regularized BRS operator (compare to (\ref{2.18}))
\eq
{\cal D}_{\Lao} =  
  \sum_{\tau}\langle J_{\tau},\, \si_{0,\Lao}
\frac{\delta}{\delta\gamma_{\tau}}\rangle +
 \langle\bar{\eta}^a,\, \si_{0,\Lao}
\frac{\delta}{\delta\omega^a}\rangle
+  \langle(\frac{1}{\alpha} \partial_{\nu} \frac{\delta}{\delta
j^a_{\nu}} - m \frac{\delta}{\delta b^a} ),\,\si_{0,\Lao}\eta^a
\rangle\ .
\label{dreg}
\eqe
ii) Secondly we define the terms emerging from the BRS-noninvariance 
of the action to form the insertion $\Lllo_1$ with ghost number 1
\eq
\Lllo_1\,\vep:= -\de_{BRS} (Q^{\Lao}\,+\,\Lllo)\ .
\label{llo1}
\eqe
Due to the regularizing factor $\si_{0,\Lao}$ in (\ref{brsre})
the insertion $\Lllo_1$ is not a local operator. Using (\ref{llo1})
we introduce the generating functional
\eq
Z^{0,\Lao}_{\chi}(K):=\,\int \, d\mu_{\Lao}(\Phi) \, 
e^{- {1\over\hbar} (\Lllo \,+\,\chi\,\Lllo_1)\,+\,{1\over\hbar}
 \langle \Phi ,\,K \rangle}\,
\label{chizet}
\eqe
for $\chi \in \bbbr$. Now the VSTI (\ref{vsti0}) can be rewritten as
\eq
{\cal D}_{\Lao}\,\tilde{Z}^{0,\Lao}(K,\xi)|_{\xi \equiv 0}\,=\,
{d \over d\chi}Z^{0,\Lao}_{\chi}(K)|_{\chi=0} \ .
\label{vsti1}
\eqe
The modified functionals from (\ref{tzet}, \ref{chizet}) permit to
define the generating functionals of the corresponding CAG with the
respective insertions
\eq
\tilde{Z}^{0,\Lao}(K,\xi)\,=\,
e^{ {1\over\hbar} P(K)}\,e^{- {1\over\hbar}
(I^{0,\Lao}+  \tilde{L}^{0,\Lao}(\vp_{\tau},c,\bc;\xi)) },
\label{tl}
\eqe
\eq
Z^{0,\Lao}_{\chi}(K)\,=\,
e^{ {1\over\hbar} P(K)}\,e^{- {1\over\hbar}
(I^{0,\Lao}+  L^{0,\Lao}_{\chi}(\vp_{\tau},c,\bc)) },
\label{chil}
\eqe
with the relations
\eq
\vp_{\tau}(x)\!=\!\!\int\!\! dy\,\Col_{\tau}(x-y) J_{\tau}(y),\
c^a(x\!)=-\!\int\!\! dy\,\Sol(x-y) \eta^a(y),\
\bc^a(x)\!=-\!\int\!\! dy\,\Sol(x-y) \etab^a(y)
\label{fields}
\eqe
between the variables of the $Z$ and $L$ functionals.
Introducing the shorthand 
\eq
 D_{\tau}\,=\,\biggl( (-\De+m^2)\de_{\mu\nu}-{1-\al \over \al}\pa_{\mu}
\pa_{\nu}\,,\ -\De+M^2\,,\ -\De+\al m^2\equiv D \biggr) 
\label{detau}
\eqe
for the inverted nonregularized propagators and also 
(remember (\ref{1ein}))
\eq
L_1:=L^{0,\Lao}_1\,=\,{d \over d\chi}L^{0,\Lao}_{\chi}|_{\chi=0} \,,
\quad
L:=L^{0,\Lao}\,=\,\tilde{L}^{0,\Lao}|_{\xi\equiv 0}\
(= \,L^{0,\Lao}_{\chi}|_{\chi=0})\ ,
\label{loo1}
\eqe
since we will mostly regard the theory with $\La$ set to 0 in this section, 
we obtain from (\ref{vsti1}) via (\ref{tl}, \ref{chil}, \ref{fields}) 
the VSTI for the connected amputated functions CAG 
\eq
L_1\,=\,
\langle  c^a,\, D({1\over \al}\pa_{\nu} A_{\nu}^a -m\,B^a)\rangle\,-\,
\langle   c^a,\,\si_{0,\Lao}(\pa_{\nu} {\de L \over \de A_{\nu}^a}
 -m {\de L \over \de B^a})\rangle
\,+\,\sum_{\tau} \langle\vp_{\tau},\,D_{\tau} L_{\ga_{\tau}}\rangle
\,-\,\langle \bc^a,\,D\, L_{\om_a} \rangle
\,.
\label{vicag}
\eqe
Since we also have to regard the proper vertex functions we define in
an intermediate step the generating functional of connected
nonamputated Green functions\footnote{noting again that vacuum
functionals should only appear before taking the infinite volume limit} 
\eq 
e ^{\,{1 \over \hbar}\,\tilde{W}(K,\,\xi)}\,=\,
{ \tilde{Z}(K,\xi)   \over \tilde{Z}(0,0)}
\label{wola}
\eqe
(leaving out again the upper indices $0,\Lao$).
From this we derive using (\ref{vsti1}, \ref{chil}, \ref{fields})
\eq
{\cal D}_{\Lao}\,\tilde{W}(K,\xi)|_{\xi=0}\,=\,-L_1(\vp_{\tau},c,\bc)\,.
\label{vstiw}
\eqe
The Legendre transform of  $\tilde{W}$ now leads us to the generating
functional of the proper vertex functions. We set
\eq
\tilde{\Ga}(\uvp_{\tau},\ubc,\uc;\xi)\,+\,
\tilde{W}(J_{\tau},\eta,\etab;\xi)\,=\,
\int dy\, \bigl\{ \sum_{\tau} \uvp_{\tau} J_{\tau}\,+\,\ubc\,\eta \,+\,
\etab \,\uc \bigr\}
\label{lege}
\eqe
with the relations
\eq
J_{\tau}(x)\,=\,{\de \tGa \over \de \uvp_{\tau}(x)}\,,\
 \uvp_{\tau}(x)\,=\,{\de \tilde{W}\over \de J_{\tau}(x)}\,,\
\label{relfel}
\eqe
\[
\eta ^a(x)\,=\,{\de \tGa \over \de \ubc^a(x)}\,,\
\ubc^a(x)\,=\,-{\de \tilde{W} \over \de \eta ^a(x)}\,,\qquad\,
\etab^a(x)\,=\,-{\de \tGa \over \de \uc^a(x)}\,,\
\uc^a(x)\,=\,{\de \tilde{W} \over \de \etab^a(x)}\ .
\]
Note that (\ref{lege}) says that $J_{\tau},\ldots$ may be viewed 
as a formal power series in $\hbar$ with coefficients depending on 
the classical fields $\uvp_{\tau},\ldots$ These series may be
inverted to express  $\uvp_{\tau},\ldots$ as series in terms of
$J_{\tau},\ldots$ 
As a consequence of (\ref{lege}) the relations 
\eq
{\de \tGa \over \de \ga_{\tau} }\,+\,
{\de \tilde{W} \over \de \ga_{\tau} }\,=\,0
\eqe
and an analogous one for the derivative w.r.t. the source $\omega^a$ hold. 
Similarly as before we write 
\eq
\Ga\,=\, \tGa|_{\xi\equiv 0}\,,\quad \Ga_{\ga_{\tau}(x)}\,=\, 
{\de \tGa \over \de \ga_{\tau}(x) }|_{\xi\equiv 0}\ .
\eqe
Then the VSTI for the proper vertex functions emerging from (\ref{vstiw}) 
(where the upper indices $\La=0,\ \Lao$ in 
(\ref{gasti},\ref{ga1l1},\ref{vatra}) are understood) read
\eq
\sum_{\tau}\langle {\de \Ga \over \de \uvph_{\tau}},\,
\si_{0,\Lao}  \Ga_{\ga_{\tau}}\rangle
\,-\, \langle {\de \Ga \over \de \uc^a},\,
\si_{0,\Lao}  \Ga_{\om^a}\rangle\,
-\,\langle ({1 \over \al}\pa_{\nu} \uA_{\nu}^a\,-\,m\,  \uB^a),\,\si_{0,\Lao} 
{\de \Ga \over \de {\ubc}^a} \rangle 
\,=\, \Ga_1(\uvph,\ubc,\uc)
\label{gasti}
\eqe
with
\eq
\Ga_1(\uvph,\ubc,\uc)\,=\,L_1(\vp,\bc,c)
\label{ga1l1}
\eqe
and 
\eq
\vp_{\tau}(x)\!=\!\int\!\! dy\,C_{\tau}(x-y)\,
{\de \Ga \over \de \uvph_{\tau}(y)}\,,\
c^a(x)=\!-\int\!\! dy\,S(x-y)\,{\de \Ga \over \de \ubc^a(y)}\,,\
\bc^a=\!\int\!\! dy\,{\de \Ga \over \de \uc^a(y)}\, S(y-x)\,.
\label{vatra}
\eqe

\subsection{Flow Equations and Renormalizability of Vertex functions}

In this section we shortly comment on  flow equations for
proper vertex functions. Such FE have been analysed  previously 
in [KKSc] for $\phi^4_4$-theory, to prove  analyticity statements
in Minkowski space. They have been derived and applied before in the
literature, see e.g. [BAM1, Wet].
 Writing (\ref{tl}, \ref{wola}, \ref{lege}) 
with general $\La$ instead of $\La=0$ we may derive FE similarly 
as in the previous chapter by deriving w.r.t. $\La$.
Deriving (\ref{wola}) we obtain 
\eq
\pa_{\La}\tilde{W}^{\La,\Lao}(K,\xi)\,=\,
\pa_{\La} P^{\La,\Lao}(K)\,-\,\pa_{\La}\tLll(\vp_{\tau},c,\bc)\,,
\label{lawe}
\eqe
and (\ref{lege}) then implies
\eq
\pa_{\La} \tGa ^{\La,\Lao}\,+\, \pa_{\La}\tilde{W}^{\La,\Lao}=0\ .
\label{laga}
\eqe
Combining both equations and using the FE derived from (\ref{wilsin})
for the functional $\tLll$ we obtain the FE for $\tGa ^{\La,\Lao}$~:
\[
\pa_{\La} \tGa ^{\La,\Lao}(\uvph_{\tau},\ubc,\uc)\,-\,
{1 \over 2} \sum_{\tau} \int_p \uvph_{\tau}(p)
\,\pa_{\La} (C^{\La,\Lao}_{\tau}(p))^{-1}\uvph_{\tau}(-p)\,+\,
\int_p \ubc^a(p)\,\pa_{\La} (S^{\La,\Lao}(p))^{-1}\uc^a(-p)
\]
\eq
\,=\,
\hbar \,(\pa_{\La}
\De(\La,\Lao))\,\tilde{L}^{\La,\Lao}(\vp_{\tau},c,\bc)\ .
\label{floga}
\eqe
The functional on the r.h.s. has to be viewed as depending on the
(classical) fields $\uvph_{\tau},\ \ubc,\ \uc\,$. In momentum 
space the fields 
$\vp_{\tau},\ \bc,\ c$ are given in terms of those through 
\[
\vp_{\tau}(p)=(2\pi)^4 \,C^{\La,\Lao}_{\tau}(p)\,
{\de \tGa^{\La,\Lao} \over \de \uvph_{\tau}(-p)}\,,\
c^a(p)\,=\!-(2\pi)^4 \,S^{\La,\Lao}(p)\,{\de \tGa^{\La,\Lao} \over 
\de \ubc^a(-p)}\,,\
\]
\[
\bc^a(p)\,=\,(2\pi)^{4} \,S^{\La,\Lao}(p)\,{\de \tGa^{\La,\Lao} \over 
\de \uc^a(-p)}
\] 
corresponding  to (\ref{vatra}).
The r.h.s of (\ref{floga}) is expressed in terms of 
$\uvph_{\tau},\ \ubc,\ \uc\,$
\footnote{ Note that $\De(\La,\Lao)$ in (\ref{floga}) is still the one
in terms of the fields $\,\vp_{\tau},\ \bc,\ c$.}  
 using the following relations (and the chain rule) 
\[
(2\pi)^{-4} \, (C^{\La,\Lao}_{\tau}(p))^{-1}\uvph_{\tau}(p)\,=\,
-\,{\de \tLll \over \de \vp_{\tau}(-p)}\,+\,
{\de \tGa ^{\La,\Lao} \over \de \uvph_{\tau}(-p)}\,,\
\]
\eq 
(2\pi)^{-4} \, (S^{\La,\Lao}(p))^{-1}\uc^a(p)\,=\,
{\de \tLll \over \de \bc^a(-p)}\,-\,
{\de \tGa ^{\La,\Lao} \over \de \ubc^a(-p)}\,,\ 
\label{impl}
\eqe
\[
(2\pi)^{-4} \, (S^{\La,\Lao}(p))^{-1}\ubc^a(p)\,=\,
-\,{\de \tLll \over \de c^a(-p)}\,+\,
{\de \tGa ^{\La,\Lao} \over \de \uc^a(-p)}\,. 
\]
The inverted propagators  appearing in (\ref{floga}, \ref{impl})
remain only at the tree level, they cancel
at loop order $\ge 1$.\\
Considering first the functional without insertions we may again
inductively bound the functions $\pa^w \Ga^{\La,\Lao}_{l,n}$ proceeding as in
Ch.3 upwards in $l$
(note the factor of $\hbar$ on the r.h.s.), for given $l$ upwards in $|n|$,  
and for given $l$, $|n|$ downwards in the number of momentum derivatives.
The induction starts from the tree order vertex functional
\[
\Ga^{\La,\Lao}_{l=0}\,=\,
\frac12 \sum_{\tau}\int_p \uvph_{\tau}(p)\,(C^{\La,\Lao}_{\tau}(p))^{-1}
\uvph_{\tau}(-p)\,-\,
\int_p \ubc^a(p)\,(S^{\La,\Lao}(p))^{-1}
\uc^a(-p)
\]
\eq
\,+\,(\Ga^{\La,\Lao}_3\,+\,\Ga^{\La,\Lao}_4)_{l=0}\,+\,
L_{irr}^0|_{l=0}\ .
\label{treega}
\eqe
The tree level three and four point functions from the third term 
are  given in App.A, the last term is the tree level contribution to the 
irrelevant extension of $L^0$ in (\ref{irr}, \ref{wert}).  
Imposing b.c. analogous to those imposed on the CAG from Ch.3.2 
in  A1), A2) we may then derive the bounds\\[.1cm] 
{\bf Proposition 5}~: 
\eq
|\pa^w  \Ga^{\La,\Lao}_{l,n}(\vec{p})| \le\,
(\La+m)^{4-|n|-|w| }\,{{\cal P}_1}(log {\La +m \over m})\,
{{\cal P}_2}({|\vec{p}| \over \La +m})
\label{propoga}
\eqe
with the same comments as for Proposition 1.\\[.1cm]
We again skip the proof.
Finally we note that to obtain the analogous renormalizability
statements for vertex functions with one insertion 
the FE (\ref{floga}) has to be derived w.r.t. the corresponding 
source. Again a FE  linear in terms of the inserted vertex functions,
 but involving also the noninserted ones, emerges. Its solutions  
are bounded in the same way as the corresponding CAG from Ch.3.

Since the analysis of the STI is more transparent in terms of the
vertex functions,  the renormalization conditions will be imposed 
on those. We may then directly infer the finiteness of the theory from
the results of this section.  We could also calculate from
the b.c. on the vertex functions those for the CAG, which then also
satisfy A1),A2) and conclude on the finiteness by Ch.3, so that 
we might have skipped this section altogether, paying instead more
attention on how to calculate b.c. on $L$ from those for $\Ga$ and
vice versa.  
Generally speaking it seems to us that FE for vertex functions
are useful in their own right. Nevertheless the CAG should perhaps be viewed
as the "primary objects" of interest, since the FE for them takes 
a closed functional form. This closed form is of fundamental importance 
for the analysis of the linear  relations among the STI and thus crucial
 for the proof of the Theorem and in particular Lemma 2 below.



\subsection{Violated Slavnov-Taylor Identities for the bare functional
$L^0$}
In this section we use again the abbreviations
\eq
\De\,=\,\De(0,\Lao)\,,\,\,
L\,=\,L^{0,\Lao},\,\,\ti{L}\,=\,\ti{L}^{0,\Lao},
\,\,L^0\,=\,L^{\Lao,\Lao},\,\, \ti{L^0}\,=\,\ti{L}^{\Lao,\Lao},\,\,
L_1^0\,=\,L_1^{\Lao,\Lao}\,\,.
\label{4.2.1}
\eqe
Our starting point are the VSTI (\ref{vicag}).
By commuting the functional differential operator 
appearing on the rhs of (\ref{vicag}) with the renormalization group
flow we will obtain the VSTI in terms of $L^0$. We introduce some further
abbreviations:  
\eq
{\de \over \de R^a(x)}\,=\, 
{1 \over \al} \pa_{\nu} {\de \over \de A_{\nu}^a(x)}
\,-\, m{\de \over \de B^a(x)}\,,\,\,\,\,
X\,=\,\langle D c^a,\,({1\over \al}\pa_{\nu} A_{\nu}^a -m\,B^a)\rangle\,,
\label{4.2.3}
\eqe
\[
Y\,=\,\langle   c^a,\,\si_{0,\Lao}{\de \over \de R^a} \rangle
\,-\,\sum_{\tau} \langle\vp_{\tau},\,D_{\tau}{\de \over \de \ga_{\tau}}\rangle
\,+\,\langle \bc^a,\,D\, {\de \over \de \om^a} \rangle \,.
\]
Now we can write (\ref{vicag}) in the form
\eq
L_1\,=\,e ^{{1\over \hbar}\ti{L}}\,(X+\hbar Y) \,
e ^{-{1\over  \hbar}\ti{L}}|_{\xi\equiv 0}\,\ .
\label{4.2.4}
\eqe
The last two factors may be rewritten as
(remember (\ref{wilsin}))
\eq
(X+\hbar Y)\,e ^{-{1\over \hbar} \ti{L}}\,=\,
e ^{{1\over \hbar} I}\,
e ^{\hbar\,\De}\,e ^{-\hbar\,\De}\,(X+\hbar Y)\,
e ^{\hbar\,\De}\,e ^{-{1\over  \hbar}\ti{L^0}}
\label{4.2.5}
\eqe
\[
\,=\,e ^{{1\over \hbar} I}\,
e ^{\hbar\,\De} \biggl( X+\hbar Y - \hbar [\De,X+\hbar Y]
+ {\hbar^2 \over 2}
[\De,[\De,X+\hbar Y]] \biggr)\,e ^{-{1\over  \hbar}\ti{L^0}}
\,\,.
\]
We have to calculate the commutators
\alpheqn
\eq 
[\De,Y]\,=\, -
\langle   {\de \over \de {\bc}^a},\,\si_{0,\Lao} S\,{\de \over \de
R^a} \rangle
\,-\,\sum_{\tau} \langle {\de \over \de \vp_{\tau}},\,\si_{0,\Lao} 
{\de \over \de \ga_{\tau}}\rangle
\,+\,\langle {\de \over \de c^a},\,\si_{0,\Lao} \, {\de \over \de
\om^a} \rangle \,,
\label{4.2.5a}
\eqe
\eq
[\De,X]\,=\, \langle   c^a,\,\si_{0,\Lao} {\de \over \de
R^a} \rangle
\,-\,\langle {\de \over \de {\bc}^a},\,\si_{0,\Lao} 
({1 \over \al} \pa_{\nu}  A_{\nu}^a
\,-\, m\, B^a) \rangle \,,
\label{4.2.5b}
\eqe
\eq
[\De,[\De,X]]\,=\, -2\,
\langle {\de \over \de {\bc}^a},\,\si_{0,\Lao}  S \,{\de \over \de
R^a} \rangle \,\,.
\label{4.2.5c}
\eqe
\reseteqn
From these relations we obtain
\eq
(X+\hbar Y)\,e ^{-{1\over \hbar} \ti{L}}\,=\,e ^{{1\over \hbar} I}\,
e ^{\hbar\,\De}\biggl\{
\langle   c^a,\,D\,({1 \over \al} \pa_{\nu}  A_{\nu}^a
\,-\, m \, B^a) \rangle \,-\,
\langle {\de \ti{L^0}\over \de {\bc}^a},\,\si_{0,\Lao}  
({1 \over \al}\pa_{\nu} A_{\nu}^a\,-\,m\, B^a) \rangle \,
\label{4.2.6}
\eqe
\[
+\,\sum_{\tau} \langle \vp_{\tau},\,D_{\tau} 
{\de  \ti{L}^0 \over \de \ga_{\tau}}\rangle
\,-\, \langle {\bc}^a,\,D \,{\de \ti{L}^0\over \de
\om^a} \rangle \,
+\,\sum_{\tau} \langle {\de \ti{L}^0 \over\de\vp_{\tau}},\,\si_{0,\Lao} 
{\de  \ti{L}^0 \over \de \ga_{\tau}}\rangle
\,-\, \langle {\de\ti{L}^0 \over \de c^a},\,
\si_{0,\Lao} {\de  \ti{L}^0 \over \de \om^a}\rangle
\biggr\}e ^{-{1\over \hbar} \ti{L}^0} \,\,.
\]
Note that due to the form of $\ti{L}^0$ the contribution
\[
\hbar \sum_{\tau} \langle {\de  \over\de\vp_{\tau}},\,\si_{0,\Lao} 
{\de   \over \de \ga_{\tau}}\rangle\,\ti{L}^0\,-\,
\hbar \, \langle {\de \over \de c^a},\,
\si_{0,\Lao} {\de   \over \de \om^a}\rangle \,\ti{L}^0\
\]
vanishes and thus may be omitted  in the parentheses in (\ref{4.2.6}). 
On the other hand using  (\ref{4.2.4}, \ref{loo1}) 
we can also express 
$(X+\hbar Y) \,e ^{-{1\over  \hbar}\ti{L}}|_{\xi \equiv 0}$ 
in terms of $L^0_1$: 
\eq
(X+\hbar Y) \,e ^{-{1\over  \hbar}\ti{L}}|_{\xi \equiv 0}\,=\,
L_1\,e ^{-{1\over  \hbar}\ti{L}}|_{\xi \equiv 0}
\label{4.2.7}
\eqe
\[
\,=\,
-\hbar\, {d \over d\chi}e ^{-{1\over \hbar}L_{\chi}}|_{\chi=0}
=
\bigl(-\hbar {d \over d\chi}\,e ^{{1\over \hbar} I}\,
 e^{\hbar \De}\,
e ^{-{1\over \hbar}L_{\chi}^0}|_{\chi=0}\bigr)
=e ^{{1\over \hbar} I}\,
e^{\hbar \De}\,L_1^0\,e ^{-{1\over \hbar}L^0}\,.
\]
Remember that $\ti{L}|_{\xi \equiv 0}\,=\,L_{\chi}|_{\chi=0}\,=\,L$.
Equality of (\ref{4.2.6}) for $\xi \equiv 0$ 
and (\ref{4.2.7}) and invertibility of 
$\exp{\hbar \Delta}\,$ (in perturbation theory) now obviously give
\eq
\langle   c^a,\,D\,({1 \over \al} \pa_{\nu}  A_{\nu}^a
\,-\, m\, B^a) \rangle \,-\,
\langle {\de L^0 \over \de {\bc}^a},\,\si_{0,\Lao}  
({1 \over \al}\pa_{\nu} A_{\nu}^a\,-\,m\,  B^a) \rangle \,
\label{4.2.8}
\eqe
\[
+\,\sum_{\tau} \langle \vp_{\tau},\,D_{\tau} 
L^0_{\ga_{\tau}}\rangle
\,-\, \langle {\bc}^a,\,D \,L^0_{\om^a} \rangle \,
+\,\sum_{\tau} \langle {\de L^0 \over \de\vp_{\tau}},\,\si_{0,\Lao} 
L^0_{\ga_{\tau}}\rangle
\,-\, \langle {\de L^0 \over \de c^a},\,
\si_{0,\Lao}  L^0_{\om^a}\rangle
\,=\, L^0_1\,\,.
\]
(\ref{4.2.8}) is the VSTI for the bare functional $L^0$. It turns out
that it plays -unexpectedly- a
prominent role in the analysis of how the STI can be restored.  
Since we impose renormalization conditions in momentum space we 
also express (\ref{4.2.8}) through the  Fourier transformed 
fields (using the conventions from Ch.2) 
\eq
L_1^0\,=\, 
\int_p 
 c^a(p) (p^2+ \al\,m^2) \{ -{i \over \al} p_{\nu}  A_{\nu}^a(-p)
\,-\, m\, B^a(-p) \}
\label{4.2.9}
\eqe
\[
\,-\,
(2\pi)^4 \int_p
 {\de L^0 \over \de {\bc}^a(p)}\,  
\{ {i \over \al}p_{\nu} A_{\nu}^a(p)\,-\,m\,  B^a(p)\}\,\si_{0,\Lao}(p^2) 
\]
\[
+\int_p A_{\mu}^a(p)[(p^2+m^2)\de_{\mu \nu}+{1-\al \over \al} p_{\mu} p_{\nu}]
L^0_{\ga_{\nu}^a(p)}
\,+\,\int_p h(p)(p^2+M^2)L^0_{\ga(p)}
\]
\[
\,+\,\int_p B^a(p)(p^2+\al m^2)L^0_{\ga ^a(p)}
-\,\int_p{\bc}^a(p)(p^2+\al m^2)L^0_{\om^a(p)} 
\]
\[
+\,
(2\pi)^4 \int_p\,\si_{0,\Lao}(p^2)
\biggl\{ { \de L^0 \over \de  A_{\la}^a(p)}L^0_{\ga_{\la}^a(-p)}
\,+\,{ \de L^0 \over \de  h(p)}L^0_{\ga(-p)}
\,+\,{ \de L^0 \over \de  B^a(p)}L^0_{\ga ^a(-p)}
\,-\, {\de L^0 \over \de c^a(p)} 
L^0_{\om^a(-p)}\biggr\}\,\,.
\]








\subsection{Choice of Renormalization Conditions and
Restoration of the Slavnov-Taylor-Identities}
We have derived the STI in the previous two subsections 
for all  three functionals $\Ga,\ L,\ L^0$.
In fact the $ L$-functional is only needed as a connecting link
between the other two.
As we mentioned before this threefold description  will be required
to recognize the linear 
interdependences among  the STI projected onto the relevant parts
of the various functionals. 
For this purpose we also need 
termwise equivalence relations among the relevant parts
of $\Ga$ and $L^0$. 
These termwise equivalence relations are simplified, if 
{\it we assume that the renormalization conditions for the  functionals
 $\Ga$  or $L$ are chosen such that}:
\eq
\ka:=\,{\de \Ga \over \de \uh(x)}|_{\uP\equiv 0}\,=\,0\,
\Longleftrightarrow \,{\de L \over \de h(x)}|_{\Phi\equiv 0}\,=\,0\,.
\label{rest}
\eqe
The condition (\ref{rest}) on the absence of tadpoles,
although probably not indispensable,  
simplifies the subsequent formulae, and it is not really a
physical restriction, but rather one on the parametrization of the
theory. 
Here and in the following we use the shorthand notation 
\[
\pa^w  \de^n_{\Phi} F|_{0} 
\]
to denote the derivative of the functional $F$ (which might be $L$ or
$\Ga$) w.r.t. $n$ fields $\Phi$, evaluated at $\Phi \equiv 0$,
followed by removing the global $\de$-function and performing the
derivatives $\pa^w$. When we write 
\[
\pa^w  \de^n_{\Phi} F|_{0,0} 
\] 
we set in addition all momenta to 0 afterwards, and 
\eq
\pa^w  \de^n_{\Phi} F|_{0,0,l} 
\eqe
is the $l$-th order coefficient in the loop expansion of the previous
expression. We now state\\[.1cm]
{\it Lemma 1:}  Under the assumption (\ref{rest})  
we have:\\
If for given $l,\, n,\, w$ and for all $l',\, n',\, w'$ with 
$l' <l$ and   $(n', \, w')\subseteq (n,\, w)$ or with 
$l' = l$ and  $(n', \, w')\subset (n,\, w)$
we have $\pa ^{w'}\de ^{n'}_{\uP} 
\Ga_1|_{0,0,l'}\,=\,0 $,
then\footnote{We use the set theoretic relations for the multiindices
$n,\,w$ though strictly speaking they are sequences.
The symbol $\subset$ means by definition strict inclusion.}
\eq
\pa ^w\de ^n_{\uP} \Ga_1|_{0,0,l}\,=\,0 \,\Longleftrightarrow \,
\pa ^w\de ^n_{\Phi} L_1|_{0,0,l}\,=\,0 \,.
\label{cond5}
\eqe
{\it Proof:} (\ref{cond5}) follows from (\ref{ga1l1},
\ref{vatra}) on noting that  all  propagators 
are finite and nonvanishing and that all
possible factorizations appearing when we 
apply the chain rule in going from  the $L$- to the $\Ga$-functions 
or vice versa vanish due to the conditions on the lower dimension
terms. \qed
Lemma 1 suggests that we satisfy the STI for the relevant
terms proceeding upwards in the number of fields and momentum
derivatives.
The subsequent comparison of the VSTI for $L$ (\ref{vicag}) and  $L^0$
(\ref{4.2.8}) shows that we also should proceed upwards in the number
of loops. Before proceeding to the termwise comparison 
it is instructive to note quite generally that from 
\eq
L_1\,=\,{d \over d\chi}L_{\chi}|_{\chi=0}
\ \mbox{ and }\  
L^0_1\,=\,{d \over d\chi}L^0_{\chi}|_{\chi=0}
\label{ll0}
\eqe
it follows similarly  as in (\ref{4.2.7}) that
\eq
L_1\,e ^{-{1 \over \hbar} (L+I)}
\,=\,e ^{\hbar \De} L_1^0\,e ^{-{1 \over \hbar} L^0}\,,
\label{ll1}
\eqe
and from the (perturbative) invertibility of $e ^{\hbar \De}$ we then obtain
the relation
\eq
L_1\,=\,0 \Longleftrightarrow L_1^0\,=\,0\ .
\label{l0l1}
\eqe
Our goal is to arrange for renormalization conditions
such that the relevant terms in $\Ga_1$ vanish, proceeding
inductively in the number of loops $l$. 
These relevant terms are listed  in App.C, $(I-XXIX)$.
By the statements from Ch.2 and 4.2 and App.C  there are  no nonvanishing 
relevant terms in $\Ga_1$ and $L_1^0$ at the tree level
 in the limit $\Lao \to \infty$  
(this limit remains formal before we have stated how to
 renormalize the theory in agreement with the STI). 
Since in the relevant part of the VSTI there are contributions 
stemming from $\dsi$ (\ref{dsi}) for
finite $\Lao$ which 
might conspire to give finite contributions in the VSTI 
when combining with divergent terms,    
our strategy is to compensate for them by introducing 
irrelevant terms in the bare action $L^0$. 
In this respect it is important to note that the termwise identities (I-XXIX) 
take the same form for $\Ga$ and  $L^0$ apart from the crucial fact that\\
i)  $L^0$ contains only those irrelevant terms 
we are going to introduce explicitly,\\
and from the fact that\\
ii)  there appear additional terms in  (\ref{4.2.8}) as compared 
to (\ref{gasti})
which just replace those 0-loop terms, excluded in $L^0$
by its definition\footnote{these terms contribute only when 
performing up to three field derivatives.}, so that as 
a consequence the termwise identities look as
before (when ignoring the irrelevant terms).\\
We will shortly denote the relevant terms in $L^0$  
by adding a sub- or superscript 0 to the corresponding term appearing in 
$\Ga$.  In the same way we denote ($I -XXIX$) written for $L^0$
as ($I^0-XXIX ^0$). In a number of STI the irrelevant terms 
introduced in $L^0$ below (\ref{irr}, \ref{wert}) 
will make appearance, namely in $III,\, V,\, VII,\, VIII$.
For those terms the STI for $L^0$  are rewritten explicitly 
in App.C including these terms.
We use similar notation as in App.A and App.C, in particular the shorthand
\eq
\dsi~:=\,\dot{\si}_{0,\Lao}(0)~:=\, {d \si_{0,\Lao}( p^2) \over d
  p^2}|_{p^2=0}~=\,- {\al m^4+(1+\al)m^2M^2 \over \Lao^6}
\label{dsi}
\eqe
 and add the following contribution\footnote{We remark that when
 working with a regulator as in footnote 7, we could spare the detour 
(\ref{irr},\ref{wert}), because then $\dsi$ would be zero.} 
to $L^0$
\eq
L^0_{irr}\,=\,
\int_p\int_q  \bigg\{
 \epsilon^{rst} A^r_{\mu}(p) A^s_{\nu}(q) B^t(-p-q) \,
[\,\de_{\mu\nu}\,(p^2-q^2)\, i_{10}^{AAB}\,+\, 
(p_{\mu}p_{\nu}-q_{\mu}q_{\nu}) i^{AAB}_{20}\,]
\label{irr}
\eqe
\[ 
 + \,\bar{c}^r(p) c^r(q) h(-p-q) \,[\,p^2 \,i_{10}^{\bar{c}ch}
\,+\, q^2 \,i_{20}^{\bar{c}ch}\,+\,pq\,i_{30}^{\bar{c}ch}\,]
\]
\[
 +\, \epsilon^{rst} \bar{c}^r(p) c^s(q) B^t(-p-q) 
\,[\,p^2 \,i_{10}^{\bar{c}cB}
\,+\, q^2 \,i_{20}^{\bar{c}cB}\,+\,pq\,i_{30}^{\bar{c}cB}\,]\,\,.
\]
We have presented $L^0$ directly in momentum space, where we perform
the analysis of the STI. The letter $i$ was chosen to remind of
'irrelevant', and we listed all terms of the respective field content 
allowed by the global
symmetries, which are of second order in the momenta.
The constants $i^{\,.\,.\,.}$ will be chosen as follows:
\eq
2\,i_{10}^{AAB}m\, R_4^0\,=\, -\dsi\,\de m_0^2\, 
 g\, R_2^0 \,,\quad
i_{20}^{AAB}\,=\,0\,,
\label{wert}
\eqe
\[
m(i_{10}^{\bar{c}cB}\,-\,i_{30}^{\bar{c}cB}) \,=\,
-\dsi \,\bigl[ m\,F^{\bc cB}_0\,-\, \Si_0^{B\!B} 
\frac12 g\,R_6^0\bigr]\,,\quad
{m \over 2}\,i_{30}^{\bar{c}ch}\,=\,\dsi \, \Si_0^{hh} \, 
{g \over 2}\,R_3^0\,,
\]
\[ 
{m \over 2}(2i_{10}^{\bar{c}ch}\,-\,i_{30}^{\bar{c}ch})\,=\,
-\dsi \bigl[ m\,F_0^{\bc ch} \,+ \,\Si_0^{B\!B} \, {g \over 2}\,R_5^0\bigr]
\,,\quad
{m \over 2}(2i_{20}^{\bar{c}ch}\,-\,i_{30}^{\bar{c}ch})\,=\,-2\,\dsi
F^{BBh}_0\, m\,R_4^0\,,
\]
\[
 m\,R_4^0(2\,i_{10}^{\bar{c}cB}\,-\,i_{30}^{\bar{c}cB})=
\dsi \,\Si_0^{\bc c} \,
g\,R_7^0\,,\quad
i_{20}^{\bar{c}cB}=\,-\dsi F^{\bc cB}_0\,.
\]
These relations are written in terms of the linear combinations 
which appear in the respective STI and are needed to verify them.  
By the  general results of Ch.3 and Ch.4.2 the theory stays finite when adding
such "irrelevant" dimension 5 terms to the bare action, under the
condition that these terms can be bounded by 
$\Lao ^{-1} {\cal P}_1(log(\Lao/m))\,{\cal P}_2({|\vec{p}| \over \Lao})$. 
If the relevant terms appearing in (\ref{wert}) obey A1)(\ref{rand}),  
this bound is obvious from the fact that 
$\, \dsi = O(\Lao ^{-6})$. After this modification of the bare action
we may state our\\[.1cm]
{\bf Induction hypothesis}: For $l \ge 1$ and all $l'\le l-1$\\ 
i) we assume that the theory to order $l'$ has been renormalized 
according to A1)(\ref{rand}) and A2) for the $\Ga$ (or equivalently for
the $L$) functional.\\  
ii) Furthermore we assume 
\eq
 \pa ^w\de ^n_{\uP}  \Ga_{1}|_{0,0,l'}\,=\,0\,,\
\, \pa ^w\de ^n_{\Phi}  L^0_{1}|_{0,0,l'}\,=\,0\,,\,\,
\mbox{ for }\,(n,w)  \,\mbox{ with }\, |n|+|w| \le 5 \ .
\label{ll}
\eqe
iii) Finally we assume that 
\eq
|\pa ^w\de ^n_{\Phi}  L^{\Lao,\Lao}_{1}|_{0,l'}\,|\le\,O(\Lao ^{5-|n|-|w|})\,
{\cal P}_1(log{\Lao \over m})\,
{\cal P}_2(\frac{|\vec{p}|}{\Lao})\;,
\,\mbox{ for }\,(n,w)  \,\mbox{ with }\,|n|+|w| > 5\ .
\label{lliii}
\eqe
All these statements are fulfilled at the tree level by our
assumptions on the tree level action. 
The rest of this section is devoted to prove the\\[.1cm]
{\bf Theorem:} {\it The induction hypothesis holds at loop order $l$}.
\\[.1cm] 
{\it Proof}~: 
At loop order $l$  we first prove the crucial\\[.1cm]
{\it Lemma 2:} For given $\,(n,w)$ with $\,|n|+|w| \le 5$
under the assumptions (\ref{rest}, \ref{ll}) and if
\eq
\pa ^{w'}\de ^{n'}_{\Phi}  L_{1}|_{0,0,l}\,= \,0\,,\quad
\pa ^{w'}\de ^{n'}_{\Phi}  L^0_{1}|_{0,0,l}\,= \,0
\,\,\mbox{ for }\,(n',w') \subset (n,w) 
\,\mbox{ and }\,n' \subset n
\label{co3}
\eqe 
the following equality holds: 
\eq
\pa ^w\de ^n_{\Phi}  L_{1}|_{0,0,l}\,= \,
\pa ^w\de ^n_{\Phi}  L^0_{1}|_{0,0,l}\,.
\label{lleq}
\eqe 
{\it Proof:} Due to the induction assumption ii), Lemma 1 and
(\ref{co3}) we find
\eq
\biggl[(- \hbar) {d \over d\chi}\pa ^w\de ^n_{\Phi} 
\,e ^{-{1 \over \hbar} L_{\chi}}\biggr]|_{0,0,\chi=0,l}
\,=\,
\pa ^w\de ^n_{\Phi}\, L_1|_{0,0,l}
\label{ll2}
\eqe
noting that  factorized terms 
give vanishing contribution, since $|n|+|w| \le 5$.
On the other hand we also obtain (cf. (\ref{ll1}))
\eq
\biggl[\,(- \hbar ) {d \over d\chi}\pa ^w\de ^n_{\Phi} \,
e ^{-{1 \over \hbar} L_{\chi}}\biggr]|_{\Phi\equiv 0,\chi=0,\,l}
\,= \,
\biggl[\pa ^w\de ^n_{\Phi}\,( e ^{ \hbar \De}\, L^0_1\,e ^{- \hbar \De}\, 
e ^{-{1 \over \hbar} L}\,)\biggr]|_{\Phi\equiv 0,\,l}\,\,.
\label{com}
\eqe
Note that here we do not yet restrict to vanishing momenta $\vec{p}$,
but assume that the momenta of the fields appearing in the derivatives
to be called $p_1,\ldots p_{|n|}$ have been chosen nonexceptional
\footnote{i.e. no subsum vanishes}. Later we take $\vec{p} \to 0$
\footnote{We point out that (\ref{ll2})
should strictly speaking also be viewed as being obtained first
for nonexceptional $\vec{p}$, where
correction terms appear, which then smoothly tend to 0 for $\vec{p} \to 0$,
so that we need not pay attention to them.}. 
We may rewrite the term $e^{ \hbar \De}\, L^0_1\,e ^{- \hbar \De}\,$
as
\eq
e ^{ \hbar \De}\, L^0_1\,e ^{- \hbar \De}\,=\,L_1^0\,+\,
\sum_{\nu=1}^5 \,{\hbar^{\nu} \over \nu~!}\,[\De\,,\,L_1^0]^{\nu}\,\,,
\label{comm}
\eqe
with the definition
\eq
[\De\,,\,\cdot \,]^{\nu}~:= 
\underbrace{[\De\,,\,[\ldots\,[\De\,,\,\cdot\,]\ldots]]}_{\nu \,\,  times}\,\,.
\label{comd}
\eqe
In (\ref{comm}) we used that $L_1^0$ is of degree 5 in the fields.
We may then define 
\eq
P_1^0\,e ^{-{1 \over \hbar} L}\,=\,\bigl(\,L_1^0\,+\,
\sum_{\nu=1}^5 \,{\hbar^{\nu} \over \nu~!}\,[\De\,,\,L_1^0]^{\nu}
\,\bigr)\,e ^{-{1 \over \hbar} L}\,,
\label{P}
\eqe 
and recognize $P_1^0$ as given by the sum over the contributions
from the connected amputated diagrams containing \\
i) exactly one vertex from  $L_1^0$\\
ii) up to 5 vertices from $-L$, which are all directly linked
to the vertex from $L_1^0$ via a propagator from $\De$\\
iii) multiplied by the monomial in the fields produced by the derivatives
from $\De$ acting on the respective term in $(-L)$, multiplied by the
respective power of $\hbar$ and a combinatoric factor to be read from
(\ref{P}).\\
We now have to regard 
\eq
\pa ^w\de ^n_{\Phi} 
(P_1^0\,e ^{-{1 \over \hbar} L})|_ {\Phi\equiv 0,\,l}\,\,. 
\label{termp}
\eqe
After performing the field 
and momentum derivatives and 
after splitting off the global $\de(p_1+\ldots +p_{|n|})$-function 
we let all momenta go to 0  so that then
\eq
\pa ^w\de ^n_{\Phi} 
(P_1^0\,e ^{-{1 \over \hbar} L})|_{0,0,\,l}\,\,. 
\label{termp1}
\eqe
is given by\\
{\it the sum over all $l$-loop connected amputated diagrams
containing exactly one vertex from $L_1^0$, $|n|$ external lines 
of the kind specified in $\de ^n_{\Phi}$, up to 5 vertices from
$\,-L$ directly linked to the one from $L_1^0$ via a propagator,
and weighed with a combinatoric factor as above. 
The  functions are derived w.r.t.  external momenta as indicated 
through $\pa ^w$ and taken at 0 external momenta in the end.}\\[.1cm]
Note that the restriction on the momenta  avoids the production 
of disconnected terms by momentum conservation.
Now remembering  (\ref{rest}) and the fact that
$L$ does not contain 0-loop two point functions 
we can use the induction hypothesis (\ref{ll}) and (\ref{co3}) 
to conclude that all 
contributions to (\ref{termp1}) vanish apart from the term 
\eq
\pa ^w\de ^n_{\Phi} L_1^0|_{0,\,0,\,l}\,=\,
\pa ^w\de ^n_{\Phi} L_1^{\Lao,\Lao}|_{0,\,0,\,l}\,.
\label{term0}
\eqe
Any other contribution would require nonvanishing
$\pa ^{w'}\de ^{n'}_{\Phi} L_1^0|_{ 0,\,0,\,l'}$
with $l' <l$ or $(n',w') \subset (n,w)$ and $n' \subset n$. 
The term (\ref{term0})
 then equals $\pa ^w\de ^n_{\Phi} L_1|_{\Phi\equiv 0,\,0,\,l}$
by (\ref{ll2}, \ref{com}) and subsequent comments.           \qed
 
After these preparations we present the renormalization conditions at 
$l$-loop order, lower orders being already fixed by induction. 
This means we fix the 37 relevant terms of the theory and  the 
7 normalization parameters $R_i$  appearing in the BRS
transformation at order $\hbar^l\,$\footnote{We mostly leave out the index 
$l$ of the loop order for readibility in the rest of this subsection.}:\\
A) We fix $\ka=0$ (\ref{rest}), and we choose freely in $\Ga$ the 8 
terms\footnote{cf. App.A (\ref{usiulsi}) for the notation}
\eq  
\ulSi_{trans}\,, \ \ulSi_{long}\,,\ \uSi^{BB}, \ \uSi^{\bc c}, \ 
\Si^{AB}, \ F^{BBh}, \ R_2, \ R_3\,.
\label{A}
\eqe
This then fixes uniquely the corresponding terms in $L$.
In fact we could interchange $R_1$ with $\ulSi_{long}$,
$R_4$ with $\Si ^{AB}$, and/or $F^{AAA}$ with $R_2$  in (\ref{A}).
We made the previous choice since it simplifies the check of the STI.\\
{\it This means that we may choose freely all field normalizations
with the exception of $h$\footnote{remember that $h,\,B^a$ stem from
  the same complex scalar doublet (\ref{2.5})}, 
one global normalization for the BRS-transformations and the two
couplings through $F^{AAA}$ and $F^{BBh}$. Our simplifying assumption
$\ka=0$ (\ref{rest}) is related to the freedom in choosing 
the vacuum expectation value of the Higgs field.}\\
B) We fix in $\ti{L^0}$ the following relevant terms: 
\eq
R_6^0\,=\, R_7^0\,=\,R_2^0\,,\quad 
 R_5^0\,=\, {(R_2^0)^2 \over R_3^0}.
\label{B}
\eqe
This means that  $R_6 ^0,\,R_7^0$  are fixed to equal  $R_2 ^0$,
which in turn is uniquely given at $l$ loop order by our free choice 
of $R_2$, and by lower loop order constants fixed before. 
Similarly $R_5 ^0$ is fixed through $R_3^0$  and  $R_2 ^0$ at $l$ loop order. 
Remember again that, by the FE for 1PI functions, an $l$-loop contribution
depends only on lower loop order terms and the $l$-loop boundary
condition for the term in question.\\
C) All those $r_0$-terms in $L^0$ having no tree correspondence 
are chosen equal to zero (11 terms to be read from App.A),  i.e.  
\eq
r_{20}^{hBA},\,\ldots,\, r_0^{\bc c \bc c}\,=\,0\,. 
\label{C}
\eqe
D) Furthermore we fix in $L_0$ the following relevant terms
\eq
 F_{0}^{BBA}\,=\,-\, {R_3^0 \over 2 R_2^0}\,F_{10}^{hBA}\,,\,\,
F_{0}^{\bc c B}\,=\,-\,{R_3^0 \over R_2^0}\, F_{0}^{\bc c h}\,,\,\,
F_{0}^{AAhh}\,=\,{R_5^0 \over R_3^0} \, F_{10}^{AABB}\,\,.
\label{D}
\eqe
More precisely (\ref{D}) should be understood as follows:
The $F$-terms  on the rhs in (\ref{D}) 
will be uniquely fixed as functions of (a subset of) the ones fixed previously
in A)-C). Then we fix each of the three terms on the lhs as
a function of those on the rhs.

Finally also the remaining 18 relevant constants will be uniquely 
fixed as functions of the previous ones in our sweep through the STI.
Since 17 relevant terms (those from B)-D)) are fixed on the wrong
side, namely in $L^0$, 
one may wonder, how we will get a finite
theory in the end. The tool to achieve this will in fact again be the
STI, once we know they are satisfied. This is not unexpected from the
traditional use made of the STI in renormalization proofs.\\
Now we first satisfy a subset of the  STI (I-VIII) containing 
only up to three field derivatives by choosing appropriately
\eq
\de m^{2}[\Si ^{\bc c},R_1]\,(I_a ),\ \Si ^{BB}[\Si ^{\bc c},R_4]\,(II_a )\,, \
R_1\,(I_b )\,,\  R_4\,(II_b )\,, \ 
F^{AAA}[\dsi \,\de m^2]\,(III_b )\,,
\label{sweep1}
\eqe
\[
F_1^{hBA}[R_5,\,\dsi (F^{\bc ch},\,\Si^{BB})]\,(VII_c )\,,\
F^{AAh}[F_1^{hBA}](VI_b)\,,\
F^{\bc cA}_1[F^{BBA},r_2^{\bc c A}]\,(IV_b )\,,
\]
\[
F^{\bc c h}[F^{AAh},F_1^{hBA},r_2^{hBA}]\,(VI_a )\,,\
\Si ^{\bc c}[R_7,F^{\bc cB}]\,(VIII_a )\,,\
\Si ^{hh}[R_5,\Si ^{BB},F^{\bc c h}] \,(VII_a )\,,
\]
\[
\dot{\Si}^{hh}[F_1^{hBA},\dsi \,\Si ^{hh}]
\,(VII_b )\ .
\]
We wrote in brackets the STI which is satisfied by the respective
choice of a renormalization constant and in square brackets 
the other relevant constants at {\it loop order ${\it l}$}, on which 
this choice depends. 
In the square brackets we omitted the terms from (\ref{A}), 
which are freely chosen, 
and  $R_1$ and $R_4$, which by (\ref{sweep1}) depend on such terms
only. Note however that $I_b$ and $II_b$ cannot  be solved for  
$R_1$ and $R_4$ depending only on such terms, before we know that   
$I_a$ and $II_a$ hold. Therefore we indicated the dependence on 
$R_1$ and $R_4$ in the first two terms.
At this stage $R_1$ and $R_4$ can already be seen to be
finite. All other terms, depending on
constants fixed on the wrong side,  might diverge with $\Lao$. We come back 
to the finiteness problem later and
first convince ourselves that the system (\ref{sweep1}) is
consistent, i.e. solvable. This is a problem only, if a term is present
before and within  square brackets at the same time, when we
successively replace each term within  square brackets
by those on which it depends at $l$ 
loop.\footnote{E.g. at $l$ loop $R_6$  depends on 
$R_2$ only by (\ref{B}), whereas  
$F^{BBA}$ depends on $R_3, \, R_2,\,F_{1}^{hBA}\,$ by (\ref{D}).}
Checking the list we find that this happens only for 
$F^{\bc c h}$, which, when substituting $F_1^{hBA}$ from (\ref{sweep1}),
depends on itself. Solving for $F^{\bc c h}$
it appears with a coefficient $1/\al\,+\,\dsi\,m^2(R_4/R_1)$.
Since we know that $R_1,\,R_4$ are finite, this coefficient does not vanish
for $\Lao$ large.\footnote{$\al$ is supposed to be finite, 
but $\al \to \infty$ may be taken after $\Lao \to \infty$.}
As a result we may replace  $F^{\bc c h}[F_1^{hBA},r_2^{hBA}]$ 
 by 
\eq
F^{\bc c h}[r_2^{hBA}]\,.
\label{fbcc}
\eqe
After this change one rapidly realizes the solvability of (\ref{sweep1}).

Now we impose renormalization conditions for the remaining 6 relevant 
terms  by satisfying the following
relations among ($I-XXIX$) for $\Ga$. 
\eq
F^{BBBB}\,(X)\,,\,\,  F^{BBhh}\,(XX)\,,\,\,
F^{hhhh}\, (XIX)\,,\,\,  F^{hhh}\, (IX)\,,\,\,
F_1^{AABB}\,(XIII_2)\,,\,\, F_1^{AAAA}\,(XIV_c)\,\,. 
\label{renco}
\eqe
The order is important for the first four terms, for the last two it
is arbitrary. 
Again we wrote in parentheses the relation which is satisfied by
and which fixes the respective renormalization condition.

At this stage the $37+7$ relevant parameters are completely fixed. 
All the remaining relations among the STI will now be verified for $L^0$.
Since there are no dimension 3 terms left, we start with the dimension
4 terms which have not yet been
verified. $IV_a^0$ is the only relation left among 
$(I ^0-VIII ^0$):
Using (\ref{C}) it takes the form  
\eq
2m\,R_4^0 \,F_0^{BBA}+
g/2 \, R_6^0 \,\Si_0^{AB} + 1/\al\,F_0^{\bc c B}=0 \,.
\label{4a}
\eqe
From (\ref{sweep1}) we know  $VI_a$ and $VI_b$ to be true.
Lemma 2 then implies $VI ^0_a$ and $VI ^0_b$ to be true
as well. These relations together with
(\ref{B}, \ref{C}, \ref{D}) then allow to verify (\ref{4a}).
Now 
\eq  
XI^0\,,\,\,XII^0\,,\,\,XIII^0_1
\label{di4}
\eqe
 are the last relations of dimension $\le 4$ to be analysed. 
They follow directly from (\ref{B}, \ref{C}, \ref{D}).
By Lemma 2 we pass from $L_1^0$ to $L_1$
for $XI,\,XII,\,XIII_1$.\\ 
{\it Therefore Lemma 1 now tells us  
that all terms in $\Ga_1$ of
dimension $\le 5$ vanish iff they vanish in $L_1$, and  Lemma 2 tells us  
that all terms in $L^0_1$ of
dimension $\le 5$ vanish iff they vanish in $L_1$.}

Among the  relations containing 
4 or more field derivatives the following ones
\eq
XIV^0_a\,,\,\, XIV^0_b\,,\,\, XIV^0_d\,,\,\, XIV^0_e\,,\,\,
XV^0_{1a}\,,\,\,XV^0_{1b}\,,\,\,
XV^0_{2a}\,,\,\,XV^0_{2b}\,,\,\,XV^0_{2c}\,,\,\,
XVI^0_{a}\,,\,\,XVI^0_{b}\,,
\label{rel}
\eqe
\[
XVI^0_{c}\,,\,\,
XVII^0_{a}\,,\,\,XVII^0_{b}\,,\,\,
XVIII^0_{a}\,,\,\,XVIII^0_{b}\,,\,\,XVIII^0_{c}\,,\,\,
XXI^0\,,\,\,XXII^0-XXIX^0\,\,
\]
remain to be verified. Only those written in (\ref{two}) are not immediately 
obvious from (\ref{B}, \ref{C}, \ref{D}). They can be verified
using the relations we wrote in parentheses
\eq
XV_{1a}^0\, (\ref{B},\, VI_b^0,\, XIII_2^0)\,,\,\,
XVII_{a}^0\, (\ref{D},\, XIII_2^0,\, VI_b^0)\,.
\label{two}
\eqe
We have not yet checked the following five relations 
of dimension 5 involving three fields only~: 
$\,III_a,\ V,\ VII_d,\   VIII_b,\ VIII_c\,,$
which are the most delicate ones. They contain terms multiplied by
 $\dsi$. For $III_a$ this is true when inserting $F^{AAA}$ 
from  $III_b$. We first forget about these 
corrections  and at the same time about the modification of 
$L_0$ by (\ref{irr}, \ref{wert}) 
and convince ourselves that the 5 STI
are fulfilled in this case.
To do so we use the following relations
\eq
I_b^0,\ II_b^0,\ IV_b^0,\   VII_c^0\,.
\label{inp}
\eqe
First we can verify $V^0$  using   
(\ref{D}, \ref{B}) and $VII_c^0$. Next we regard $III_a ^0$
and realize that it  amounts to show that
\eq
g\,R_2^0(1+\ulSi_{{\rm long},0})\,=\, F^{\bc cA}_{1,0}\,.
\label{3a}
\eqe
The lhs equals $\,g\,(R_2^0/R_1^0)(1+\uSi^{\bc c})$ by $I ^0_b\,$,
the rhs equals 
$\,-g/m\,R_2^0\,\Si_0^{AB}-\,4\,R_4^0\, F_0^{BBA}\,$ by $IV ^0_b$. 
Using $II^0_b$ for the first and $V^0$ for the second term we recognize
now that (\ref{3a}) holds.
$VII ^0_d$ follows directly from (\ref{B}), and $VIII ^0_b$ 
follows similarly as $III ^0_a$ from  $IV^0_b$, $II ^0_b$ and $V^0,$  
$VIII ^0_c$ follows from (\ref{B}). 
Now we also take into account the correction terms~:
Those relations  among ($I^0-XXIX^0$)
 which are affected by $L^0_{irr}$ (\ref{irr}) are listed explicitly 
in App.C. On inspection one realizes that the choice (\ref{wert})
exactly cancels all terms $\sim \dsi$ in ($I^0-XXIX^0$).
 
Thus all STI are fulfilled in our theory, and item ii) of the induction 
hypothesis is satisfied to loop order $l$. What remains to show
is that the theory defined up to $l$-loop order 
is finite for $\Lao \to \infty$.
As we noted, apart from the  9 evidently
finite  constants in (\ref{A}), finiteness of    
$R_1$ and $R_4$ can be inferred from (\ref{sweep1}).
To proceed further it is important to note that all irrelevant
terms appearing in the STI apart from those in  (\ref{irr},
\ref{wert}) are a priori finite at $l$-loop since they only depend on
the renormalization conditions at order $l'\le l-1$. 
The next step is then to convince oneself of the fact that
$(F^{AAA})^{0,\Lao}$ has a finite limit for $\Lao \to \infty$.
As we see from $III_b$ the finiteness of  $F^{AAA}$ follows, if
we can show that $\dsi\,\de m^2 $ has a finite limit.
From (\ref{B}, \ref{C}, \ref{D}) it is evident that all relevant
parameters fixed on the wrong side (at $\La=\Lao$) satisfy 
the bound assumed in the Corollary from Ch.3.2.
From this Corollary (adapted to the $\Ga$-functional)
and the induction hypothesis we therefore 
conclude that $\de m^2$ is bounded by $\Lao ^2 \,{\cal P}(log{\Lao
  \over m})$, whereas $\dsi\,\sim \Lao ^{-6}$. This proves the
finiteness of $F^{AAA}$.\footnote{ Using the STI we may in fact show
at this stage that $\de m^2$ diverges at most logarithmically.}
Then we  go through the STI as follows:
\eq
F_1^{AAAA}\,(XIV_c),\,\,
r_1^{AA\bc c}\,(XIV_b),\,\,
r_2^{AAAA}\,(XIV_a),\,\,
r_2^{AA\bc c}\,(XIV_e),\,\,
r_1^{BB\bc c}\,(XV_{1b}),
\label{sweep2}
\eqe
\[
r^{\bc c\bc c}\,(XVIII_b)\,,\,\,
r_2^{\bc c A}\,(XVIII_c)\,,\,\,
r_2^{AABB}\,(XXII)\,.
\]
In parentheses we wrote the STI from which the finiteness of the
respective relevant term may be inferred. In 
$r^{\bc c\bc  c}\,(XVIII_b)$ note that $XVIII_b$
does not depend on $r_2^{\bc c A}$ at $l$-loop order. We now infer from
$VII_d$ that
\eq 
r_2^{hBA}\,=\,2m \dsi \,\bigl[ F_0^{BBh}\,{R^0_4 
\over R_1}\bigr ]|_l\,+\,finite \,\,
\mbox{ has a finite limit for }\,\, \Lao \to \infty\,.
\label{imp}
\eqe
Here the first contribution stems from the irrelevant term 
${m \over 2}(2i_{20}^{\bar{c}ch}\,-\,i_{30}^{\bar{c}ch})
=\,-2m\,\dsi \,F^{BBh}_0\, R_4^0$
in (\ref{wert}). In $VII_d$ this contribution appears among the
irrelevant terms and originates from the b.c. at $\La=\Lao$. Note
that $F^{BBh}_0,\, R_4^0$ diverge at most linearly with $\Lao$ using the
results from Ch.3.2 and Ch.4.2.
Disposing of the finiteness of $R_2$ and  $r_2^{hBA}\,$
finiteness follows now also for\\
$
R_6 \,(XVI_a),\ R_7 \,(XVIII_a),\  
r^{hh\bc c}\,(XVII_b),\,\,
r^{hB\bc c}\,(XXVIII),\
r_2^{BB\bc c}\,(XXVII),\
F^{BBBB}\,(X)\,.
$\\
Similarly as in (\ref{imp}) we may now conclude from $V$ 
the finiteness of $\,F^{BBA}\,(V)\,$.
Next we pass through the following finiteness chain\\
$F_1^{\bc c A}(IV_{b}),\,\,
F_1^{AABB}(XV_{1a}),\,\,
F^{AAh}(XIII_2),$
$F_1^{hBA}(VI_{b}),\,\,
F^{\bc cB}(IV_a),\,\,
\Si^{\bc c}(VIII_a),\,\,
\Si^{BB}(II_a)$,\\ $\de m^2(I_a),\ $
and then we can establish finiteness of 
$\,F^{\bc ch}\,(VI_a)\,,\,\,
R_5 \,(VII_c \mbox{ or } \,XVI_b)\,$.\\
Finally it is easy to convince oneself of the finiteness of the remaining
constants\\
$\Si^{hh}\,(VII_a)\,,\,\,
\uSi^{hh}\,(VII_b)\,,\,\,
F^{BBhh}\,(XX)\,,\,\,
F^{hhh}\,(IX)\,,\,\,
F^{hhhh}\,(XIX)\,,\,\,
F^{AAhh}\,(XVII_a)\,.
$\\
In regarding the previous series of finiteness statements it 
is interesting to note that it is first extracted for
 the pure gauge sector and last 
for the terms involving the $h$ field.\footnote{This is reminiscent of
the fact that the radiative corrections in the scalar boson sector are
more rapidly  divergent, namely quadratically, than all other ones.}
By now   all of the 44 relevant
constants are known to be finite, and thus item i) of the 
induction hypo\-thesis 
to loop order $l$ is verified.\footnote{The smoothness assumption 
directly follows from the smoothness of the
regulator and from the b.c. which depend on $\Lao$ only through the 
regulator.}  
Once i) ii) are verified, item iii) immediately 
follows from the general
bounds in Ch.3.2 on noting that\\
a) from our choice of the bare action it is evident that 
$\,\pa ^w\de ^n_{\Phi}  L^0_{1}|_{0,0,l}\,=\,0\,,\,\, |n| >5\,,$\\ 
b) the irrelevant terms in $L_1^0$ generated from those introduced 
in (\ref{irr}) on BRS transformation obey the required bound
as a consequence of the previous finiteness statements\\
c) all other irrelevant terms in $L_1^0$ are 
generated by momentum derivatives acting on the regulating
factor $\si_{0,\Lao}(p)$, which automatically produces 
(more than) the required negative powers of $\Lao$.\\[.1cm]
So the induction hypothesis holds to $l$-loop order. 
This ends the proof of the Theorem. \qed  
Once the Theorem is proven, Proposition 4 tells us that the STI hold
in the limit $\Lao \to \infty$.\\[.5cm]
{\bf Concluding Remarks}\\
We have presented a renormalization proof for spontaneously 
broken Yang-Mills theory based on the Wilson
renormalization group. 
The renormalization conditions admissible in view of the STI 
could be stated explicitly in (\ref{A}) to
(\ref{renco}).\footnote{Using 
in particular (\ref{C}) it should be possible 
to derive the antighost equation of motion often used in textbooks
[FaSl], [ZiJ].} 
We tried to avoid any equivocality as regards the
analytical status of the statements we made, in particular for
which  values of the cutoffs they hold. We did not make use of
unregularized path integrals. We think that the analytical aspect
is generally somewhat neglected in the recent literature including
textbooks.  We did not attempt at generality on the symmetry
or group theoretical aspects, which have been studied extensively 
in the literature, and restricted for simplicity to the
physically interesting SU(2) case. 
We think it would be worth-while to extend the work - with the same
precision on the analytical status - to the physical consequences   
to be drawn from the STI, in particular the gauge invariance of the 
$S$-Matrix. Further interesting problems to be treated 
in this context are the renormalization of QCD and the analysis 
of anomaly problems and of the action principle.  



\subsection*{Appendix A}
Here we consider the generating functional for the proper vertex functions
\[ \Gamma(\uA,\uh,\uB,\ubc,\uc) = \sum^4_{n=1} \Gamma_n + \Gamma_{(n > 4)}, \]
$n$ counting the number of fields, and extract its relevant part, i.e. its 
local field content with mass dimension not greater than four. 
Generally we will not underline the field variable symbols in the
Appendices, though of course all $\Ga$ functional arguments should be
understood as such. In App.A and App.B the regulators are not explicited, 
apart from the subsequent comments on the two-point functions,  
where contributions arising for finite $\Lao$ are explicited. 
\vspace{0.2cm}
\noindent
1) One-point function:
\[ \Gamma_1 = \kappa \hat{h}(0). \]
2) Two-point functions: 
\[
\Gamma_2 = \int_p \Bigg\{ \frac12 A^a_{\mu}(p) A^a_{\nu}(-p) 
\Gamma^{AA}_{\mu\nu}(p) +
\frac12 h(p) h(-p) \Gamma^{hh}(p) 
 + \frac12 B^a(p)B^a(-p) \Gamma^{BB}(p) 
\]
\[
- \bar{c}^a(p) c^a(-p) 
\Gamma^{\bar{c}c}(p)+ A^a_{\mu}(p) B^a(-p) \Gamma^{AB}_{\mu}(p) \Bigg\}, 
\]
\[
\Gamma^{AA}_{\mu\nu}(p) = \delta_{\mu\nu}(m^2+\delta m^2) + 
(p^2\delta_{\mu\nu}-p_{\mu}p
_{\nu}) (1 + \Sigma_{\rm trans} (p^2))
 + \frac{1}{\alpha} p_{\mu}p_{\nu} (1 + \Sigma_{\rm long} (p^2))\,,
\]
\[
\Gamma^{hh}(p) = p^2 + M^2 + \Sigma^{hh}(p^2)\,,\quad
\Gamma^{BB}(p) = p^2 + \alpha m^2 + \Sigma^{BB}(p^2)\,, 
\]
\[
\Gamma^{\bar{c}c}(p) = p^2 + \alpha m^2 + \Sigma^{\bar{c}c}(p^2)\,,\quad
\Gamma^{AB}_{\mu}(p) = ip_{\mu} \Sigma^{AB}(p^2)\,.   
\]
Besides the  unregularized tree order  
there emerge 10 relevant 
para\-meters from the various self 
energies: $\delta m^2, \Sigma_{\rm trans}(0), \Sigma_{\rm long}(0), 
\Sigma^{hh}(0), 
\dot{\Sigma}^{hh}(0), \Sigma^{BB}(0), \dot{\Sigma}^{BB}(0),
\Sigma^{\bar{c}c}(0)$,  $\dot{\Sigma}^{\bar{c}c}(0)$ and
$\Sigma^{AB}(0)$, where we used the 
notation $\dot{\sum}(0) \equiv (\partial_{ p^2} \sum)(0)$.\\
By (\ref{lege}, \ref{relfel}, \ref{vatra}) the 0-loop-order
functional $\Ga^{0,\Lao}_{2,l=0}$ carries the inverted regulating
factor $\,(\si_{0,\Lao})^{-1}\!(p^2)=1-\dsi\,p^2+{\cal{O}}((p^2)^2)\,$
 with $\dsi \,=\,- (\al m^4+(1+\al)m^2M^2)/ \Lao^6\,$.
Therefore all self energies vanish at order $l=0\,$, whereas
\eq
\dot{\Sigma}^{hh}_{l=0}(0)\,=\,  -\dsi  \,M^2\,,\quad
\dot{\Sigma}^{BB}_{l=0}(0)\,=\,
\dot{\Sigma}^{\bc c}_{l=0}(0)\,=\, -\dsi\, \al\,\,m^2\,,
\eqe
\[\Sigma_{trans}|_{l=0}(0)\,=\, -\dsi  \,m^2\,,\quad
\Sigma_{long}|_{l=0}(0)\,=\, -\dsi \,\al\, \,m^2\,.
\]
To clearly isolate the tree level cutoff effects from the loop
contributions we introduce the notation
\eq
\ulSi(0)\,=\,\Si(0)\,-\,\Si(0)|_{l=0}\,,\quad
\uSi(0)\,=\,\dot{\Si}(0)\,-\,\dot{\Si}(0)|_{l=0}\,. 
\label{usiulsi}
\eqe
3) Three-point functions:\\ 
Only the relevant part is given explicitly: $r= {\cal O}(\hbar)$ denotes 
a relevant parameter
which vanishes in the tree order, otherwise a relevant parameter is denoted 
by $F$. Moreover, we
indicate an irrelevant part by a symbol ${\cal O}_n, \; n \in  \bbbn$, 
indicating that this
part vanishes as an $n$-th power of the momentum in the limit when all 
momenta tend to zero
homogeneously.
\begin{eqnarray*}
\Gamma_3 &=& \int_p\int_q \Bigg\{ \epsilon^{rst} A^r_{\mu}(p) A^s_{\nu}(q) 
A^t_{\lambda}(-p-q)
\Gamma^{AAA}_{\mu\nu\lambda}(p,q) \\
& & + A^r_{\mu}(p) A^r_{\nu}(q) h(-p-q) \Gamma^{AAh}_{\mu\nu}(p,q) 
+ \epsilon^{rst} B^r(p) B^s(q) A^t_{\mu}(-p-q) \Gamma^{BBA}_{\mu}(p,q) \\[0.3cm]
& & + h(p) B^r(q) A^r_{\mu}(-p-q) \Gamma^{hBA}_{\mu}(p,q) 
 + \epsilon^{rst} \bar{c}^r(p) c^s(q) A^t_{\mu}(-p-q) 
\Gamma^{\bar{c}cA}_{\mu}(p,q) \\[0.3cm]
& & + B^r(p) B^r(q) h(-p-q) \Gamma^{BBh}(p,q) 
 + h(p) h(q) h(-p-q) \Gamma^{hhh}(p,q) \\[0.2cm]
& & + \bar{c}^r(p) c^r(q) h(-p-q) \Gamma^{\bar{c}ch}(p,q) 
 + \epsilon^{rst} \bar{c}^r(p) c^s(q) B^t(-p-q) \Gamma^{\bar{c}cB}(p,q) 
\Bigg \},
\end{eqnarray*}
\begin{eqnarray*} \begin{array}{llllll}
\Gamma^{AAA}_{\mu\nu\lambda}(p,q) &=& \delta_{\mu\nu} i(p-q)_{\lambda} 
F^{AAA} + {\cal O}_3, \quad &
F^{AAA} &=& - \frac12 g + r^{AAA}, \\[0.2cm]
\Gamma^{AAh}_{\mu\nu}(p,q) &=& \delta_{\mu\nu} F^{AAh} + {\cal O}_2, &
F^{AAh} &=&  \frac12 mg + r^{AAh}, \\[0.2cm]
\Gamma^{BBA}_{\mu}(p,q) &=& i(p-q)_{\mu} F^{BBA} + {\cal O}_3, &
F^{BBA} &=&  - \frac14 g + r^{BBA}, \\[0.2cm]
\Gamma^{hBA}_{\mu}(p,q) &=& i(p-q)_{\mu} F_1^{hBA} &
F_1^{hBA} &=&  \frac12 g + r_1^{hBA}, \\
& & + i(p+q)_{\mu} r^{hBA}_2 + {\cal O}_3,\\[0.2cm]
\Gamma^{\bar{c}cA}_{\mu}(p,q) &=& ip_{\mu} F_1^{\bar{c}cA} + iq_\mu 
r^{\bar{c}cA}_2 
+ {\cal O}_3, &
F_1^{\bar{c}cA} &=& g + r_1^{\bar{c}cA}, \\[0.2cm]
\Gamma^{BBh}(p,q) &=& F^{BBh} + {\cal O}_2, &
F^{BBh} &=&  \frac14 g \frac{M^2}{m} + r^{BBh}, \\[0.2cm]
\Gamma^{hhh}(p,q) &=& F^{hhh} + {\cal O}_2, &
F^{hhh} &=&  \frac14 g \frac{M^2}{m} + r^{hhh}, \\[0.2cm]
\Gamma^{\bar{c}ch}(p,q) &=& F^{\bar{c}ch} + {\cal O}_2, &
F^{\bar{c}ch} &=&  - \frac12 \alpha gm + r^{\bar{c}ch}, \\[0.2cm]
\Gamma^{\bar{c}cB}(p,q) &=& F^{\bar{c}cB} + {\cal O}_2, &
F^{\bar{c}cB} &=&  \frac12 \alpha gm + r^{\bar{c}cB}. 
\end{array}
\end{eqnarray*}
The 3-point functions $AAB$ and $BBB$ have no relevant local content.\\[.3cm]
4) Four-point functions: 
With parameters $r$ and $F$ defined as before
\begin{eqnarray*}
\Gamma_4|_{\rm rel} &=& \int_k \int_p \int_q \bigg\{ \epsilon^{abc} 
\epsilon^{ars} A^b_{\mu}(k)
A^c_\nu(p)A^r_\mu(q) A^s_\nu(-k-p-q) F_1^{AAAA} \\[0.2cm]
& & + A^r_\mu(k) A^r_\mu(p) A^s_\nu(q) A^s_\nu(-k-p-q) r_2^{AAAA} \\[0.2cm]
& & + A^a_\mu(k)A^b_\mu(p) \bar{c}^r(q) c^s(-k-p-q) 
(\delta^{ab}\delta^{rs}r_1^{AA\bar{c}c}
+ \delta^{ar}\delta^{bs}r_2^{AA\bar{c}c}) \\[0.2cm]
& & + A^a_\mu(k)A^b_\mu(p) B^r(q) B^s(-k-p-q) (\delta^{ab}\delta^{rs}F_1^{AABB}
+ \delta^{ar}\delta^{bs}r_2^{AABB}) \\[0.2cm]
& & + B^a(k)B^b(p) \bar{c}^r(q) c^s(-k-p-q) 
(\delta^{ab}\delta^{rs}r_1^{BB\bar{c}c}
+ \delta^{ar}\delta^{bs}r_2^{BB\bar{c}c}) \\[0.2cm]
& & + h(k)h(p)h(q)h(-k-p-q) F^{hhhh} \\[0.2cm]
& & + B^r(k)B^r(p)h(q)h(-k-p-q) F^{BBhh} \\[0.2cm]
& & + B^r(k)B^r(p)B^s(q)B^s(-k-p-q) F^{BBBB} \\[0.2cm]
& & + A^r_\mu(k)A^r_\mu(p)h(q)h(-k-p-q) F^{AAhh} \\[0.2cm]
& & + h(k)h(p)\bar{c}^r(q)c^r(-k-p-q) r^{hh\bar{c}c} \\[0.2cm]
& & + \bar{c}^a(k)c^a(p)\bar{c}^r(q)c^r(-k-p-q) r^{\bar{c}c\bar{c}c} \\[0.2cm]
& & + \epsilon^{rst}h(k)B^r(p)\bar{c}^s(q)c^t(-k-p-q) r^{hB\bar{c}c} \bigg\},
\end{eqnarray*}
\begin{eqnarray*}\begin{array}{llllll}
F_1^{AAAA} &=& \frac14 g^2 + r_1^{AAAA}, \quad &
F_1^{AABB} &=& \frac18 g^2 + r_1^{AABB}, \\
F^{hhhh} &=& \frac{1}{32} g^2 \left( \frac Mm \right)^2 + r^{hhhh}, &
F^{BBhh} &=& \frac{1}{16} g^2 \left( \frac Mm \right)^2 + r^{BBhh}, \\
F^{BBBB} &=& \frac{1}{32} g^2 \left( \frac Mm \right)^2 + r^{BBBB}, &
F^{AAhh} &=& \frac18 g^2 + r^{AAhh}.
\end{array} 
\end{eqnarray*}
Hence, in total $\Gamma$ involves $1 + 10 + 11 + 15 = 37$ relevant parameters.








\subsection*{Appendix B}
We also have to consider the vertex functions with operator insertions 
stemming from the 
BRS-transforms. These insertions have mass dimension $\le 2$. \\
Only the respective relevant part of the four vertex functions with 
insertions is listed:
\begin{eqnarray*}
\Gamma_{\gamma^a_\mu(p)}|_{\rm rel} &=& -ip_\mu c^a(-p)R_1 + \epsilon^{arb} 
\int_q A^r_\mu(q)
c^b(-p-q) gR_2, \\
\Gamma_{\gamma(p)}|_{\rm rel} &=& \int_q B^r(q)c^r(-p-q) (- \frac12 gR_3), \\
\Gamma_{\gamma^a(p)}|_{\rm rel} &=& m c^a(-p)R_4 + \int_q h(q)c^a(-p-q) 
\frac12 gR_5 +
\epsilon^{arb} \int_q B^r(q) c^b(-p-q) \frac12 gR_6, \\
\Gamma_{\omega^a(p)}|_{\rm rel} &=& \epsilon^{ars} \int_q c^r(q) c^s(-p-q) 
\frac12 gR_7.
\end{eqnarray*} 
There appear 7 relevant parameters 
\[ R_i = 1 + r_i, \quad r_i = {\cal O}(\hbar), \quad i = 1,...,7. \]
All other 2-point functions, and the higher ones, of course, are of 
irrelevant type.




\subsection*{Appendix C}
Here we present the 53 conditions which result upon requiring that the
functional $\Gamma_1$, (\ref{ga1l1}), has a vanishing local part for
(mass) dimensions smaller or equal to five
\[ \Gamma_1(\uA,\uh,\uB,\ubc,\uc)|_{\dim \le 5} \begin{array}{c}
 ! \\ = \\ \\
\end{array} 0 .\]
Into most of these conditions also irrelevant contributions enter which
are not given explicitly but are simply indicated by "irr". To recognize
the local origin, we keep the momentum factors arising. The
$\delta$-distribution emerging from the functional derivatives and
forcing the sum of the corresponding momenta to zero is not written.
Relations explicitly rewritten for $L^0$ carry a zero
in the numbering. In those, the irrelevant terms from (\ref{irr},\ref{wert})
are the only ones appearing and 
are written explicitly. 

The STI for $\Ga$ are supposed to be written
for the case $\La=0$, $\Lao\le \infty$. Note that they take 
different form for $\Lao < \infty$ and $\Lao \to \infty$ only, if $\dsi$
appears, which is the case in $I_b,\,II_b,\,III_b,\,V,\,VII_b,\,VII_c,
\,VII_d,\,VIII_b,\,VIII_c\,$.
For the $L^0$-functional we write for the loop level two-point
functions $\Si_0$ instead of $\Si$ and $\ulSi_0$, $\uSi_0$ instead of 
$\ulSi$, $\uSi$.\\[0.2cm]
Two fields
\begin{enumerate}
\item[I)] $\; \delta_{A^a_\mu(q)} \delta_{c^r(k)} \Gamma_1|_0$
\begin{enumerate}
\item[a) $0 \, \stackrel {!} {= } \,$]$   q_\mu \left\{ - (m^2+\delta
m^2)R_1 + \sum^{AB}(0) m R_4 + m^2
+ \frac{1}{\alpha} \sum^{\bar{c}c}(0) \right\} $,
\item[b) $0  \, \stackrel {!} {= } \,$]$  q^2 q_\mu \Big\{ - {1\over
      \al}(1 + \ulSi_{\rm long} (0)) R_1 
+{1\over  \al} (1 + \uSi^{\bar{c}c}(0)) - 
\dsi [\de m^2 R_1- \sum^{AB}(0)m R_4-{1\over  \al}
\sum^{\bar{c}c}(0)] +{\rm irr} \Big\} $.
\end{enumerate}
\item[II)] $\; \delta_{B^a(q)} \delta_{c^r(k)} \Gamma_1|_0$
\begin{enumerate}
\item[a) $0  \, \stackrel {!} {= }$]$ m (\alpha m^2+\sum^{BB}(0))R_4
- m (\alpha m^2+ \sum^{\bar{c}c}(0)) + \kappa
(- \frac12 g)R_3$,
\item[b) $0   \stackrel {!} {=}$]$q^2  \Big\{ - \sum^{AB}(0) R_1
+ m(1 + \uSi^{BB}(0))R_4-m (1 + \uSi^{\bar{c}c}(0)) -\dsi
m[\Si^{\bar{c}c}(0)-\Si^{BB}(0)R_4]
 + {\rm irr} \Big\} $.
\end{enumerate}
\end{enumerate}
\vspace{0.2cm}
\noindent
Three fields
\begin{enumerate}
\item[III)] $\; \delta_{A^r_\mu(p)} \delta_{A^s_\nu(q)} \delta_{c^t(k)}
\Gamma_1|_0$
\begin{enumerate}
\item[a) $ 0    \stackrel {!} {= }($]$\!\!\!p_\mu p_\nu - q_\mu q_\nu)\!
\Big\{\!\! -\! 2 F^{AAA}R_1\! - \! \frac{1}{\alpha}
(F_1^{\bar{c}cA}\! - r_2^{\bar{c}cA})\! +\!
\left[\frac{1}{\alpha} ( 1\! +
\ulSi_{\rm long}(0))\!-\!
(1\!+\ulSi_{\rm trans}(0))\right]gR_2\!+\!{\rm irr}\Big\},$
\item[b) $0   \stackrel {!} {= }    ($]$\!\! p^2-q^2)\delta_{\mu\nu}
\left\{ 2 F^{AAA} R_1 +  (1 + \ulSi_{\rm trans} (0)) g R_2
 +\dsi\,\de m^2 g R_2  + {\rm irr} \right\} $,
\item[$b^0)$ $0 \! \stackrel {!} {= }\! ($]$\!\!p^2-q^2) \delta_{\mu\nu}
\left\{ 2 F_0^{AAA} R^0_1 +  (1 + \ulSi_{0,\rm trans} (0)) g R^0_2
 +\dsi\,\de m_0^2\, g R^0_2  + 2\,i_{10}^{AAB}m\,R_4^0 \right\} $,
\end{enumerate}
\item[IV)] $\; \delta_{A^r_\mu(p)} \delta_{B^s(q)} \delta_{c^t(k)}
\Gamma_1|_0$
\begin{enumerate}
\item[a) $0   \, \stackrel {!} {= } \,$]$   p_\mu \left\{ 2F^{BBA} mR_4 +
\frac12 g \sum^{AB}(0) R_6 + \frac{1}{\alpha}
F^{\bar{c}cB} - m r_2^{\bar{c}cA} + {\rm irr} \right\}$,
\item[b) $0   \, \stackrel {!} {= } \,$]$ q_\mu \left\{ g \sum^{AB}(0) R_2
+ 4F^{BBA} mR_4 + m (F_1^{\bar{c}cA} - r_2^{\bar{c}cA}) + {\rm irr}
\right\}$,
\end{enumerate}
\item[V)] $ \delta_{B^r(p)} \delta_{B^s(q)} \delta_{c^t(k)} \Gamma_1|_0
$
\begin{enumerate}
\item[] $\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!
0 \! \stackrel {!} {= }\! (p^2\!-q^2)\!\left\{ 2R_1 F^{BBA} +
(1 + \uSi^{BB}(0)) {g \over 2} R_6 
 -\dsi[m\, F^{\bc c B}\!-\Si^{BB}(0){g\over 2} R_6]
+ {\rm irr} \right\} $,
\end{enumerate}
\item[$V^0)$] 
\begin{enumerate}
\item[] $\!\!\!\!\!\!\!\!\!\!\!\!\!\!
0 \! \stackrel {!} {= } \!(p^2\!-q^2)\! \left\{ 2R^0_1 F_0^{BBA} +
(1 + \uSi_0^{BB}(0)) {g \over 2} R^0_6 
 -\dsi[m\, F_0^{\bc c B}\!-\Si_0^{BB}(0){g\over 2} R^0_6]
- m(i_{10}^{\bc c B}\!- i_{30}^{\bc c B})\!\right\} $,
\end{enumerate}
 \item[VI)] $\; \delta_{A^r_\mu(p)} \delta_{h(q)} \delta_{c^t(k)}
\Gamma_1|_0$
\begin{enumerate}
\item[a) $0 \, \stackrel {!} {= }\, $]$   p_\mu \Big\{ -2R_1F^{AAh}+
mR_4 (F_1^{hBA} - r_2^{hBA}) 
+ \sum^{AB}(0) \frac12 gR_5 - \frac{1}{\alpha} F^{\bar{c}ch}  + {\rm
irr} \Big\}$,
\item[b) $0   \, \stackrel {!} {= } \,$]$ q_\mu \left\{ -2 R_1F^{AAh} +
2mR_4  F_1^{hBA} + {\rm irr} \right\}$,
\end{enumerate}
\item[VII)] $\; \delta_{h(p)} \delta_{B^s(q)} \delta_{c^t(k)}
\Gamma_1|_0$
\begin{enumerate}
\item[a) $0   \, \stackrel {!} {= } \,$]$   (M^2 + \sum^{hh}(0)) (-
\frac12 g R_3) + 2m F^{BBh} R_4 + mF^{\bar{c}ch}
 + (\alpha m^2 + \sum^{BB}(0)) \frac12 g R_5 $,
\item[b) $0   \, \stackrel {!} {= } \,$]$ p^2 \Big\{ F_1^{hBA}R_1 - (1 +
\uSi^{hh}(0)) \frac12 gR_3 
 -\dsi \, \sum^{hh}(0) \frac12 gR_3 + {\rm irr}
\Big\}$,
\item[$b^0)$ $0   \, \stackrel {!} {= } \,$]$ p^2 \Big\{ F_{1}^{hBA}R^0_1 - (1 +
\uSi_0^{hh}(0)) \frac12 gR^0_3 
 -\dsi \, \sum_0^{hh}(0) \frac12 gR^0_3 + \frac12\, m\,i_{30}^{\bc c h}
\Big\}$,
\item[c) $0   \, \stackrel {!} {= } \,$]$ q^2 \Big\{- F^{hBA}_1R_1 + (1 +
\uSi^{BB}(0)) \frac12 gR_5 
 +\dsi [mF^{\bar{c}ch} + \sum^{BB}(0)\frac12 gR_5] + {\rm irr} \Big\}$,
\item[$c^0)$ $0   \, \stackrel {!} {= } \,$]$ q^2 \Big\{- F_{10}^{hBA}
  R^0_1 + (1 + \uSi_0^{BB}(0)) \frac12 gR_5^0 
 +\dsi [mF_0^{\bar{c}ch} + \sum_0^{BB}(0)
\frac12 gR^0_5] + \frac12\, m\,(2i_{10}^{\bc c h}-i_{30}^{\bc c h})\Big\}$,
\item[d) $0   \, \stackrel {!} {= } \,$]$ k^2 \left\{ r_2^{hBA} R_1
 +\dsi\,
 2m F^{BBh}R_4 + {\rm irr} \right\}$,
\item[$d^0)$ $0   \, \stackrel {!} {= } \,$]$ k^2 \left\{ r_{20}^{hBA} R^0_1
 +\dsi\,
 2m F_0^{BBh}R^0_4 + \frac12\, m\,(2i_{20}^{\bc c h}-i_{30}^{\bc c h})
 \right\}$,
\end{enumerate}
\item[VIII)] $\; \delta_{c^t(q)} \delta_{c^s(p)} \delta_{\bar{c}^r(k)}
\Gamma_1|_0$
\begin{enumerate}
\item[a) $0   \, \stackrel {!} {= } \,$]$   2m F^{\bar{c}cB}R_4 - (\alpha
m^2 +  \sum^{\bar{c}c}(0))  g R_7$,
\item[b) $0   \, \stackrel {!} {= } \,$]$ k^2 \Big\{ F_1^{\bar{c}cA}R_1 -
r_2^{\bar{c}cA}R_1 - (1 + \uSi^{\bar{c}c}(0)) gR_7
 -\dsi \, \sum^{\bar{c}c}(0))  gR_7 + {\rm
irr} \Big\}$,
\item[$b^0)$ $0   \, \stackrel {!} {= } \,$]$
 k^2 \Big\{ F_{10}^{\bar{c}cA}R^0_1 -
r_{20}^{\bar{c}cA}R^0_1 - (1 + \uSi_0^{\bar{c}c}(0)) gR^0_7
 -\dsi\sum_0^{\bar{c}c}(0))  gR^0_7 + 
 m\,R_4^0(2i_{10}^{\bc c B}-i_{30}^{\bc c B})
 \Big\}$,
\item[c) $0   \, \stackrel {!} {= } \,$]$ (p^2 + q^2) \Big\{
r_2^{\bar{c}cA}R_1  +\dsi\, mF^{\bar{c}cB} R_4 + {\rm
irr} \Big\}$.
\item[$c^0)$ $0   \, \stackrel {!} {= } \,$]$ (p^2 + q^2) \Big\{
r_{20}^{\bar{c}cA}R^0_1  +\dsi\, mF_0^{\bar{c}cB} R^0_4 +
 m\,R_4^0 \,i_{20}^{\bc c B} \Big\}$.
\end{enumerate}
\end{enumerate}
\noindent
Four fields
\begin{enumerate}
\item[IX)] $\; \delta_{h(p)} \delta_{h(q)} \delta_{B^1(k)}
\delta_{c^1(l)} \Gamma_1|_0$ \\
\\
$0   \, \stackrel {!} {= } \, 6 F^{hhh} (- \frac12 gR_3) + 4 F^{BBhh} m
R_4 + 2 F^{BBh} g R_5 + 2m r^{hh\bar{c}c} + {\rm irr} $.
\item[X)] $\; \delta_{B^1(k)} \delta_{B^1(p)} \delta_{B^2(q)}
\delta_{c^2(l)} \Gamma_1|_0$ \\
\\
$0   \, \stackrel {!} {= } \, - F^{BBh} gR_3 + 8F^{BBBB} m R_4
 + m \left( 2r_1^{BB\bar{c}c} +r^{BB\bar{c}c}_2 \right) + {\rm irr} $.
\item[XI)] $\; \delta_{h(l)} \delta_{\bar{c}^3(k)} \delta_{c^1(p)}
\delta_{c^2(q)} \Gamma_1|_0$ \\
\\
$0   \, \stackrel {!} {= } \, 2r^{hB\bar{c}c}  mR_4 + F^{\bar{c}cB} g
R_5 + F^{\bar{c}ch} gR_7  + {\rm irr} $.
\item[XII)] $\; \delta_{c^2(k)} \delta_{\bar{c}^2(l)} \delta_{c^1(p)}
\delta_{B^1(q)} \Gamma_1|_0$ \\
\\
$0   \, \stackrel {!} {= } \,  F^{\bar{c}ch} (- \frac12 g R_3) +
(2r_1^{BB\bar{c}c}- r_2^{BB\bar{c}c}) m R_4 
 + F^{\bar{c}cB} (\frac12 gR_6 - gR_7) + 2m r^{\bar{c}c\bar{c}c} + {\rm
irr} $.
\item[XIII)$_1$] $\; \delta_{A^1_\mu(k)} \delta_{A^2_\nu(p)}
\delta_{B^1(q)} \delta_{c^2(l)} \Gamma_1|_0$ \\
\\
$0   \, \stackrel {!} {= } \,  2r_2^{AABB} R_4 + r_2^{AA\bar{c}c}  +
{\rm irr} $.
\item[XIII)$_2$] $\; \delta_{A^1_\mu(k)} \delta_{A^1_\nu(p)}
\delta_{B^2(q)} \delta_{c^2(l)} \Gamma_1|_0$ \\
\\
$0   \, \stackrel {!} {= } \,  -F^{AAh}g R_3 + 4F_1^{AABB} m R_4  + 2m
r_1^{AA\bar{c}c}  + {\rm irr} $.
\item[XIV)] $\; \delta_{A^1_\mu(p)} \delta_{A^1_\nu(q)}
\delta_{A^2_\rho(k)} \delta_{c^2(l)} \Gamma_1|_0$
\begin{enumerate}
\item[a)] $0   \, \stackrel {!} {= } \, 2 \delta_{\mu\nu} l_\rho \Big\{
4 (F^{AAAA}_1 + r_2^{AAAA}) R_1 
+ 2F^{AAA} gR_2 + \frac{1}{\alpha} r_1^{AA\bar{c}c} + {\rm irr}
\Big\}$,
\item[b)] $0   \, \stackrel {!} {= } \, \delta_{\mu\nu} (p_\rho +
q_\rho)  \left\{ \frac{2}{\alpha} r_1^{AA\bar{c}c} + {\rm irr}
\right\}$,
\item[c)] $0   \, \stackrel {!} {= } \, (\delta_{\mu\rho}l_\nu +
\delta_{\nu\rho}l_\mu) \left\{ - 4 F_1^{AAAA}R_1 - 2 F^{AAA} gR_2 + {\rm
irr} \right\}$,
\item[d)] $0   \, \stackrel {!} {= } \, (\delta_{\mu\rho}p_\nu +
\delta_{\nu\rho}q_\mu) \left\{ 0 + {\rm irr} \right\}$,
\item[e)] $0   \, \stackrel {!} {= } \, (\delta_{\mu\rho}q_\nu +
\delta_{\nu\rho}p_\mu)  \left\{ - \frac{1}{\alpha} r_2^{AA\bar{c}c} +
{\rm irr} \right\}$.
\end{enumerate}
\item[XV)$_1$] $\; \delta_{B^1(p)} \delta_{B^1(q)} \delta_{A^2_\mu(k)}
\delta_{c^2(l)} \Gamma_1|_0$
\begin{enumerate}
\item[a)] $0   \, \stackrel {!} {= } \, l_\mu  \Big\{ 4 F_1^{AABB}R_1 +
2F^{BBA} gR_6 + {\rm irr} \Big\}$,
\item[b)] $0   \, \stackrel {!} {= } \, k_\mu \left\{ r_1^{BB\bar{c}c} +
{\rm irr} \right\}$,
\end{enumerate}
\item[XV)$_2$] $\; \delta_{B^1(p)} \delta_{B^2(q)} \delta_{A^1_\mu(k)}
\delta_{c^2(l)} \Gamma_1|_0$
\begin{enumerate}
\item[a)] $0   \, \stackrel {!} {= } \, p_\mu  \left\{ -2r_2^{AABB}R_1 +
2F^{BBA} gR_2 + F_1^{hBA} gR_3 + {\rm irr} \right\}$,
\item[b)] $0   \, \stackrel {!} {= } \, q_\mu \left\{-2 r_2^{AABB}R_1 -
2F^{BBA} gR_2 + 2F^{BBA}gR_6 + {\rm irr} \right\}$,
\item[c)] $0   \, \stackrel {!} {= } \, k_\mu \Big\{-2 r_2^{AABB}R_1 +
F_1^{hBA} \frac12 gR_3 + r_2^{hBA}\frac12 gR_3
 + F^{BBA} gR_6 - \frac{1}{\alpha} r_2^{BB\bar{c}c} + {\rm irr}
\Big\}$,
\end{enumerate}
\item[XVI)] $\; \delta_{h(p)} \delta_{A_{\mu}^1(k)} \delta_{B^2(q)}
\delta_{c^3(l)} \Gamma_1|_0$
\begin{enumerate}
\item[a)] $0   \, \stackrel {!} {= } \, p_\mu  \left\{ F_1^{hBA}g(R_6
-R_2) - r_2^{hBA} gR_2 + {\rm irr} \right\}$,
\item[b)] $0   \, \stackrel {!} {= } \, q_\mu \left\{F_1^{hBA}gR_2 -
r_2^{hBA} gR_2 + 2F^{BBA}gR_5 + {\rm irr} \right\}$,
\item[c)] $0   \, \stackrel {!} {= } \, k_\mu \Big\{F_1^{hBA} \frac12
gR_6 - r_2^{hBA} \frac12 gR_6 + F^{BBA} gR_5 
 - \frac{1}{\alpha} r^{hB\bar{c}c} + {\rm irr} \Big\}$,
\end{enumerate}
\item[XVII)] $\; \delta_{h(p)} \delta_{h(q)} \delta_{A^1_\mu(k)}
\delta_{c^1(l)} \Gamma_1|_0$
\begin{enumerate}
\item[a)] $0   \, \stackrel {!} {= } \, l_\mu  \left\{ 4F^{AAhh}R_1 -
F_1^{hBA} gR_5 + {\rm irr} \right\}$,
\item[b)] $0   \, \stackrel {!} {= } \, k_\mu \left\{r_2^{hBA}gR_5 +
\frac{2}{\alpha} r^{hh\bar{c}c}  + {\rm irr} \right\}$.
\end{enumerate}
\item[XVIII)] $\; \delta_{A^2_\mu(k)} \delta_{c^2(p)} \delta_{c^1(q)}
\delta_{\bar{c}^1(l)} \Gamma_1|_0$
\begin{enumerate}
\item[a)] $0   \, \stackrel {!} {= } \, l_\mu  \left\{
F_1^{\bar{c}cA}g(R_2 -R_7) + \frac{2}{\alpha} r^{\bar{c}c\bar{c}c} +
{\rm irr} \right\}$,
\item[b)] $0   \, \stackrel {!} {= } \, p_\mu \Big\{
2r_1^{AA\bar{c}c}R_1 + r_2^{\bar{c}cA} g(R_2 - R_7) $
$ + \frac{2}{\alpha} r^{\bar{c}c\bar{c}c} + {\rm irr} \Big\} $,
\item[c)] $0   \, \stackrel {!} {= } \, q_\mu \Big\{ -r_2^{AA\bar{c}c}
R_1 - r_2^{\bar{c}cA} gR_7  + \frac{2}{\alpha} r^{\bar{c}c\bar{c}c} +
{\rm irr} \Big\}$.
\end{enumerate}
\end{enumerate}

\vspace{0.2cm}
\noindent
Five fields
\begin{enumerate}
\item[XIX)] $\; \delta_{h(p)} \delta_{h(q)} \delta_{h(k)}
\delta_{B^1(l)} \delta_{c^1(l^\prime)} \Gamma_1|_0$ \\
\\
$0   \, \stackrel {!} {= } \, -2F^{hhhh}R_3 + F^{hhBB} R_5 + {\rm irr}
$.
\item[XX)] $\; \delta_{h(p)} \delta_{B^1(q)} \delta_{B^1(k)}
\delta_{B^2(l)} \delta_{c^2(l^\prime)} \Gamma_1|_0$ \\
\\
$0   \, \stackrel {!} {= } \, -F^{BBhh} R_3 + 2F^{BBBB} R_5 + {\rm irr}
$.
\item[XXI)] $\; \delta_{A^1_\mu(k)} \delta_{A^1_\nu(p)} \delta_{h(k)}
\delta_{B^2(l)} \delta_{c^2(l^\prime)} \Gamma_1|_0$ \\
\\
$0   \, \stackrel {!} {= } \, -F^{AAhh} R_3 + F_1^{AABB} R_5 + {\rm irr}
$.
\item[XXII)] $\; \delta_{A^1_\mu(k)} \delta_{B^1(p)}
\delta_{c^1(l^\prime)} \delta_{A^2_\nu(q)} \delta_{B^3(l)} \Gamma_1|_0$
\\
\\
$0   \, \stackrel {!} {= } \, r_2^{AABB} (R_6- 2R_2) + {\rm irr} $.
\item[XXIII)] $\; \delta_{A^1_\mu(k)} \delta_{B^1(q)}
\delta_{A^2_\nu(p)} \delta_{c^2(l^\prime)} \delta_{h(l)} \Gamma_1|_0$ 
\\
\\
$0   \, \stackrel {!} {= } \, r_2^{AABB} R_5  + {\rm irr} $.
\item[XXIV)] $\; \delta_{A^3_\mu(k)} \delta_{A^3_\nu(p)}
\delta_{\bar{c}^2(q)} \delta_{c^3(l)}  \delta_{c^1(l^\prime)}
\Gamma_1|_0$ \\
\\
$0   \, \stackrel {!} {= } \, r_2^{AA\bar{c}c} R_2  + r_1^{AA\bar{c}c}
R_7 + {\rm irr} $.
\item[XXV)] $\; \delta_{A^3_\mu(k)} \delta_{\bar{c}^3(q)}
\delta_{A^2_\nu(p)}  \delta_{c^3(l)} \delta_{c^1(l^\prime)}
\Gamma_1|_0$ \\
\\
$0   \, \stackrel {!} {= } \, r_2^{AA\bar{c}c} (3R_2 - R_7) + {\rm irr}
$.
\item[XXVI)] $\; \delta_{B^1(p)} \delta_{B^1(q)} \delta_{\bar{c}^1(k)}
\delta_{c^2(l)}  \delta_{c^3(l^\prime)}  \Gamma_1|_0$ \\
\\
$0   \, \stackrel {!} {= } \, r_2^{BB\bar{c}c} (R_6 - R_7)  -
r_1^{BB\bar{c}c} R_7 + {\rm irr} $.
\item[XXVII)] $\; \delta_{B^1(p)}  \delta_{\bar{c}^1(k)} \delta_{B^2(q)}
\delta_{c^3(l)}  \delta_{c^1(l^\prime)}  \Gamma_1|_0$ \\
\\
$0   \, \stackrel {!} {= } \, -r^{hB\bar{c}c} R_3  + r_2^{BB\bar{c}c}
(3R_6 - 2R_7) + {\rm irr} $.
\item[XXVIII)] $\; \delta_{h(p)}  \delta_{h(q)} \delta_{\bar{c}^1(k)}
\delta_{c^2(l)} \delta_{c^3(l^\prime)}   \Gamma_1|_0$ \\
\\
$0   \, \stackrel {!} {= } \, r^{hB\bar{c}c} R_5  + r^{hh\bar{c}c} R_7 +
{\rm irr} $.
\item[XXIX)] $\; \delta_{h(p)}  \delta_{B^1(q)} \delta_{c^1(l)}
\delta_{\bar{c}^2(k)} \delta_{c^2(l^\prime)}   \Gamma_1|_0$ \\
\\
$0   \, \stackrel {!} {= } \, 2r^{hh\bar{c}c} R_3  - 2r_1^{BB\bar{c}c} R_5
+ r_2^{BB\bar{c}c} R_5 + r^{hB\bar{c}c} (-R_6 + 2R_7) + {\rm irr} $.
\end{enumerate}
These 53 conditions are fulfilled in the (tree) order $\hbar^0$
for $\La=0$ and $\Lao \le \infty$. For finite $\Lao$ we have also 
to take into account the tree order irrelevant contribution 
from (\ref{irr}, \ref{wert}) to the classical action.\\[.3cm]
\noindent
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\end{document} 
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