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\begin{document}
\title{ Renormalizability Proof for QED based on Flow Equations}
\author{Georg Keller \thanks{Now at Institut f\"ur Theoretische
Physik, ETH H\"onggerberg, CH-8093 Z\"urich, Switzerland.
Supported by the Swiss National Science Foundation and by the Ambrose
Monell Foundation.}
\\
School of Mathematics\\
Institute for Advanced Study\\
Princeton NJ 08540, USA
\and
Christoph Kopper\\
Institut f\"ur Theoretische Physik\\
Universit\"at G\"ottingen \\
Bunsenstra\ss e 9\\
D 37073 G\"ottingen Germany}
\maketitle
\begin{abstract}
We prove the perturbative renormalizability
of Euclidean $QED_4$ using flow equations, i.e. with the aid of
the Wilson renormalization group adapted to perturbation theory.
As compared to $\Phi^4_4$ the additional difficulty to overcome
is that the regularization violates gauge invariance. We prove that
there exists a class of renormalization conditions such that the
renormalized Green functions satisfy the QED Ward identities
and such that they are infrared finite at nonexceptional momenta.
We give bounds on the
singular behaviour at exceptional momenta (due to the massless
photon) and comment on the adaptation to the case when the fermions
are also massless.
\end{abstract}
\newpage
\section{Introduction }
\vskip.5cm \noindent
About twenty years ago Wilson and his collaborators published their
ideas on the renormalization group and effective Lagrangians [1],
which have stimulated the progress of quantum field theory
and statistical mechanics ever since. In 1984 Polchinski [2] showed
that these ideas are suited for a treatment of the renormalization
problem of perturbative field theory which does not make any use of
Feynman diagrams and in particular sidesteps the complicated
analysis of the divergence/convergence properties of the general
bare or renormalized Feynman diagram. Instead
he showed that the problem can be solved by bounding
the solutions of a system of first order differential
equations, the flow equations, which are a reduction of the Wilson
flow equations to their perturbative content.\\
The present paper is part of a programme of the authors with the
aim to show that the Polchinski method is suited to prove
(in the sense of mathematical physics) the perturbative
renormalizability of any by naive power counting renormalizable
theory of physical interest. Polchinski's original proof for
Euclidean massive $\Phi^4_4$ was restricted to unphysical
renormalization conditions (because they were imposed on the
Green functions with an additional (large) infrared cutoff),
and it was achieved by estimating the
solutions of three types of flow equations for different quantities
successively. In our first paper we redid Polchinski's proof with
two essential modifications: By showing the effective Lagrangian
to be the generating functional of the perturbative connected
amputated Green functions (CAG) we could include any renormalization
conditions (r.c.). Recently the construction of the analytical
minimal subtraction scheme was performed explicitly [8].
By improving Polchinski's induction hypothesis for bounding
the solutions of the flow equations we could reduce the proof
to one type of flow equations (FE) only [3].
The method was then applied to prove the renormalization of composite
operators, the Zimmermann identities,
and the existence of the short distance
expansion [4,5]. It turned out particularly suited for studying questions
of convergence of the regularized theory to the renormalized one
which go under the name of Symanzik's improvement programme [6];
see also [7], where the same question is analyzed in Polchinski's
original framework. A recent proof by one of the authors also
established a de Calan-Rivasseau bound for the large orders of
perturbation theory - i.e. local Borel summability - for massive
$\Phi^4_4$, which shows that the FE method works beyond questions
of perturbative finiteness [9]. In recent years there has also
been increasing interest in the FE method from a more
phenomenological point of view, i.e. with the aim to find new
approximation schemes for the system of FE which differ from
standard perturbation theory. In this case the FE are mostly
presented and analysed in different form, namely for one particle
irreducible Green functions. For example critical exponents
for $\Phi^4_4$-type theories have been calculated in
[10]. It has also been applied to the problem of bound states
and vacuum condensates [11], see also
[12].\\
If the FE are supposed to be suited for a renormalizability proof
of, say, the standard model, it is necessary to cope with gauge
theories. Gauge symmetries constitute a particular problem, since
our framework crucially makes use of momentum space
cutoffs, which necessarily violate gauge invariance,
or - on the level of Green functions - the Ward identities (WIs).
The problem is less severe for an Abelian gauge theory as
QED due to the absence of photon self-interactions.
Nevertheless it necessitates the introduction of new counter terms
to render the Green functions finite. The theory including these new
counter terms will be called a
fermion photon theory in the following. It contains more free
parameters than QED.
We studied the renormalizability of QED in a recent letter [13].
There it was shown that there is a unique choice for the r.c.
corresponding to the new counter terms such that the WIs are restored
in the renormalized theory. This proves the
renormalizability of perturbative Euclidean QED. In this paper
we want to give a complete and fully rigorous proof of the
renormalizability of perturbative QED.
In particular we shall not make use of the nonexistent path
integral measures to derive the WIs and their violation.
And we want to go beyond the previous letter in that we do not
restrict any more to a theory with a massive photon.
The method of dealing with theories with massless particles has
been developed previously for massless $\Phi^4_4$ [14] and shall be
applied to QED now, where we still restrict to the Euclidean
framework, however.
The renormalization of QED using noninvariant regularizations
has also been studied by Feldman, Hurd, Rosen and Wright [15],
Hurd [16] (this paper is closest in spirit to ours), Rosen and
Wright [17]. These papers are based on the Gallavotti-Nicol\`o
tree- formalism and they also include de Calan-Rivasseau
type bounds on the large orders of perturbation theory
and certain statements on Minkowski-space theory. Their
method is still closer to
Feynman diagram based proofs than Polchinski's
method. They work in position space and they do not make
explicit statements on the IR singularities for
exceptional momentum configurations. Our method permits
to analyze these singularities (see Proposition 6). \\
We proceed as follows. In section 2 we introduce the FE
framework and the Lagrangian which for a special choice
of counter terms will be proven later on to define perturbative QED
in a general covariant gauge. In section 3 we prove the
renormalizability of our $O(4)$- and charge conjugation invariant
fermion photon theory, in which however for general counter terms or
r.c. the WIs are violated. The reader unfamiliar with the
subject is recommended to
first read the papers [3] or [4], and [14] where we describe the same
procedure more extensively and where the line of thought is not
burdened by the heavy notation required due to the QED symmetry
structure. In section 4 we explicitly derive the violated WIs (vWIs)
for the regularized theory as relations between CAGs. We show that
there is a unique choice for the r.c. corresponding to those
counter terms which manifestly vanish in invariantly regularized
QED such that - for cutoff to infinity - the QED WIs are restored.
In the last section we comment on the modifications
necessitated when one regards massless fermions or does not
renormalize at zero momentum.
\section{ The Fermion-Photon Theory - Definition and Flow Equations}
As usual in the FE framework we start by defining the regularized
propagators, here for photon and fermion, in Euclidean space. We set
for $m>0$
\eq
(D^{\Lambda_0}_{\Lambda} (k))_{\alpha \beta} :=
\frac{1}{k^2} [(\delta_{\alpha \beta} - \frac{k_{\alpha}k_{\beta}}
{k^2})+\frac{1}{\lambda}\frac{k_{\alpha}k_{\beta}}
{k^2}]\,(R(\Lambda_0,k)\,-\,R(\Lambda ,k))
\eqe
\[
S^{\Lambda_0}_{\Lambda} (p)\,:=
\,\frac{1} {/\!\!\!p+m}(R_m(\Lambda_0,p)\,-\,R_m(\Lambda ,p))
\]
with $/\!\!\!p=p_{\mu} \gamma_{\mu}$, $\{ \gamma_{\mu},\gamma_{\nu}
\}\,=\, -2\delta_{\mu \nu}$ for the Euclidean Dirac matrices.
The functions $R_a$ for $a \ge 0$ are characterized as follows:
\eq
R_a(\Lambda ,k) \,=\, K(\frac{k^2+a^2}{\Lambda^2})\,, \quad
R(\Lambda ,k) \,:=\,R_0(\Lambda ,k) \,, \;(\Lambda ,k) \neq (0,0)
\;\,\mbox{for}\;\,a=0\,.
\eqe
Here $\; 0\le \Lambda \le \Lambda_0 \le \infty \, ,
$
and $K$ satisfies
\begin{equation}
K \in C^{\infty}[0,\infty) ,\quad 0 \le K \le 1 ,
\quad K(x) = 1 \quad \mbox{for}\quad x \le 1 , \quad
K(x) = 0 \quad\mbox{for}\quad x \ge 4 \,.
\end{equation}
>From (2),(3) we find $R \in C^{\infty}(\R_+\times\R^4)$
and $R_m \in C^{\infty}([0,\infty)\times\R^4)$.
We also have for $\Lambda > 0$ and for $0 \le |k^2+a^2| \le \Lambda^2$
or
$4 \Lambda^2 \le |k^2+a^2| \,$
\begin{equation}
\partial^w R_a(\Lambda,k) = 0\,,\;w\neq 0,\;\;\;
\partial^w \partial_{\Lambda}R_a(\Lambda,k) = 0 \,,
\end{equation}
where the multiindex $w$ indicates momentum derivatives
\[ \partial^w = \partial^{w_1}\cdots \partial^{w_4} =
\frac{\partial^{w_1}}{\partial k_1^{w_1}} \cdots
\frac{\partial^{w_4}}{\partial k_4^{w_4}}
\quad\mbox{for}\; k= (k_1,\ldots ,k_4), \; w_i \in \N_0 \,.\]
Replacing $R$ by $R_m$ (which is an improved version of the $R$
used in [5]) for the massive fermions allows to obtain better
statements on the IR behaviour later on, but $R_m$ does not serve
as an IR
regulator and therefore it should not be used for the massless
photon.
As can be seen in (1) we restrict to a general covariant gauge. The
regularization breaks gauge invariance and consequently the WI's,
but not $O(4)$-invariance.
Due to this breaking of the WI's our interaction Lagrangian will also
have to contain terms of dimension $\le 4$ which for invariant
regularization need not be introduced- due to the WI's.
We define $L^{\Lambda_0}(A,\psib ,\psi)$ as
\eq
L^{\Lambda_0} :=
\int dx \{ \frac{z_3}{4}F_{\mu \nu}^2 \,+\, \frac{\delta \lambda}{2}
(\partial A)^2 \,+\, \frac{\delta \mu^2 }{2}
A^2 \,+\, z_4 (A^2)^2 \,-\,z_2 \psib i /\!\!\! \partial \psi \,+\,
\delta m \psib \psi \,+\,e(1+z_1) \psib /\!\!\!\! A \psi \} \, .
\eqe
The notation is rather standard (including the summation convention),
but we set
\eq
z_i :=Z_i-1, \quad 1\le i \le 3
\eqe
as compared to standard textbooks. The WI's for invariantly
regularized QED would then imply $z_1=z_2$, $\delta \lambda =0$,
$z_4=0$, $\delta \mu^2 =0$. The parameters
$z_i$, $\delta \lambda$, $\delta \mu^2$ are formal power series in
the coupling $e$. Apart from $z_1$ they have to be assumed to
be of at least first order in $e$. For standard r.c. all constants
are even of second order in $e$ (see below (33)).
The perturbative Green functions are obtained from (5) by the
standard rules which imply that $\psib,\psi$ are viewed as
independent elements of an infinite-dimensional (formal)
Grassmann-algebra; $A_{\mu}(x)$ may be viewed as an element of
${\cal S}(\R ^4)$. \\
As regards their transformation properties under $O(4)$ and charge
conjugation $C$, we impose
\eq
O(4):\,\, \psi'(x') = S(\Lambda)\psi (x),\;
\psib '(x') = \psib (x) S(\Lambda),\;
A'_{\mu}(x') = \Lambda_{\mu \nu} A_{\nu}(x)
\mbox{ with } x'_{\mu} = \Lambda_{\mu \nu} x_{\nu} \,,
\eqe
\eq
C:\,\,
\psi'(x) = -C^{-1} \psib^T (x),\;
\psib' (x) = \psi^T (x) C\, ,\;
A'_{\mu}(x) = -A_{\mu}(x)
\,,
\eqe
where the charge conjugation matrix $C$ fulfills
\eq
C\gamma_{\mu} C^{-1} = -\gamma_{\mu}^T \quad \mbox{(e.g. } C=\gamma_0
\gamma_2).
\eqe
$\Lambda$ and $S(\Lambda)$ are the vector and spinor representations
of $O(4)$ respectively. Using (8),(9) and $|\det\Lambda|=1$ as well as
the canonical assignments $dimA=1,$ $dim\psi =dim\psib =3/2$ we find
\\
\noindent
{\bf Lemma 1}: $L^{\Lambda_0}$ is the integral of the most general
local polynomial of dimension $\le 4$
in the fields $A, \psi ,\psib $ and their derivatives which is $O(4)$
and charge conjugation invariant and fully symmetric under
permutations of the $A$-fields.\\
\noindent
The procedure to derive the FE's is analogous to that employed for
$\Phi^4_4$ [3,4].
We introduce the source functions
\eq
J_{\mu}(x),\; \eta(x),\; \etab (x)
\eqe
for the $A,\psib,\psi$-fields and find for the generating functional
of the perturbative regularized Green functions formally given by
\[
\int \, DA\,D\psib D\psi \;
e^{-\frac{1}{2} }\,
e^{<\psib,S^{\Lambda_0}_{\Lambda} \psi>}\,
e^{-L^{\Lambda_0} \,+\, \int \,
J\cdot A\,+\, \psib \cdot \eta \,+\, \psi \cdot \etab}
\]
the following rigorous formula
\eq
Z^{\Lambda_0}_{\Lambda} (J,\etab, \eta) :=
e^{-L^{\Lambda_0}(\delta_J,-\delta_{\eta},\delta_{\etab}) }\,
e^{1/2 }
\,e^{ <\etab,S^{\Lambda_0}_{\Lambda}\eta>} \,.
\eqe
We assume $J_{\mu} \in {\cal S}(\R^4)$ and
$\eta, \etab$ to be Grassmann variables
and we demand that all sources have the same $O(4)$ and $C$ transformation
properties as the respective fields. We employed the usual notation
\eq
= \int d^4x\,d^4y\,
J_{\mu}(x) (D^{\Lambda_0}_{\Lambda})_{\mu \nu}(x-y) J_{\nu}(y)
\,= \int \frac{d^4k}{(2\pi)^4}
J_{\mu}(-k) (D^{\Lambda_0}_{\Lambda})_{\mu \nu}(k) J_{\nu}(k)
\eqe
and similarly for $<\etab,S^{\Lambda_0}_{\Lambda} \eta>$.
We set $J_{\mu}(x)
\,= \int \frac{d^4k}{(2\pi)^4}
e^{ikx}\,J_{\mu}(k)$.
\\
Then we introduce the functional Laplace operator
\eq
\triangle(\Lambda,\Lambda_0)
\,=\,
\triangle'(\Lambda,\Lambda_0)
\,+\,
\triangle''(\Lambda,\Lambda_0)
\eqe
\[
\triangle'(\Lambda,\Lambda_0)
\,=\, <\delta_A,D^{\Lambda_0}_{\Lambda}\delta_A>\,,\;
\triangle''(\Lambda,\Lambda_0)
\,=\,<\delta_{\psi},S^{\Lambda_0}_{\Lambda}\delta_{\psib}>\,,
\]
and find \\
{\bf Proposition 2}:
\eq
Z^{\Lambda_0}_{\Lambda} (J,\etab, \eta) =
e^{1/2 }
\,e^{ <\etab,S^{\Lambda_0}_{\Lambda}\eta>}
\;
(e^{\triangle(\Lambda,\Lambda_0)} \;
e^{-L^{\Lambda_0}(A,\psib,\psi)})|_{A=D^{\Lambda_0}_{\Lambda}J,
\psi=S^{\Lambda_0}_{\Lambda}\eta,
\psib=\etab S^{\Lambda_0}_{\Lambda}} \;.
\eqe
{\sl Proof}: In the proof we omit $\Lambda, \Lambda_0$ in
$D,S,\triangle$. It may be performed in two steps.
In the first step one shows
\eq
e^{-L^{\Lambda_0}(\delta_J,-\delta_{\eta},\delta_{\etab}) }\,
e^{1/2 }
\,=\,
e^{1/2 } \, (e^{\triangle'}\,
e^{-L^{\Lambda_0}(A,-\delta_{\eta},\delta_{\etab}) })|_{A=DJ}\,.
\eqe
We omit the proof of (15) here since it is analogous to the proof of
the corresponding statement for $\Phi^4_4$ [3,4]. The way of
proceeding may also be inferred from the treatment of the fermionic
part, which is performed explicitly now. (14) follows immediately
using (15), if we can show that
\eq
(e^{\triangle''}\,
e^{-L^{\Lambda_0}(A,\psib,\psi) })|_{\psi=S\eta,\psib=\etab S}\,=\,
e^{-<\etab,S\eta>}\,
e^{-L^{\Lambda_0}(A,-\delta_{\eta},\delta_{\etab}) }\,
e^{<\etab,S\eta>}\,.
\eqe
So we want to prove (16). We write in the proof for the r-th order
perturbative contribution to $L^{\Lambda_0}$
\[
L^{\Lambda_0}_r(\psib,\psi) \,=\, <\psib,M_r \psi>\,+\,B_r
\]
with suitable local $A-$dependent $M_r(x)$. $B_r$ contains the
$\psib,\psi$-independent terms and is not of importance here.
The first step is to derive a commutation relation for the functional
differential operator $\triangle''$
\[
[\triangle''\,,\,L^{\Lambda_0}_r\,]\,=\,
a_r\,,\quad
[(\triangle'')^2\,,\,L^{\Lambda_0}_r\,]\,=\,
2a_r\,\triangle''\,+\, b_r
\]
and by induction
\eq
[(\triangle'')^n\,,\,L^{\Lambda_0}_r\,]\,=\,
na_r\,(\triangle'')^{n-1}\,+\,
n(n-1)b_r (\triangle'')^{n-2} \,,\;n\ge 2\, ,
\eqe
where
\[
a_r\,=\, -<\delta_{\psi},SM_r \psi>\,+\, <\psib,M_rS
\delta_{\psib}>\,,\quad
b_r\,=\,
-<\delta_{\psi},SM_rS\delta_{\psib}>\,.
\]
(17) implies
\eq
[e^{\triangle''}\,,\,L^{\Lambda_0}_r\,]\,=\,
(a_r\,+\, b_r )e^{\triangle''} \,.
\eqe
Now we may get rid of $a_r,b_r$ using essentially the same mechanism
by which they were produced, namely we find
\eq
L^{\Lambda_0}_r(-\delta_{\eta},\delta_{\etab})
\;e^{<\etab,S \eta>}\,=\,
e^{<\etab,S \eta>}\,(L^{\Lambda_0}_r\,+\,a_r\,+\, b_r
)|_{\psi=S\eta,\psib=\etab S} \,,
\eqe
so that (18),(19) together give
\[
(e^{\triangle''}\,L^{\Lambda_0}_{r_1}
\ldots
\,L^{\Lambda_0}_{r_k})
|_{\psi=S\eta,\psib=\etab S}
\,=\,
e^{-<\etab,S \eta>}\,
e^{<\etab,S \eta>}\,
\prod_{i=1}^k \,(L^{\Lambda_0}_{r_i}\,+\,a_{r_i}\,+\, b_{r_i})
|_{\psi=S\eta,\psib=\etab S} \,=\,
\]
\[
=\, e^{-<\etab,S \eta>}\,
L^{\Lambda_0}_{r_1}(-\delta_{\eta},\delta_{\etab})
\ldots
L^{\Lambda_0}_{r_k}(-\delta_{\eta},\delta_{\etab})
\;
e^{<\etab,S \eta>}\, ,
\]
from which (16) immediately follows on expanding
$e^{-L^{\Lambda_0}}$. (Note that a factor $e^{\triangle''}$
on the r.h.s. of the second equation is replaced by 1,
when we regard the equations as equations
for functionals, not operators.)
\qed
We may then introduce the generating functionals
$W^{\Lambda ,\Lambda_0} (J,\etab, \eta) $ of the (nontrivial,
regularized) connected
Green functions, and $L^{\Lambda ,\Lambda_0} (A,\psib, \psi) $
of the (nontrivial, regularized) connected amputated Green functions (CAG),
given by
\eq
e^{-(W^{\Lambda ,\Lambda_0} (J,\etab, \eta)+f.i.) } \,= \,
Z_{\Lambda}^{\Lambda_0} (J,\etab, \eta) \, ,
\;
e^{-(L^{\Lambda ,\Lambda_0} (A,\psib, \psi)+f.i.)} \,=\,
e^{\triangle (\Lambda ,\Lambda_0)} \,
e^{-L^{\Lambda_0} (A,\psib, \psi)} \,,
\eqe
where $f.i.$ (for field-independent)
is defined such that
\[
W^{\Lambda ,\Lambda_0}(0,0,0) \,=\,L^{\Lambda ,\Lambda_0}(0,0,0)
\,=\, 0\,.
\]
Thus $f.i.$ also depends on $\Lambda ,\Lambda_0$, and the volume
has to be kept
finite as long as we deal with $f.i.$. Since we are not
interested in $f.i.$ we spare ourselves being precise here and refer to [3,4]
instead. Note that
\eq
L^{\Lambda_0 ,\Lambda_0} \, \equiv \, L^{\Lambda_0} \,.
\eqe
The FE is then obtained by taking derivatives w.r.t. $\Lambda$
on both sides of (20),$2^{nd}$ equ. We obtain
\eq
\partial_{\Lambda }L^{\Lambda ,\Lambda_0} \,=\,
\partial_{\Lambda }\triangle(\Lambda ,\Lambda_0) \,- \,
1/2<\delta_A L^{\Lambda ,\Lambda_0},(\partial_{\Lambda }
D_{\Lambda}^{\Lambda_0}) \delta_A L^{\Lambda ,\Lambda_0}>
\eqe
\[
\,+ \,
<\delta_{\psi} L^{\Lambda ,\Lambda_0},(\partial_{\Lambda }
S_{\Lambda}^{\Lambda_0}) \delta_{\psib} L^{\Lambda ,\Lambda_0}> \,- \,
\partial_{\Lambda }
f.i.
\]
To proceed further we expand $L$ in terms of powers of external
fields and orders of perturbation theory in momentum space
\eq
L^{\Lambda ,\Lambda_0} \,=\,
\sum_{r\ge 1}\, e^r \, L^{\Lambda ,\Lambda_0}_r
\eqe
and
\eq
L^{\Lambda ,\Lambda_0}_r \,=
\sum_{m+n >0}
\int \frac{d^4k_1}{(2\pi)^4}
\ldots
\frac{d^4p_{2n-1}}{(2\pi)^4}\,
({\cal L}^{\Lambda,\Lambda_0,r}_{m,2n})_{\mu_1 \ldots \mu_m
i_1 \ldots i_n j_1 \ldots j_n } (k_1,\ldots k_m,p_1,\ldots
p_{2n-1}) \, \times
\eqe
\[
\times
A_{\mu_1}(k_1) \ldots A_{\mu_m}(k_m)
\psib_{i_1}(p_1) \ldots \psib_{i_n}(p_n)
\psi_{j_1}(p_{n+1}) \ldots \psi_{j_n}(p_{2n})\,
\;,\]
where for $n \neq 0$ we set
$
p_{2n}:= -k_1-\ldots k_m-p_1-\ldots -p_{2n-1} \,.
$
We did not write explicitly the case $n=0$ where by momentum
conservation $k_m=-k_1-\ldots-k_{m-1}$. \\
The following symmetry properties of the
${\cal L}^{\Lambda,\Lambda_0,r}_{m,2n}$ follow from the
properties of $L$ and $\triangle$:
\begin{itemize}
\item[(i)] ${\cal L}^{\Lambda,\Lambda_0,r}_{m,2n}=0\,$, if $\;m+2n>4r$
(connectedness)
\item[(ii)] ${\cal L}^{\Lambda,\Lambda_0,r}_{2m+1,0}=0$
(charge conjugation symmetry, Furry's theorem)
\item[(iii)] ${\cal L}^{\Lambda,\Lambda_0,r}_{m,2n}$
may (and will) be chosen fully symmetric under permutations of
$(k_1,\mu_1), \ldots, $ $(k_m,\mu_m)$ and fully antisymmetric under
permutations of $(p_1,i_1), \ldots , (p_n,i_n)$ and \\
$(p_{n+1},j_1), \ldots , (p_{2n},j_n)$.
\item[(iv)] The $O(4)$-transformation properties of
${\cal L}^{\Lambda,\Lambda_0,r}_{m,2n}$
are indicated through the vector and spinor indices
$\mu_1 \ldots i_1 \ldots j_1 \ldots $
\item[(v)] ${\cal L}^{\Lambda,\Lambda_0,r}_{m,2n}$
is in $C^{\infty}((0,\Lambda_0] \times \R^{4(m+2n-1)})$
as a function of $\Lambda, k_1 \ldots p_{2n-1}$, due to the
smoothness
of the regularized propagators.
\end{itemize}
The FE for the coefficient functions
${\cal L}^{\Lambda,\Lambda_0,r}_{m,2n}$
is then obtained from (22),(23) by identifying
the coefficients of $m$ photon and $n$ fermion and
antifermion fields in (22). We obtain:
\[
(\partial_{\Lambda}{\cal L}^{\Lambda,\Lambda_0,r}_{m,2n})
_{\mu_1 \ldots \mu_m
i_1 \ldots i_n j_1 \ldots j_n } (k_1,\ldots k_n,p_1,\ldots
p_{2n-1}) \, = \,
\]
\[
=\,-{m+2 \choose 2}
\,\int \, \frac{d^4k}{(2\pi)^4} \partial_{\Lambda}R(\Lambda ,k)\,
D_{\mu \nu}(k) \; \times
\]
\[
\times \,({\cal L}^{\Lambda,\Lambda_0,r}_{m+2,2n})
_{\mu \nu \mu_1 \ldots \mu_m i_1 \ldots i_n j_1 \ldots j_n }
(k,-k,k_1,\ldots k_n,p_1,\ldots p_{2n-1}) \; + \,
\]
\eq
+\,(n+1)^2 (-1)^n
\,\int \, \frac{d^4p}{(2\pi)^4} \partial_{\Lambda}R_m(\Lambda,p)
S_{ji}(p) \;\times
\eqe
\[
({\cal L}^{\Lambda,\Lambda_0,r}_{m,2n+2})_{\mu_1 \ldots \mu_m
i i_1 \ldots i_n j j_1 \ldots j_n } (k_1,\ldots k_n,-p,p_1,\ldots
p_n,p,\ldots ,p_{2n-1}) \, + \,
\]
\[
+\, \sum_{m'+m''=m+2,n'+n''=n,r'+r''=r} \;\;
\frac{m'\,m''}{2} (-1)^{n'n''} [
\partial_{\Lambda}R(\Lambda ,k')\,
D_{\mu \nu}(k') \; \times
\]
\[
({\cal L}^{\Lambda,\Lambda_0,r'}_{m',2n'})_{\mu \mu_1 \ldots
\mu_{m'-1}
i_1 \ldots i_{n'} j_1 \ldots j_{n'} } (k',k_1,\ldots
k_{m'-1},p_1,\ldots,p_{n'},
p_{n+1}, \ldots p_{n+n'-1}
) \, \times \,
\]
\[
({\cal L}^{\Lambda,\Lambda_0,r''}_{m'',2n''})_{\nu \mu_{m'} \ldots
\mu_m
i_{n'+1} \ldots i_n j_{n'+1} \ldots j_n } (-k',k_{m'},\ldots
k_m,p_{n'+1},\ldots,p_n,
p_{n+n'}, \ldots p_{2n-1})]_{SAS} \;+\,
\]
\[
+\, \sum_{m'+m''=m,n'+n''=n+1,r'+r''=r} \;\;
n'\,n'' (-1)^{(n'+1)n''}
[\partial_{\Lambda}R_m(\Lambda ,p')\,
S_{ji}(p') \times
\]
\[ ({\cal L}^{\Lambda,\Lambda_0,r'}_{m',2n'})_{\mu_1 \ldots
\mu_{m'}
i_1 \ldots i_{n'} j j_1 \ldots j_{n'} } (k_1,\ldots
k_{m'},p_1,\ldots,p_{n'},p,'
p_{n+1}, \ldots p_{n+n'-1}
) \, \times \,
\]
\[
({\cal L}^{\Lambda,\Lambda_0,r''}_{m'',2n''})_{\mu_{m'+1} \ldots
\mu_m i
i_{n'+1} \ldots i_n j_{n'+1} \ldots j_n } (k_{m'+1},\ldots
k_m,-p',p_{n'+1},\ldots,p_n,
p_{n+n'}, \ldots p_{2n-1})]_{SAS} \;.
\]
The momenta $k',p'$ are determined by momentum conservation.
$SAS$ indicates symmetrization w.r.t. photon and antisymmetrization
w.r.t. i) fermion and ii) antifermion momenta and indices.
Many of the details of (25) are not important for us. The important
points are the following: \\
(i) The r.h.s. contains only $\cal L$ terms for which either
$r$ is of smaller value than that of the l.h.s. or, if not, $m+n$
is of larger value than that of the l.h.s. This together with (i)
after (24) fixes the induction scheme through which we will estimate the
solutions of (25).\\
(ii) The induction Ansatz will be determined by the power counting
w.r.t. $\Lambda$ for the differentiated regularized propagators.
For a complete estimate of the solutions we will (as always)
need the equations generated from (25) by taking $|w| \in \N$
momentum derivatives.
As regards
notation, we set
\eq w\in {\N}^{4(m+2n-1)}_0 \qquad w=(w_1, \ldots ,w_{m+2n-1})
\qquad w_i = (w_{i,1}, \ldots ,w_{i,4})
\eqe
\[ \quad |w| = \sum |w_{i_{\nu}}|
\qquad \partial^w = \partial^{w_{1,1}}\cdots \partial^{w_{m+2n-1,4}} =
\frac{\partial^{w_{1,1}}}{\partial k_{1,1}^{w_{1,1}}} \cdots
\frac{\partial^{w_{m+2n-1,4}}}{\partial p_{2n-1,4}^{w_{m+2n-1,4}}}
\]
\section{ UV- and IR-finiteness of the fermion-photon theory }
\subsection{UV-finiteness}
The proof of UV- and IR-finiteness proceeds similarly as in $\Phi^4_4$
[3,4,14]. We start with the UV-problem. That means we choose a scale
$\Lambda_1 \, > \, 0$, for simplicity $\Lambda_1 \, = \, 1$, and
want to show that $\lim_{\Lambda_0 \to \infty} \, {\cal
L}^{1,\Lambda_0,r}_{m,2n}$ exists for all $m+n>0,\; r \ge 1$ and
arbitrary (bounded) momenta. The proof requires that we fix all terms
of (mass) dimension $\le 4$ which are not automatically zero due to
the symmetry structure of the theory, through renormalization
conditions (r.c.) at $\Lambda_1$. The symmetry structure, i.e.
invariance under the Euclidean group and charge conjugation-
has been fixed through the structure of $L^{\Lambda_0}$ (the
particular values of the $z_i$, $\delta \lambda$,
$\delta \mu^2$, $\delta m $ are not yet fixed) and through
$\triangle$.
Since we are dealing with a partially massless theory, the CAG
without IR regularization can generally be expected to
exist in momentum space only, if certain restrictions on the r.c. are
obeyed and if the momentum configuration is nonexceptional
(see [14,18]). More precise statements will follow. As long as we
keep $\Lambda \ge 1$ we need not care about these restrictions, but
we will choose the renormalization points such that the notation
is as simple as possible and such that we need not change them when we
go down to $\Lambda=0$, namely it turns out that all renormalizations
for the photon Green functions
${\cal
L}_{m,0}\,,m \le 4$ should be performed
at zero momentum in order to obtain reasonably simple IR bounds.
The photon mass term has to vanish at $0$
momentum and $\Lambda=0$ for the theory to exist.
In massless QED the renormalization conditions for the
$\partial^w{\cal L}_{m,2n}$ with $m+3n+|w|=4$
have to be imposed
at nonvanishing momenta, however (ch.5).
We noted already that due to C-invariance
\eq
{\cal L}^{\Lambda,\Lambda_0,r}_{m,0} \equiv 0
\quad \mbox{for}\, m \, \mbox{odd, in particular for m=1,3}.
\eqe
>From $O(4),C$-invariance of $L^{\Lambda,\Lambda_0}$ and permutation
(anti)symmetry of ${\cal L}^{\Lambda,\Lambda_0,r}_{m,2n}$ we obtain
for the remaining terms at zero momentum and for given
$\Lambda,\Lambda_0,r$ (which we suppress)
\eq
({\cal L}_{2,0})_{\mu \nu}(0)
\, \sim \, \delta_{\mu \nu} \, , \;
\partial_{\rho}({\cal L}_{2,0})_{\mu \nu}(0)
\, = \, 0 \, ,\;
\partial_{\sigma}\partial_{\rho}
({\cal L}_{2,0})_{\mu \nu}(0)
\, = \, c\, \delta_{\mu \nu}\delta_{\rho \sigma} \,+\,
c'\,(\delta_{\mu \rho}\delta_{\nu \sigma}+
\delta_{\mu \sigma}\delta_{\nu \rho}) \, ,
\eqe
\eq
({\cal L}_{4,0})_{\mu \nu \rho \sigma}(0)
\, \sim \, f_{\mu \nu \rho \sigma} := 1/3
( \delta_{\mu \nu}\delta_{\rho\sigma} \,+\,
\delta_{\mu \rho}\delta_{\nu\sigma}+
\delta_{\mu \sigma}\delta_{\nu \rho}) \, ,
\eqe
\eq
({\cal L}_{0,2})_{i j}(0)
\, \sim \, \delta_{ij} \, ,
\quad
\partial_{\mu} ({\cal L}_{0,2})_{i j}(0)
\, \sim \, (\gamma_{\mu})_{ij} \, ,
\eqe
\eq
({\cal L}_{1,2})_{\mu i j}(0)
\, \sim \, (\gamma_{\mu})_{ij}\, .
\eqe
Thus 7 independent constants fix the
terms of dimension $\le 4$.
The structure of $L^{\Lambda_0}$ determines the b.c. for
$\Lambda = \Lambda_0$. (5) tells us that at \\
$\underline{\Lambda = \Lambda_0}$:
\eq
\partial^w ({\cal L}^{\Lambda_0,\Lambda_0,r}_{m,2n})
(k_1,\ldots ,p_{2n-1}) \equiv 0, \quad \mbox{if } m+3n+|w| \ge 5.
\eqe
We impose at \\
$\underline{\Lambda = 1},\, r\ge 2$:
\begin{itemize}
\item[(i)] $\; ({\cal L}^{1,\Lambda_0,r}_{2,0})_{\mu \nu}(0)
\, =\, \frac{1}{2} \, \delta \mu_r^{2,1} \delta_{\mu \nu} \, ,
$
\item[(ii)] $\; \partial_{\sigma}
\partial_{\rho}
({\cal L}^{1,\Lambda_0,r}_{2,0})_{\mu \nu}(0)
\, = \, \frac{1}{2} \, \delta \lambda^1_r \,
(\delta_{\mu \rho}\delta_{\nu \sigma} \,+\,
\delta_{\mu \sigma}\delta_{\nu \rho})\,+ \,
\frac{1}{4}\, z^1_{3r} (2\delta_{\mu \nu}\delta_{\sigma \rho} \,- \,
\delta_{\mu \rho}\delta_{\nu \sigma} \,-\,
\delta_{\mu \sigma}\delta_{\nu \rho})
$\\
(so that $\frac{1}{2} \, z_3$ corresponds to $ c, $ and $ \frac{1}{2}
\, \delta \lambda $ corresponds to $c'+1/2\, c$ in (28))
\eq \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \mbox{(iii)}
\; ({\cal L}^{1,\Lambda_0,r}_{4,0})_{\mu \nu \rho \sigma}(0)
\, = \, z^1_{4,r} \, f_{\mu \nu \rho \sigma}
\, , \qquad \qquad \qquad\qquad \qquad \qquad
\quad \qquad\qquad \qquad \qquad
\eqe
\item[(iv)]$ \; ({\cal L}^{1,\Lambda_0,r}_{0,2})_{i j}(0)
\, = \, \delta m^1_r \, \delta_{ij} \, ,
$
\item[(v)]$\;
\partial_{\mu} ({\cal L}^{1,\Lambda_0,r}_{0,2})_{i j}(0)
\, = \,- z^1_{2r} \, (\gamma_{\mu})_{ij} \, ,
$
\item[(vi)]$ \; ({\cal L}^{1,\Lambda_0,r}_{1,2})_{\mu i j}(0)
\, = \, z^1_{1,r-1}(\gamma_{\mu})_{ij}\,.
$
\end{itemize}
$\underline{\Lambda = 1},\, r=1 $:
$\qquad ({\cal L}^{1,\Lambda_0,1}_{1,2})_{\mu i j}(0)
\, = \, (\gamma_{\mu})_{ij} \, . $
We assume (and this is standard) that all renormalization constants
apart from $z^1_1$ vanish for $r=1$. This somewhat simplifies the notation.
Leaving out this restriction is possible, but not of much interest.
It is easy
to see from the FE (and obvious to anyone acquainted with QED)
that the ${\cal L}^{\Lambda,\Lambda_0,1}_{m,2n}$ then also vanish
for $\Lambda \neq 1$ for $m \neq 1$ or $n \neq 1$.
Apart from this restriction all constants are completely arbitrary numbers
which later on will be uniquely fixed by the r.c. which we impose at
$\Lambda =0$. They are of course assumed to be independent of
$\Lambda_0$. \\
To prove the UV finiteness of the fermion-photon theory we
introduce the (by now standard) (semi-)norms $\|\;\|_{(a,b)}$ defined
as
\eq
\| \partial^z \,f \|_{(a,b)} \,=\,
\sup_{x_1,\ldots ,x_n,w,i_1,\ldots ,i_l,|x_i| \le max(a,b)}
|\partial^w f_{i_1,\ldots ,i_l} (x_1,\ldots ,x_n)| \;,
\eqe
where $z=|w|$, $z \in \N_0$, $a,b \ge 0$ and for any system of
sufficiently smooth functions
$f_{i_1,\ldots ,i_l }:\R^n \to \C$ with $i_j$ running through
some finite set.
We find for $1 \le \Lambda \le \Lambda_0$ and any fixed
$B>0$ and $a\ge 0$:
\eq
\|\partial^z R_a(\Lambda,\cdot )\|_{(2\Lambda,B)} \, \le \,
c(z) \Lambda^{-z},\quad
\|\partial^z \partial_{\Lambda}R_a(\Lambda,\cdot )\|_{(2\Lambda,B)} \, \le \,
c(z) \Lambda^{-z-1}
\eqe
\eq
\|\partial^z R_a(\Lambda,\cdot )\, f_1\|_{(2\Lambda,B)} \, \le \,
c(z) \Lambda^{-2-z},\,z>0\,, \;\;
\|\partial^z \partial _{\Lambda} R_a(\Lambda,\cdot )\, f_1\|_{(2\Lambda,B)}
\, \le \,
c(z) \Lambda^{-3-z},
\eqe
\eq
\|\partial^z R_a(\Lambda,\cdot )\, f_2\|_{(2\Lambda,B)} \, \le \,
c(z) \Lambda^{-1-z},\,z>0\,,\;\;
\|\partial^z \partial _{\Lambda} R_a(\Lambda,\cdot )\, f_2\|_{(2\Lambda,B)}
\, \le \,
c(z) \Lambda^{-2-z},
\eqe
\eq \mbox{where } f_1(k)=k^{-2},
\;
f_{2,ij}(p)=(\frac{1}{/\!\!\!p+m})_{ij},\;
c(z) \mbox{ is some suitable constant.}
\eqe
For $0< \Lambda \le 1$
we also find
\eq
|\partial^w R(\Lambda,k)| \, \le \,
c(z) (\sup (\Lambda,|k|))^{-z}, \;\;
|\partial^w \partial_{\Lambda}R(\Lambda,k)| \, \le \,
c(z) (\sup (\Lambda,|k|))^{-z-1}.
\eqe
\[
|\partial^w R_m(\Lambda,p)| \, \le \,
c(z) ,\;
|\partial^w \partial_{\Lambda}R_m(\Lambda,p)| \, \le \,
c(z) \,.
\]
($c(z)$ also depends on the mass $m$, which we do not indicate
since $m$ is fixed).\\
Now we may state the UV-renormalizability of the fermion-photon
theory through \\
\noindent
{\bf Proposition 3}: For $1\le \Lambda \le \Lambda_0 < \infty$
we have the following estimates
\begin{itemize}
\item[(i)]
$\|\partial^z \,{\cal L}^{\Lambda,\Lambda_0,r}_{m,2n}\, \|_{(2\Lambda,B)}
\, \le \,
\Lambda^{4-m-3n-z} Plog\Lambda \;\;$ (UV-boundedness) \\
\item[(ii)]
$\|\partial_{\Lambda_0}\partial^z \,
{\cal L}^{\Lambda,\Lambda_0,r}_{m,2n}\,\|_{(2\Lambda,B)}
\, \le \,
(\frac{\Lambda}{\Lambda_0})^2
\Lambda^{3-m-3n-z} Plog\Lambda_0 \;\;$ (UV-renormalizability), \\
\end{itemize}
where we denote (as usually) by $Plog\Lambda$ a polynomial in
$log\Lambda$ with nonnegative coefficients independent of
$\Lambda,\Lambda_0$, but depending on $m,n,r,z,B$.\\
\noindent
{\sl Proof}: We proceed in the standard way [3,4] by induction on
$r$. For given $r$ we descend in the values of $m+2n$
(remember (i) after (24))
and then in the
values of $z$ for fixed $m+2n$ starting from some arbitrary
$z_{\max}$. \\
$\underline{r=1}$: \\
(a) $m+3n+|w| \ge 5$:
$\partial^w \,{\cal L}^{\Lambda,\Lambda_0,1}_{m,2n}\, \equiv \, 0$
from the b.c. (32) and the FE. \\
(b) $m+3n+|w| \le 4$:
The r.c. (33) (plus subsequent comments) and the FE tell us that
\[
({\cal L}^{\Lambda,\Lambda_0,1}_{1,2})_{\mu ij}(0)\,=\,
(\gamma_{\mu})_{ij}
\]
Using (a) and Taylor's theorem we find
$({\cal L}^{\Lambda,\Lambda_0,1}_{1,2})_{\mu ij}(k,p)\,=\,
(\gamma_{\mu})_{ij}
$. (33) and the FE also tell us that all other
${\cal L}^{\Lambda,\Lambda_0,1}_{m,2n}$ vanish.
Thus (i),(ii) are true for $r=1$.\\
$\underline{r-1 \to r}$: \\
We assume to have verified the bound (i) for any
$m,n,z$ and $r' \le r-1$ for $r \ge 2$ and for
$r$ and all $m',n',z'$ with $m'+2n' >m+2n$
or with $m'+2n'=m+2n$ and $z'>z$.
We prove it now for $r$ and $(m,n,z)$ and start with \\
(a) $m+3n+z \ge 5$: As for $\Phi^4_4$ we may write an estimated FE
which is in shorthand notation (leaving out indices and
collecting all $\Lambda,\Lambda_0$-independent constants into one $c$)
\[
\|\partial^z \partial_{\Lambda}\,{\cal L}^{\Lambda,\Lambda_0,r}_{m,2n}\,
\|_{(2\Lambda,B)}
\, \le \,
c\,\{ \, \int_{\Lambda}^{2\Lambda}\, dt (\,
\|\partial^z \,
{\cal L}^{\Lambda,\Lambda_0,r}_{m+2,2n}\,\|_{(2\Lambda,B)}
\, + \,
t \, \|\partial^z \,
{\cal L}^{\Lambda,\Lambda_0,r}_{m,2n+2} \,\|_{(2\Lambda,B)})
\]
\eq
+ \, \sum \,
(\, \Lambda^{-3-z'''}
\|\partial^{z'} \,
{\cal L}^{\Lambda,\Lambda_0,r'}_{m',2n'}\,\|_{(2\Lambda,B)}
\, \, \,
\|\partial^{z''} \,
{\cal L}^{\Lambda,\Lambda_0,r''}_{m'',2n''}\,\|_{(2\Lambda,B)} \,)
\, + \,
\eqe
\[
+ \, \sum \,
(\, \Lambda^{-2-z'''}
\|\partial^{z'} \,
{\cal L}^{\Lambda,\Lambda_0,r'}_{m',2n'}\,\|_{(2\Lambda,B)}
\, \, \,
\|\partial^{z''} \,
{\cal L}^{\Lambda,\Lambda_0,r''}_{m'',2n''}\,\|_{(2\Lambda,B)}\,)
\,\}
\]
The sums are over the same values as in (25) and additionally over
all $z',z'',z''' \ge 0$ with $z'+z''+z''' = z$. We used (35)-(37).
The bound (i) then follows from (32) and on integration of (40) from
$\Lambda_0$ to $\Lambda$, since the r.h.s. of (40) is bounded using
(i), by induction. \\
(b) $m+3n+z \le 4$: \\
(b1) $m+3n+z = 4$: Use the r.c. (33), the FE and (a) to verify
\[
|\partial^w \,{\cal L}^{\Lambda,\Lambda_0,r}_{m,2n}(0) \,|
\, \le \,
\Lambda^{4-m-3n-|w|} Plog\Lambda\,,
\quad 1 \le \Lambda \le \Lambda_0
\]
for any choice of indices.\\
Once this has been achieved we may pass on to arbitrary momenta
using the
Schl\"omilch formula as in $\Phi^4_4$:
\eq
f(p) \,=\, f(0) \,+\, p_{\mu} \int_0^1 \,d\lambda \, \partial_{k_{\mu}}
f(k)\, , \;\, k= \lambda p.
\eqe
The integrated derivative has $m+3n+z=5$ and is thus already bounded
by induction for bounded momenta. So (i) can again be verified.\\
(b2) $m+3n+z = 3,\,m+3n+z = 2$ are then subsequently verified in the
same manner. Note that we have to proceed in this order to be able
to estimate the integrated derivative on the r.h.s. of (41) by
induction. \\
For (ii) we
do not give an explicit proof, but refer to $\Phi^4_4$.
The essential points are the following: \\
1. Differentiate both sides of the FE w.r.t. $\Lambda_0$ and write again
an estimated form of this equation corresponding to (40). \\
2. Use the same induction scheme as before to estimate the r.h.s.
of this estimated FE. \\
3. Use the $\Lambda_0$-independence of the r.c.
to realize that the boundary terms
$\partial_{\Lambda_0}\partial^w \,
{\cal L}^{1,\Lambda_0,r}_{m,2n}\,$ vanish for $m+3n+|w| \le
4$ at zero momentum. This is the important change as compared to
the proof of (i). (At $\Lambda = \Lambda_0$ we use as before the b.c.
(32)). Use again (41) to go away from zero momentum. From this it is
then straightforward to verify the bound (ii).
\qed
\\[.3cm]
Referring to earlier papers [3,6] we note in passing that a
statement like (ii) also holds if we soften the requirements of
$\Lambda_0$-independent r.c. and/or $\partial^w \,
{\cal L}^{\Lambda_0,\Lambda_0,r}_{m,2n}\,=\,0$ for $m+3n+|w| \ge 5$
to only requiring that these terms are suppressed by powers of
$\Lambda_0$ according to their power counting dimension.
This freedom may also be used to improve on the rate of convergence
in (ii), see [6].
\subsection{IR-Finiteness}
Now we turn to the IR part of the problem. Proposition 3 tells us
that for $\Lambda \ge 1$
the
${\cal L}^{\Lambda,\Lambda_0,r}_{m,2n}$
exist for $\Lambda_0 \to \infty$ and
for arbitrary indices and momenta bounded in modulus by $B$.
Looking at $0 < \Lambda \le 1$ we want to show that for suitable
b.c. the
${\cal L}^{\Lambda,\Lambda_0,r}_{m,2n}$
exist for $\Lambda \to 0$, if the external momenta are chosen
nonexceptional, i.e. no partial sums vanish.
We again proceed in analogy to $\Phi^4_4$ [14].
I.e. we first define an IR index $g$ for any configuration of $m$
photon and $2n$ fermion momenta, taylored such that we can prove
inductively with the help of the FE
\eq
|{\cal L}^{\Lambda,\Lambda_0,r}_{m,2n}(k,p)|
\le \Lambda^{-2g} Plog \Lambda^{-1} \,, \;\,
\mbox{if } \Lambda \to 0
\eqe
for any exceptional momentum configuration. Afterwards we can prove
finiteness for nonexceptional momenta. All momenta are from now
on supposed to be bounded by $B$. \\
The proof of a formula as (42) with the use of the FE can work
only if the IR indices of the momentum sets on the r.h.s. of the FE
obey sufficiently strong bounds in terms of the index of the momentum
set appearing on the l.h.s. We need the following definitions to
proceed in this direction:\\
\noindent
{\sl Definition 1}: A set of photon and fermion-antifermion momenta
$\{k_1,\ldots,k_m,p_1,\ldots p_{2n} \}$ denoted also as
$\{q_1,\ldots,q_{m+2n} \}$ \footnote{We regard $q_i$ and
$q_j (i \ne j)$ as different entities, even if $q_i=q_j$ as elements
of $ {\R}^4$, since they belong to different fields or external lines.
$q_i$ may be thought of as a mapping $i \mapsto q_i$, we do not
develop this point explicitly, however. }
is called admissible w.r.t. QED, if \\
$(i) \; m+3n>2 \quad (ii) \; m$ even, if $n=0$ $\quad
(iii) \;\sum_i k_i\,
+\,\sum_j p_j \,=\,0.$ \\
{\sl Definition 2}: An admissible momentum set (a.m.s.) $Q$ is called
exceptional, if there exists $Q_1 \subset Q $, $\emptyset \neq
Q_1 \neq Q$, such that $\sum_{Q_1} q_i \, =0$.
Otherwise it is called nonexceptional. $\sum _{Q_1}:=
\sum_{q_i\in Q_1}$.\\
{\sl Definition 3}: A partition $Z(Q)$ of an a.m.s. $Q$ is a system
of nonempty subsets
$E_{\nu} \subset_{\neq} Q $, $\nu = 1,\ldots ,N$
with
\[ (i) \quad Q= \bigcup^N_{\nu = 1}E_{\nu} \qquad
(ii) \quad E_{\nu} \cap
E_{\mu} =\emptyset,\; \mbox{ if }\; \nu \ne \mu \qquad
(iii) \quad \sum_{E_{\nu}}q_i =0 \]
$(iv) \quad E_{\nu}$
contains the same number of fermion and antifermion (fe-afe) momenta.
\\
For any partition $Z(Q)$ we define the subsets and numbers
\[
A(Z)\,=\,\{E_{\nu} \in Z |\; E_{\nu} \mbox{ consists of a single photon
momentum } \}, \; a :=\, |A| \,,
\]
\eq
B(Z)=\{ E_{\nu} \in Z |\; E_{\nu} \mbox{ consists of only $\ge2$ photon
momenta} \} \,,
\;b :=\, |B| \,,
\eqe
\[
D(Z)=Z \setminus (A(Z) \cup B(Z))\,, \; d=|D|\,.
\]
{\sl Definition 4}: The IR index $g_Z(Q)$ of a partition $Z(Q)$ is
defined as follows:
\[g_Z(Q) \,= \, \sup(0,\frac{a}{2}\,+b+\frac{3}{2}\,d-2) \]
The IR-index $g$ of an a.m.s. $Q$ is defined to be
\[g(Q) \,= \,0\,, \mbox{ if no $Z(Q)$ exists and }
\; g(Q)\,=\, \max_{Z(Q)} \; g_Z(Q) \mbox{ otherwise. }
\]
So in particular $g(Q)=0$, if $Q$ is nonexceptional.
As a motivation for Definition 4 note that by naive power
counting one photon contributes one power in the
IR cutoff to the IR singularity. This explains $\frac{a}{2}$, $b$.
The momenta in $D$ contribute more since they may flow across a
subdiagram
into one one-particle-reducible photon line and then contribute again
via this line. On inspecting examples one finds that a constant
($=2$) may be subtracted. Using Definition 5 below we will obtain better
IR bounds than with Definition 4 which is not optimal
in this respect, but this requires additional
effort.
For an a.m.s. $Q$ and pairs $\{ k,-k\}$, $\{p,-p\}$
of photon and fermion-antifermion momenta we finally define
the sets
\[
Q_A=\{ k,-k\} \cup Q, \quad Q_F=\{p,-p\} \cup Q. \]
And for
$\emptyset \ne Q_1 \subset_{\ne} Q, \quad Q_2 := Q\setminus Q_1$
we set
\eq
Q'_A =Q_1 \cup \{k' \},\quad Q'_F =Q_1 \cup \{p' \}, \quad
Q''_A =Q_2 \cup \{k'' \},\quad Q''_F =Q_2 \cup \{p''\}
\eqe
where the new momenta $k',k'',p',p''$ have values
\[
k'=-\sum_{Q_1}\,q_i\,,\; p'=-\sum_{Q_1}\,q_i\,,\;k''=-k'\,,\; p''=-p'.
\]
After these definitions we can now prove
\\
{\bf Lemma 4}: Let $Q$ be an a.m.s. Suppose
$Q_A,Q'_A,Q''_A,Q_F,Q'_F,Q''_F$ (44) are also a.m.s. Then
we have
\begin{itemize}
\item[(a1)] $\; g(Q_A) \le g(Q)+1, \;$ if all $q_i$ vanish or
-for $\sup |q_i| >0$- if $\; |k| \le \eta$,
where $ \eta >0$ is defined as
\eq
\eta (Q) := \frac{1}{2} \inf_J \,\eta_J,
\eqe
and the $\inf$ is over all sets $J$ with $J\subset_{\ne} \{1, \ldots
,m+2n\}$
such that $| \sum_{i \in J} q_i | =: \eta_J >0$.
\item[(a2)] $\;g(Q_A) \, \le \, g(Q)\,+ \, \frac{3}{2} \,, $
\item[(b1)] $\; g(Q'_A)\, +\, g(Q''_A)\,+ \,1 \,\le \,g(Q),\;$ if
$\; k'=0 \,,$
\item[(b2)] $\; g(Q'_A)\, +\, g(Q''_A)\, \le \, g(Q) \,,$
\item[(c)]
$\; g(Q_F) \, \le \, g(Q)\,+\, \frac{3}{2} \,,$
\item[(d)]
$\; g(Q'_F)\, +\, g(Q''_F) \,\le \,g(Q)\,.$
\end{itemize}
{\sl Remarks}: \\
In the proof we will denote the IR indices of $Q_A,Q_F,Q'_A, \ldots$
by $g_A,g_F,g'_A,\ldots $ and suitable partitions
of these sets maximizing $g_Z$ will be denoted by
$Z_A,Z_F,Z'_A,\ldots$. \\
{\sl Proof}: \\
(a1) $g_A \le 1$: trivial.$\quad g_A \ge 1$: \\
(1) Assume there exists $E \in Z_A$ with $E \supset \{k,-k\}$,
and set $Z:=(Z_A \setminus \{E\}) \cup \{ E \setminus \{k,-k\}\}$,
and verify that $g_Z(Q) \ge g_A-1$, whether $\{ E \setminus \{k,-k\}\}$
is empty or not. \\
(2) If $E$ as in (1) does not exist, then for suitable
$\nu ,\nu'$, $k\in E_{\nu}$, $-k \in E_{\nu'}$, and we set
$Z:=(Z_A \setminus \{E_{\nu},E_{\nu'} \} \cup \{ (E_{\nu}\setminus
\{k\}),(E_{\nu'}\setminus \{-k\}) \}$. Again $g_Z(Q) \ge g_A-1$,
whatever $E_{\nu}, E_{\nu'}$ are. Note that the sum over
the momenta in $E_{\nu} \setminus \{k\}$ still vanishes due to the
supplementary condition on $k$ which here implies $k=0$ . \\
(a2): The proof is as for (a1) except for the last case, where we
have to set
$Z:=(Z_A \setminus \{E_{\nu},E_{\nu'} \} \cup \{ (E_{\nu} \setminus
\{k\}) \cup (E_{\nu'} \setminus \{-k\})\}$ so that $g_Z(Q) \ge
g_A\,-\,3/2$.\\
(b1) If $g'_A,g''_A=0$, set $Z=\{Q_1,Q_2\}$ (44), so that $g_Z(Q)=1$.
Now observe that for any a.m.s. a partition $Z$ maximizing
$g_Z$ may always
be chosen such that $a=|A(Z)|$ is maximal. For the rest of the proof
we shall assume $a$ to be maximal in any maximizing partition to
appear.
Now if $g'_A >0, \,g''_A=0$ set
$Z=(Z'_A \setminus \{\{\ k'\} \}) \cup \{ Q_2 \}$, which gives
$g_Z=g'_A-1/2+3/2$. Finally for $g'_A,g''_A >0$ set
$Z:=(Z'_{A} \setminus \{\{\ k'\} \}) \cup
(Z''_{A} \setminus \{\{\ k''\} \})$ so that
$g_Z=g'_A-1/2+g''_A-1/2+2$. \\
(b2)
$g'_A,g''_A=0$ is trivial.
For $g'_A >0, \,g''_A=0$ take away from $Z'_A$ the set $E'$
containing $k'$ and replace it by
$(E' \setminus \{ k' \}) \cup \{ Q_2 \}$ to verify (b2).
For $g'_A,g''_A >0$ (b2) is verified on defining
$Z:=(Z'_{A} \setminus \{ E' \}) \cup
(Z''_{A} \setminus \{E''\} ) \cup \{ (E' \setminus \{ k' \}) \cup
(E'' \setminus \{ k'' \})\}$. Whatever $E',E''$ are, we even find
$g_Z \ge g'_A+g''_A+1/2$. \\
(c) One easily convinces oneself that a maximizing partition $Z_F$
of $Q_F$ may be chosen such that for some $\nu$: $E_{\nu}=\{p,-p\}$.
Then $g_{Z_F}(Q_F) \le g(Q)+3/2$ is obvious.\\
(d) The proof is the same as for (b2) replacing $k \to p, A \to F$.
\qed
\\
\noindent
The IR index of Definition 4 is slightly more crude than that of
[14], which however facilitated the proof of Lemma 4 considerably.
Since we want to show that all renormalizations may be performed at
zero momentum (for massive fermions) we need a somewhat sharper
version. \\
{\sl Definition 5}: For an a.m.s. $Q=\{k_1, \ldots ,k_m,p_1,
\ldots ,p_{2n} \}$ set
\eq
g_1(Q)\,= \, g(Q)-\frac{1}{2}, \quad \mbox{if}
\eqe
(i) $Q$ contains at most one fe-afe pair, and \\
(ii) $Q$ is such that $g(Q) >0$ and such that $g(Q)$ takes the maximal
value possible for the given number of photon and fe-afe momenta in $Q$.
\\
Otherwise set
\eq
g_1(Q)=g(Q).
\eqe
Now we can prove \\
{\bf Lemma 5}:
With the assumptions of Lemma 4 and the additional requirement that
none of the a.m.s.
$Q_A,Q'_A,Q''_A,Q$ appearing below consist of four photon momenta only
we have:
\begin{itemize}
\item[(a1)] $\; g_1(Q_A) \le g_1(Q)+1, \;$ if all $q_i$ vanish or
-for $\sup |q_i| >0$- if $\, |k| \le \eta \,,$
\item[(a2)] $\;
g_1(Q_A) \, \le \, g_1(Q)\,+ \, \frac{3}{2}\,,$
\item[(b1)] $\; g_1(Q'_A)\, +\, g_1(Q''_A)\,+ \,1 \,\le \,g_1(Q),\;$
if $
\; k'=0 \,,$
\item[(b2)] $\; g_1(Q'_A)\, +\, g_1(Q''_A)\, \le \, g_1(Q) \,,$
\item[(c)]
$\; g_1(Q_F) \, \le \, g_1(Q)\,+\, 2 \,,$
\item[(d1)]
$\; g_1(Q'_F)\, +\, g_1(Q''_F) \,\le \,g_1(Q),
\;$ if $p'=0 \,.$
\item[(d2)]
$\; g_1(Q'_F)\, +\, g_1(Q''_F) \,\le \,g_1(Q) \,+ \, 1/2
\,.$
\end{itemize}
\noindent
{\sl Proof}: The notation is as in the proof of Lemma 4.
We have only to look at the cases where $g_1(Q) < g(Q)$ for
the a.m.s. $Q$ appearing on the r.h.s.:\\
(a1),(a2): If $Q$ is such as in (46) (i),(ii), then $Q_A$ also
fulfills these conditions and (a) follows from Lemma 4.\\
(b1),(b2): Due to our restrictions on the sets $Q'_A,Q''_A,Q$ we
find $g_1(Q) \ge 3/2$ $(=3/2$ for the case of one fe-afe pair and 5
photons), if $g_1(Q) 0$ we only have to verify the case where
$g_1(Q) 0$ and a neighbourhood
\eq
U_{\varepsilon}(Q)=\{ \{ \hat{q}_1,\ldots ,\hat{q}_{m+2n} \} \,|\;
(q_i-\hat{q}_i)^2 \le
\varepsilon^2, \, 1 \le i \le m+2n, \, \sum_{i=1}^{m+2n} \hat{q}_i =0\},
\eqe
such that for any $\hat{Q}= \{ \hat{q}_1,\ldots ,\hat{q}_{m+2n} \}
\, \in U_{\varepsilon}(Q):
\; g(\hat{Q}) \le g(Q) $. \\
This holds since for all partitions of $Q$ all subsets $S \subset
_{\ne}Q$ which
are not an element of any $Z(Q)$ have $\sum_{q_i \in S}q_i \ne 0$.
Take $\varepsilon $ so small that all these inequalities still hold
in $ U_{\varepsilon}(Q)$.
The second is on the sets of nonexceptional momenta
$M_{m+2n}, \; $ as subsets of $\R ^{4(m+2n-1)}$: \\
\eq
M_{m+2n}:= \{ ( q_1, \ldots ,q_{m+2n-1} ) \in \R ^{4(m+2n-1)}|
\, \sum_{i \in
J}q_j
\ne 0 \mbox{ for all} \; J \subset_{\ne} \{1, \ldots ,m+2n\}\}
\eqe
(as usual $\; q_{m+2n}=-q_1- \ldots -q_{m+2n-1}$).
The sets $M_{m+2n}$ are obviously open in $\R^{4(m+2n-1)}$.
Now we prove \\
\noindent
{\bf Proposition 6}: Let $\Lambda \le 1 \le \Lambda_0 \le \infty $ and
$r \ge 1$. All (independent) momenta are assumed to be bounded by
$B>0$ (arbitrarily fixed).
\begin{itemize}
\item[(a)] The ( connected amputated ) renormalized Green functions
of the perturbative fermion photon theory,
defined through (32) and the renormalization conditions
(48), which are given as
\[ {\cal L}_{m,2n}^{\Lambda_0 ,r}(q_1, \ldots ,q_{m+2n-1})
:= \lim_{\Lambda \rightarrow 0}
{\cal L}_{m,2n}^{\Lambda , \Lambda_0, r}
(q_1, \ldots ,q_{m+2n-1}),
\]
in particular
$ {\cal L}_{m,2n}^r (q_1, \ldots ,q_{m+2n-1})
:=
{\cal L}_{m,2n}^{\infty,r }(q_1, \ldots ,q_{m+2n-1}),
$
exist in $C^{\infty}(M_{m+2n})$ (see (51)), and in
$C^{\infty}(M_{m+2n})$ we may interchange the limits:
\[
\lim_{\Lambda \to 0}\,\lim_{\Lambda_0 \to \infty}
{\cal L}_{m,2n}^{\Lambda , \Lambda_0, r} \,=\,
\lim_{\Lambda_0 \to \infty}\,\lim_{\Lambda \to 0}
{\cal L}_{m,2n}^{\Lambda , \Lambda_0, r} \,=\,
{\cal L}_{m,2n}^r
\,.
\]
Furthermore
${\cal L}_{2,0}^{\Lambda,\Lambda_0 ,r} \in C^2([0,\infty)\times
\R ^4)$,
${\cal L}_{4,0}^{\Lambda,\Lambda_0 ,r} \in C^0([0,\infty)\times\R^{12})$
and
${\cal L}_{0,2}^{\Lambda,\Lambda_0 ,r} \in C^1([0,\infty) \times \R ^4)$,
${\cal L}_{1,2}^{\Lambda,\Lambda_0 ,r} \in C^0 ([0,\infty) \times \R ^8 )$
as functions of $\Lambda$ and the (independent) momenta.
\item[(b)] Let $Q=\{q_1, \ldots ,q_{m+2n} \}$ be an a.m.s. (Def.1).
\item[(b1)] If $Q$ is nonexceptional or- for $m=2,n=0 $, if $q_1 \ne 0$ -
we have
\[
\partial^w {\cal L}_{m,2n}^{\Lambda_0 ,r}(q_1, \ldots ,q_{m+2n-1}) =
\lim_{\Lambda \rightarrow 0}
\partial^w {\cal L}_{m,2n}^{\Lambda, \Lambda_0 ,r }(q_1, \ldots ,q_{m+2n-1})
\]
uniformly in $U_{\varepsilon}(Q)$.
\item[(b2)] Assume the a.m.s. $Q$ is such that $Q=Q_a \cup Q_b$,
where $Q_a \ne \emptyset $ and $Q_b$ has the form
$Q_b= \{ q_1^{(b)}, -q_1^{(b)}, \ldots , q_l^{(b)},-q_l^{(b)} \}$,
and such that for any $E \subset Q $ with $\sum_{E} q_i = 0$
either $Q_a \subset E$ or $Q_a \cap E = \emptyset $ .
Let $Q_a = \{q_1, \ldots ,q_s \}$; we denote by $Q_{a_1}$ some
(arbitrary) subset of $s-1$ momenta of $Q_a$ and by $\partial^w_{a_1}$
any sequence of $|w|$ derivatives w.r.t. to momenta in $Q_{a_1}$.
Finally we denote by $N_{\varepsilon}(Q_b)$ the set
$\{ \hat{q}_1^{(b)}, -\hat{q}_1^{(b)}, \ldots , \hat{q}_l^{(b)},-
\hat{q}_l^{(b)} \}$ with $|\hat{q}_i^{(b)}-q_i^{(b)}|<\varepsilon$.
Then we claim
\eq
|\partial^w_{a_1} {\cal L}^{\Lambda , \Lambda_0 ,r}_{m,2n}
(\hat{Q}_b, \hat{Q}_{a_1})| \le \Lambda^{-2g_1(Q)} Plog \Lambda^{-1}
\,.
\eqe
The statement is uniform in $U_{\varepsilon}(Q_a) \cup
N_{\varepsilon}(Q_b)$. The constants in $Plog$ depend on
$\varepsilon, Q, r, m, n,
|w|$.
The notation in (52) is slightly abusive in that it requires that we
parametrize ${\cal L}^{\Lambda , \Lambda_0 ,r}_{m,2n}
$ possibly in terms of momenta differing from the standard
choice (24), and in different order.
\item[(b3)] For a general a.m.s. $Q$ we have (for $\Lambda >0)$
\eq
|\partial^w {\cal L}^{\Lambda ,\Lambda_0 ,r}_{m,2n}(\hat{q}_1, \ldots ,
\hat{q}_{m+2n-1})| \le
\Lambda^{-2g_1(Q)-|w|} Plog \Lambda^{-1}
\eqe
uniformly for $\hat{Q} \in U_{\varepsilon}(Q)$,
the constants in $Plog$ depend on $\varepsilon, Q, r, m, n,
|w|$.
\item[(c)]
For $m+3n+|w| \le 4 $ we obtain the bounds (for given $r$ and
$\Lambda >0$)
\eq
|\partial^w {\cal L}^{\Lambda ,\Lambda_0 ,r}_{2,0}(k)|
\le
\Lambda^{3-|w|} Plog \Lambda^{-1} \,, \quad
|\partial^w {\cal L}^{\Lambda ,\Lambda_0 ,r}_{4,0}(k_1,k_2,k_3)|
\le
\Lambda^{1-|w|} Plog \Lambda^{-1}\,,
\eqe
and for the r.c.(49) and $r \ge 2$ (second inequ.) also
\eq
|\partial^w {\cal L}^{\Lambda ,\Lambda_0 ,r}_{0,2}(p)| \le
\Lambda^{2-|w|} Plog \Lambda^{-1} \,, \quad
|\partial^w {\cal L}^{\Lambda ,\Lambda_0 ,r}_{1,2}(k,p)|
\le
\Lambda^{1-|w|} Plog \Lambda^{-1} \,.
\eqe
Those statements in (54),(55)
for which the r.h.s. vanishes for $\Lambda \to 0$
hold uniformly only for $|k|,|k_i|,|p| \le \Lambda$ (or
$\,O(1)\Lambda)$,
the others hold uniformly for $|k|,|k_i|,|p| \le B$.
In case of the r.c.(48) the statement (55)
holds only for $|w| \ge 2$ (first inequ.) resp.
$|w| \ge 1$ (second inequ.).
\end{itemize}
All statements in (b),(c) are uniform in $\Lambda_0$.
\\
{\sl Remarks }: We left and leave out indices on the
${\cal L}^{\Lambda,\Lambda_0}_{m,2n}(q_1, \ldots ,q_{m+2n-1}) $
whenever possible, and we abbreviate (in slightly abusive
shorthand notation) ${\cal L}^{\Lambda,\Lambda_0}_{m,2n}(Q) =
{\cal L}^{\Lambda,\Lambda_0}_{m,2n}(q_1, \ldots ,q_{m+2n-1}) $
etc.
For a given momentum set $Q=\{q_1, \ldots ,q_{m+2n} \}$
we denote by $\hat{Q}=\{ \hat{q}_1, \ldots ,\hat{q}_{m+2n} \}$
a momentum set such that $\{ \hat{q}_1, \ldots ,\hat{q}_{m+2n} \}$
is in
$U_{\varepsilon}(Q)$ and by $Q_A(k)$ or shortly $Q_A$
the set $\{k,-k,q_1, \ldots ,q_{m+2n} \}$ etc. (cf. Lemma 4).
The symbol $\varepsilon$ will always denote a positive number,
chosen sufficiently small case per case ( we do not introduce
$\varepsilon ',\varepsilon '', \ldots $) such that the respective
estimate holds uniformly in $U_{\varepsilon}(\ldots )$.
$\,\varepsilon$ depends in particular on the respective $\eta(Q)$(45).
$c,c_1, \ldots$ denote positive $\Lambda,\Lambda_0$-independent
constants.
The proof of Proposition 6
is in many aspects analogous to that of Theorem 1 in [14].
Here we are slightly shorter.
\\
{\sl Proof:}
We use the standard FE induction scheme which proceeds upwards
in $r$ and for given $r$ downwards in $l=m+3n$ using (i) after
(24) (see also Prop.3). \\
(A) $r=1 \; $: The b.c. (32) and the r.c. (48) give vanishing
${\cal L}^{\Lambda ,\Lambda_0 ,1}_{m,2n}$ apart from
${\cal L}^{\Lambda ,\Lambda_0 ,1}_{1,2}$ (see also Proposition 3).
The r.h.s. of the FE for $\partial_{\Lambda} {\cal L}^{\Lambda ,\Lambda_0
,1}_{1,2} $ vanishes identically in $\Lambda ,\Lambda_0 ,k\, ,p$.
Thus
\[
({\cal L}^{\Lambda ,\Lambda_0 ,1}_{1,2})_{\mu ij} (k,p) =
(\gamma_{\mu})_{ij} \,,\;
{\cal L}^{\Lambda ,\Lambda_0 ,1}_{m,2n} \equiv
0, \, m \neq 1 \, \mbox{or} \, n \neq 1 \, .
\]
So the proposition is true for $r=1$.
\\
(B) $r>1$: We assume the proposition to be true
for $r', \, l'$ with $r'l$. We prove it now for $r,l$.
We start with proving (b),(a) for \\
(B1) $m+3n+|w| \ge 5, \, m+3n >2$: First we prove \\
(b3): Bounding the ${\cal L}'$s on the r.h.s. of the FE
for $\partial_{\Lambda} {\cal L}^{\Lambda ,\Lambda_0
,r}_{m,2n} $ with the aid of the induction hypothesis
and using Lemma 5 c),a) we obtain
\[
|\partial^w {\cal L}^{\Lambda ,\Lambda_0 ,r}_{m,2n+2} (p,-p, \hat Q
)| \, \le \, \Lambda^{-2 g_1(Q) -4-|w|} \, Plog \Lambda^{-1}\, ,
\]
\[
|\partial^w {\cal L}^{\Lambda ,\Lambda_0 ,r}_{m+2,2n} (k,-k, \hat Q
)| \, \le \, \Lambda^{-2 g_1(Q) -2-|w|} \, Plog \Lambda^{-1}\, ,
\]
if $|k| \le \eta(Q) $ or if all momenta vanish, and generally
\[
|\partial^w {\cal L}^{\Lambda ,\Lambda_0 ,r}_{m+2,2n} (k,-k, \hat Q
)| \, \le \, \Lambda^{-2 g_1(Q) -3-|w|} \, Plog \Lambda^{-1}\, .
\]
The sums on the r.h.s. of (25) can be bounded - using Lemma 5 b),d),
(4),(39) and the induction assumption - by $\; \Lambda^{-2 g_1(Q) -1-|w|}
\, Plog \Lambda^{-1}\,$.
Here we note that in the cases where Lemma 5 b1)
cannot be applied the corresponding contributions
vanish for $\Lambda < 1/2\,\eta(Q)$ due to (4),
since $|k'|,|p'| \ge 2 \eta -
O(\varepsilon)$ in $\hat Q$. For
$\Lambda \ge 1/2\,\eta (Q)$ they can be absorbed in the constants of
$Plog$. All previous bounds are by induction assumption uniform
in the respective $U_{\varepsilon}$'s. By a standard compactness argument
the first two thus hold uniformly in
$\{(p,-p) \;| \; |p| \le \eta \} \, \times \, U_{\varepsilon}(Q)$
respectively in
$\{(k,-k) \;| \; |k| \le \eta \} \, \times \, U_{\varepsilon}(Q)$.
>From (1)-(4),(25) we thus obtain in $U_{\varepsilon}(Q)$
\[
|\partial^w \partial_{\Lambda}
{\cal L}^{\Lambda ,\Lambda_0 ,r}_{m,2n} (\hat Q
)| \, \le \,
\int_{\Lambda}^{2\Lambda} dt\; t^3 [ t^{-2g_1 -4 -|w|} Plog t^{-1}
\,+ \,
t^{-3-2g_1-2-|w|} Plogt^{-1} \, + \,
\]
\[
+\, \theta (t -\eta ) \,
\eta^{-3-2g_1 -3 -|w|} \, Plog \eta^{-1}] \, +
\, \Lambda ^{-2g_1 -1 -|w|} Plog\Lambda^{-1 } \; \le \;\,
\Lambda ^{-2g_1-1 -|w|} Plog\Lambda^{-1 }.
\]
Here $g_1= g_1(Q), \, \eta = \eta(Q)$.
Integrating now from 1 to $\Lambda < 1$ shows
\eq
|\partial^w {\cal L}^{\Lambda ,\Lambda_0 ,r}_{m,2n} (\hat Q
)| \, \le \,
\Lambda ^{-2g_1 -|w|} Plog\Lambda^{-1 } \,
+ \, |\partial^w {\cal L}^{1 ,\Lambda_0 ,r}_{m,2n} (\hat Q )|.
\eqe
The last term is independent of $\Lambda$ and uniformly bounded
in $\Lambda_0$ by Proposition 3. So it may be absorbed in the
first. Proposition 3 was proven for r.c. imposed at
$\Lambda = 1$. Now we impose them at $\Lambda = 0$. So we have
to show that both classes of r.c. are in one-to-one relation.
That this is true indeed can be seen from the FE when integrating
from $\Lambda =1$ to $\Lambda=0$. Imposing the r.c. at $\Lambda=0$
one then finds that ${\cal L}^{1 ,\Lambda_0 ,r}_{m,2n}$ fulfill
r.c. at $\Lambda=1$ of the form (33).
The way of proceeding can also be inferred from (B2) below.
Strictly speaking this
uniqueness statement is also part of the induction hypothesis.\\
(b2) If $Q$ is as in (b2), then
$ Q_F = (Q_b \cup \{p,-p\}) \cup Q_a, \; Q_A =
(Q_b \cup \{k,-k\}) \cup Q_a$ also fulfill the assumptions
of (b2), if $0< |p|,|k| \le \eta (Q)$.
The important point to note is that if the intermediate
momenta $p' ,k'$ appearing on the r.h.s. of the
FE fulfill $|p'|,|k'| \le \eta (Q)$ then
the derivatives
$\partial^w_{a_1}$ ($w\neq 0$)
applied to $k'$ (and $p'$) give zero by our assumptions.
(Note that $k',p'$ need not vanish in this case if the external momenta
are taken in $U_{\varepsilon}(Q_a) \cup
N_{\varepsilon}(Q_b)$.)
Using these facts the verification of (b2) proceeds as that of (b3).
We again have to use Lemma 5 and the induction assumption to bound
separately the regions where $\Lambda >\eta/2$ and $\Lambda \le
\eta/2$, and we also have to use again the compactness argument from
the proof of (b3). \\
(b1) For Q nonexceptional, the sets
$\hat Q_F := \{p,-p\} \cup \hat Q, \; \hat Q_A :=
\{k,-k\} \cup \hat Q ,\, 0< |p|,|k| \le \eta (Q)$, fulfill
the assumptions of (b2), furthermore the momenta $k',p'$
appearing on the r.h.s. of (25) fulfill $|k'|,|p'| \ge \eta (Q)$
in $\hat Q$. This implies as in (b2) (since $g_1(Q_F)$, $g_1(Q_A)$
$=0$)
\[
|\partial^w \partial_{\Lambda}
{\cal L}^{\Lambda ,\Lambda_0 ,r}_{m,2n} (\hat Q)|
\, \le \,
Plog \Lambda^{-1}.
\]
Integrating from 1 to $\Lambda$ proves the
existence of $\lim_{\Lambda \to 0}
\partial^w \partial_{\Lambda}
{\cal L}^{\Lambda ,\Lambda_0 ,r}_{m,2n} (\hat Q)$
uniformly in $U_{\varepsilon}(Q)$ and therefore (b1). \\
(a) now follows (for $m+3n+|w| \ge 5, \; m+3n > 2)$
from the proof of (b1), since $|w|$ may take any (finite)
value. In
particular we may interchange the limits using a
standard $\varepsilon /4$-argument for:
\[
{\cal L}^{\delta \to 0, \infty} \,-\,
{\cal L}^{0,\Lambda_0 \to \infty}\,=\,
{\cal L}^{\delta \to 0, \infty}\,-\,
{\cal L}^{\delta, \infty}\,+\,
{\cal L}^{\delta, \infty}\,-\,
{\cal L}^{\delta,\Lambda_0} \,+\,
{\cal L}^{\delta,\Lambda_0}\,-\,
{\cal L}^{0,\Lambda_0}\,+\,
{\cal L}^{0,\Lambda_0}\,-\,
{\cal L}^{0,\infty}
\]
and our knowledge on the IR and UV-limits. \\
(B2): Now we prove (b3),(b2),(b1),(a)
for $ m+3n+|w| \le 4$ or $m+3n =2$ as well as (54),(55). We start
from $m+3n=4$,$m=4$: The r.c. fix the value of
${\cal L}^{0 ,\Lambda_0 ,r}_{4,0} (0)$, which is in one-to-one
relation to ${\cal L}^{1 ,\Lambda_0 ,r}_{4,0} (0)$ through
\[
{\cal L}^{1 ,\Lambda_0 ,r}_{4,0} (0) =
{\cal L}^{0 ,\Lambda_0 ,r}_{4,0} (0) \, +\,
\int_0^1 \,dt \, \partial_t {\cal L}^{t ,\Lambda_0 ,r}_{4,0} (0)
\]
because the integrand is independent of the r.c. for
${\cal L}^{0 ,\Lambda_0 ,r}_{4,0} (0)$ to order $r$:
It is given by the r.h.s. of the FE.
Noting that for $m+3n=4$ and any $Q$ we have $g_1(Q)=0,\,g_1(Q_A) \le
1/2,\, g_1(Q_F) \le 1$ we find by induction, also using (c) to
lower order
\[
\partial^w \partial_{\Lambda}
{\cal L}^{\Lambda ,\Lambda_0 ,r}_{4,0} (\hat Q) \;
\le \; \Lambda^{-|w|} Plog \Lambda^{-1}
\]
uniformly in $U_{\varepsilon} (Q)$, including the case where all $q_i$
vanish.\\
Using a compactness argument and integrating over $\Lambda$
we then deduce
\[
{\cal L}_{4,0}^{0,\Lambda_0 ,r} \in C^0(\R^{12})\,,\;
\mbox{ and } \;
{\cal L}_{4,0}^{\Lambda ,\Lambda_0 ,r}(Q) =
{\cal L}_{4,0}^{0,\Lambda_0 ,r}(Q) \, +\,
O(\Lambda \, Plog \Lambda^{-1})
\]
uniformly in $Q=\{ \{ q_1,q_2,q_3 \}\,|\; |q_i| \le B \}$.
The statements in (a),(b1),(c) follow from the previous estimates
and (B1) (where $|w| \ge 1$ is included), the uniformity of the limit
$\Lambda \to 0$ and the Schl\"omilch formula together with the r.c.
The treatment of ${\cal L}_{1,2}^{\Lambda,\Lambda_0 ,r}$
in case of the r.c. (49)
is analogous
to that of ${\cal L}_{4,0}^{0,\Lambda_0 ,r}$
and we do not repeat the argument.
In case of the r.c.(48) ${\cal L}_{1,2}^{\Lambda,\Lambda_0 ,r}(0,0)$
may be nonzero for $\Lambda \to 0$.
But the regularity properties in (a),(c) are verified as before
using in particular the fact that $R_m$ vanishes for $\Lambda < m/2$
to bound the $\sum$-terms in the FE.
Note that the statements on ${\cal L}_{1,2}^{\Lambda,\Lambda_0 ,r}$
for the r.c. (48)
could not be verified with our method
if we regularized the fermions in the same
way as the photons using $R$ instead of $R_m$.
Now we come to $m=2,n=0$.
The statements on ${\cal L}_{2,0}$
in (a),(b1),(c) are again proven by bounding the r.h.s. of the
FE by induction for any $k$. We obtain
\eq
|\partial^w \partial_{\Lambda}{\cal L}_{2,0}^{\Lambda ,\Lambda_0
,r}(k) |\;
\le \,
\Lambda^{2-|w|} \, Plog \Lambda^{-1}\,.
\eqe
The bounds are as usually uniform in the respective
$U_{\varepsilon} \,$. (57) for $|w| \le 1$
only holds for $|k| \le \, c \,\Lambda$.
Integration over $\Lambda$ and the r.c. - or for $m+|w| \ge 5$
the b.c. at $\Lambda = 1$ (or at $\Lambda = \Lambda_0$, cf. the remark
in (B1))- the usual uniformity and compactness arguments and Taylor
expansion around zero momentum then provide the estimates in (c) and
the statements of (a). The last case to treat is $m=0,n=1$.
We again have to distinguish between the r.c. (49) and (48).
But the way of proceeding is as previously for ${\cal L}_{1,2}$.
To verify the
statements we again need the regularity of $R_m$ around 0.
\qed
We have seen in the end of the previous proof that our techniques
really require different regularizations $R,R_m$ to prove the
Proposition. Using $R $ throughout we can only prove results as
sketched in sect.5 which also hold in massless QED.
It is a straightforward exercise
to show that replacing $R_m$ by different smoothed versions of $R$
(see e.g.[5]) produces the same results on taking the limits
(one estimates the difference of the two regularized versions).
\section{ Violation and restoration of the Ward Identities }
We start with a few introductory remarks forgetting about
regularization, $\Lambda,\; \Lambda_0$ etc.
The standard QED Ward Identity (WI)
may be expressed in terms of the generating functional $Z$ (11),(14)
as
\eq
\{\lambda \Box \partial_{\mu} \delta_{J_{\mu}(x)} \,-\,ie \eta
\delta_{\eta(x)} \,+\,ie \etab \delta_{\etab(x)} \,+\,
\partial_{\mu} J_{\mu}(x) \} Z(J,\etab,\eta) \,=\,0
\eqe
In terms of $W$ with $Z=e^{-(W+f.i.)}$ we obtain
\eq
\{-\lambda \Box \partial_{\mu} \delta_{J_{\mu}(x)} \,+\,ie \eta
\delta_{\eta(x)} \,-\,ie \etab \delta_{\etab(x)} \}
W(J,\etab,\eta) \,=\,- \partial_{\mu} J_{\mu}(x)
\eqe
or
\eq
(\delta_{\chi(x)}W(J_{\mu}-\partial_{\nu} (D^{-1})_{\nu \mu} \chi
\,,\,\etab e^{-ie\chi}\,,\,e^{ie\chi}\eta))|_{\chi=0}\,=\,
- \partial_{\mu} J_{\mu}(x)
\eqe
$D^{-1}$ is the inverse photon propagator and
$\chi$ decribes the gauge transformations, we assume $\chi \in
{\cal S}(\R^4)$:
\eq
A_{\mu} \to A_{\mu}+\partial_{\mu} \chi, \; \psi \to
e^{-ie\chi}\psi, \; \psib \to e^{ie\chi}\psib\,.
\eqe
Now we look at the regularized theory (see (11),(20)).
For safeness we keep $0<\Lambda \le \Lambda_0 < \infty$.
Due to the violation of gauge invariance implied by the momentum
cutoff the WI's will also be violated. (60) leads us to define
\eq
J_{\mu}(\chi)\,=\,J_{\mu}-\partial_{\nu} (D^{-1})_{\nu \mu} \chi
,\; \etab(\chi)\,=\, \etab\, e^{-ie\chi},
\;\eta (\chi) \,=\,e^{ie\chi}\eta
\eqe
(position space arguments are suppressed), and we set
\eq
Z_{\Lambda}^{\Lambda_0} (J,\etab, \eta;\chi) \,= \,
Z_{\Lambda}^{\Lambda_0} (J(\chi),\etab(\chi), \eta(\chi))\,,\;
W^{\Lambda,\Lambda_0}_{\chi} (J,\etab, \eta) \,= \,
W^{\Lambda,\Lambda_0} (J(\chi),\etab(\chi), \eta(\chi))
\eqe
\eq
L^{\Lambda_0}_{\chi} (\delta_J,-\delta_{\eta},\delta_{\etab})
\,=\,
L^{\Lambda_0}(\delta_{J(\chi)},-\delta_{\eta(\chi)},\delta_{\etab(\chi})
\,=\,
L^{\Lambda_0}(\delta_J,-e^{-ie\chi}\delta_{\eta},
e^{ie\chi}\delta_{\etab})
\eqe
so that
\eq
Z_{\Lambda}^{\Lambda_0} (J,\etab, \eta;\chi) \,= \,
e^{-(W^{\Lambda,\Lambda_0}_{\chi} (J,\etab, \eta) \,+ \,f.i.(\chi))}
\,= \,
e^{-L^{\Lambda_0}_{\chi} (\delta_J,-\delta_{\eta},\delta_{\etab})
}\;e^{\frac{1}{2} \,+\,
<\etab(\chi),S^{\Lambda_0}_{\Lambda}\,\eta(\chi)>} \,.
\eqe
In deriving the violated WI's (vWI's) we are only interested in
contributions of first order in $\chi$.
We find
\eq
L^{\Lambda_0}_{\chi} (\delta_J,-\delta_{\eta},\delta_{\etab})
\,=\,
L^{\Lambda_0}\,+\,
e \,z_2<\chi, \partial_{\mu} \delta_{\eta} \,\gamma^{\mu}\,
\delta_{\etab})>
\mbox{ (exactly)}
\eqe
\eq
1/2\, \,=\,
1/2\, \,+\,
<\chi,\partial D^{-1}\,D^{\Lambda_0}_{\Lambda}\,J> \,+\,O(\chi^2)\,,
\eqe
\eq
\,<\etab(\chi),S^{\Lambda_0}_{\Lambda}\,\eta(\chi)> \,=\,
<\etab,S^{\Lambda_0}_{\Lambda}\,\eta> \,+\,
ie(<\etab,S^{\Lambda_0}_{\Lambda}\,\chi \,\eta> \,-\,
<\etab,\chi \,S^{\Lambda_0}_{\Lambda}\,\eta>) \,+\,
O(\chi^2)\,.
\eqe
We thus may rewrite (65) to first order in $\chi$
\[
e^{-(W^{\Lambda,\Lambda_0}_0 (J,\etab, \eta) \,+ \,f.i.(0))}
\,(-<\chi,\delta_{\chi}(W^{\Lambda,\Lambda_0}_{\chi}\,+\,f.i.)|_{\chi=0}>)
\,=\,
\]
\eq
e^{-L^{\Lambda_0}_0}[\,-e\,z_2 \,<\chi,
\partial_{\mu} \delta_{\eta} \,\gamma^{\mu}\,
\delta_{\etab})>
\,+\,
<\chi,\partial D^{-1}\,D^{\Lambda_0}_{\Lambda}\,J> \,-\,
\eqe
\[
-ie\,\{<\etab,\chi \,\delta_{\etab}> \,+\,
<\delta_{\eta},\chi \,\eta>' \,\}]
\;
e^{1/2\, \,+\,<\etab,S^{\Lambda_0}_{\Lambda}\,
\eta>} \,,
\]
where $<\delta_{\eta},\chi \,\eta>'$ means that we subtract the
contribution where $\delta_{\eta}$ applies to the $\eta$ in
$<\ldots>$.
Now we find
\[
[\,L^{\Lambda_0}_0 (\delta_J,-\delta_{\eta},\delta_{\etab}),\,
<\chi,\partial D^{-1}\,D^{\Lambda_0}_{\Lambda} \,J>\,]
\,=\,
\]
\eq
<\chi,(\partial D^{-1}\,D^{\Lambda_0}_{\Lambda})_{\mu} \,
(\,\delta_{\mu \nu}(-z_3 \Box \,+\,\delta \mu^2)\,+\,
(z_3 \,-\,\delta \lambda)
\partial_{\mu}\partial_{\nu}\,) \,\delta_{J_{\nu}}\,> \,-\,
\eqe
\[
-\,4z_4\, f_{\mu \nu \rho \sigma} \,
<\chi, (\partial D^{-1}\,D^{\Lambda_0}_{\Lambda})_{\mu}
\,\delta_{J_{\nu}}\,\delta_{J_{\rho}}\,\delta_{J_{\sigma}}\,>
\,+\,
e\,(1+z_1)\,
<\chi, (\partial D^{-1}\,D^{\Lambda_0}_{\Lambda})_{\mu}
\,\delta_{\eta}\,\gamma^{\mu}\,\delta_{\etab}\,>
.\]
Note that the commutator commutes with $L^{\Lambda_0}_0$.
For the last term we write
\eq
<\chi, \partial_{\mu} R^{\Lambda_0}_{\Lambda}
\,\delta_{\eta}\,\gamma^{\mu}\,\delta_{\etab}>
=
<\chi, \partial_{\mu}
\,\delta_{\eta}\,\gamma^{\mu}\,\delta_{\etab}>
+
<\chi, (\partial D^{-1}\,D^{\Lambda_0}_{\Lambda}
-\partial)_{\mu} \, \delta_{\eta}\,\gamma^{\mu}\,\delta_{\etab}>
\,,
\eqe
where $R^{\Lambda_0}_{a,\Lambda}(p)\,:=\,
R_a(\Lambda_0,p)-R_a(\Lambda,p)$,$R^{\Lambda_0}_{\Lambda}:=
R^{\Lambda_0}_{0,\Lambda}$.
Taking the first term on the r.h.s.of (71)
together with the two terms in curly
brackets in (69) we find on application of derivatives
\eq
[\,e\,<\chi, \partial_{\mu}
\,\delta_{\eta}\,\gamma^{\mu}\,\delta_{\etab}\,>
\,-\,
ie\{\,
<\etab,\chi \,\delta_{\etab}> \,+\,
<\delta_{\eta},\chi \,\eta>' \,\}]
\;
e^{<\etab,S^{\Lambda_0}_{\Lambda}\, \eta>} \,=\,
\eqe
\[
=\, ie\,\{<\etab,(R^{\Lambda_0}_{m,\Lambda}\,-\,1) \,\chi\, \delta_{\etab}\,>
\,+\,
<\delta_{\eta},\chi \,(R^{\Lambda_0}_{m,\Lambda}\,-\,1)\,\eta>' \,\}
\; e^{<\etab,S^{\Lambda_0}_{\Lambda}\, \eta>} \,,
\]
where we used
$S\,/\!\!\! \partial \,=\, -i1 \, +\,imS ,\;/\!\!\!\partial S \,=\, i1 \,
-\,imS$.
Using (70)-(72) in (69) and commuting $e^{-L^{\Lambda_0}_0}$
through (70) (trivial), and the curly brackets in (72), finally gives
\[
e^{-(W^{\Lambda,\Lambda_0}_0 (J,\etab, \eta) \,+ \,f.i.(0))}
\;[-<\chi,\delta_{\chi}(W_{\chi}\,+\,f.i.)|_{\chi=0}>
\,-\,
<\chi,\partial R^{\Lambda_0}_{\Lambda}\,J> \,+\,
\]
\eq
+\,ie
\{\,<\etab,r^{\Lambda_0}_{m,\Lambda}\,\chi \,\delta_{\etab}> \,+\,
<\delta_{\eta},\chi \,r^{\Lambda_0}_{m,\Lambda}\,\eta>' \,\}
W^{\Lambda,\Lambda_0}_0] \,=\,
\eqe
\[
=\, \{\,e^{-L^{\Lambda_0}_0\,+\,<\chi, O^{\Lambda_0}_{\Lambda}>}\,
\,
e^{(1/2\, \,
+\,<\etab,S^{\Lambda_0}_{\Lambda}\,\eta>)} \,
\}|_{1^{st}\mbox{ order in } \chi}
\,,
\]
with the following explanations:\\
\begin{itemize}
\item[1.] $r^{\Lambda_0}_{a,\Lambda}(p)=R^{\Lambda_0}_{a,\Lambda}(p)-1\;$,
$r^{\Lambda_0}_{a,\Lambda}(x)=\int \frac{d^4p}{(2\pi)^4}e^{ipx}\;
r^{\Lambda_0}_{a,\Lambda}(p)$.
\item[2.] $O^{\Lambda_0}_{\Lambda}$
collects the outcome of the commutators. In (73)
it carries the arguments \\
$O^{\Lambda_0}_{\Lambda} (\delta_J,
-\delta_{\eta},\delta_{\etab})$ (as $L^{\Lambda_0}_0$).
It has the form
\[
O^{\Lambda_0}_{\Lambda} (A,\psib,\psi)(x)\,=
\,e\,z_2 \,(\partial_{\mu}\psib \gamma^{\mu}\psi)(x) \,-\,
\,e\,z_1\,[\, \partial_{\mu} R^{\Lambda_0}_{\Lambda} \,
\psib \gamma^{\mu}\psi](x)
\]
\[
-\, e\,(\, \partial_{\mu} r^{\Lambda_0}_{\Lambda} \,
\psib \gamma^{\mu}\psi)(x) \,-
\, [\, \partial_{\mu} R^{\Lambda_0}_{\Lambda} \,
(\,\delta^{\mu \nu}(-z_3 \Box \,+\,\delta \mu^2)\,+\,
(z_3 \,-\,\delta \lambda)
\partial^{\mu} \partial^{\nu}\,) \,A_{\nu}\,](x)
\]
\eq
-\,4z_4 \, f_{\mu \nu \rho \sigma} \,
[ \partial_{\mu} R^{\Lambda_0}_{\Lambda}
\,A_{\nu}\,A_{\rho}\,A_{\sigma}\,](x)
\eqe
\[
+\,ie\,\{
iz_2\, \int\,d^4z \,[ \psib(z) /\!\!\!\partial_z
\, r^{\Lambda_0}_{m,\Lambda}(z-x)\,\psi(x)\,-\,
\psib(x) \,
r^{\Lambda_0}_{m,\Lambda}(x-z)
\, /\!\!\!\partial_z
\, \psi(z) ]
\]
\[
-\,\delta m \, \int\,d^4z \,[ \psib(z)
\, r^{\Lambda_0}_{m,\Lambda}(z-x)\,\psi(x)\,-\,
\psib(x) \,
r^{\Lambda_0}_{m,\Lambda}(x-z)
\, \psi(z) ]
\]
\[
-\,e(1+z_1)
\, \int\,d^4z \,[ \psib(z) /\!\!\!\!A(z)
\, r^{\Lambda_0}_{m,\Lambda}(z-x)\,\psi(x)\,-\,
\psib(x)
r^{\Lambda_0}_{m,\Lambda}(x-z)
\, /\!\!\!\!A(z)
\, \psi(z) ] \}
\]
The curly brackets correspond to the contribution of those in (72), on
commuting through as in (70).
\item[3.] The term $<\chi,O>$ has been raised to the exponent. This
is
allowed since we regard only the first order in $\chi$.
\end{itemize}
As can be seen from (74), all terms in $O$ vanish, if we formally let
$\Lambda\to 0,\Lambda_0 \to \infty$ and require
$z_1=z_2,\, z_4=0,\delta \lambda=0,\, \delta \mu^2=0$.
Thus $O^{\Lambda_0}_{\Lambda}$ collects the gauge symmetry violating
contributions, and (73) has been derived to control them, again using
flow equations. (73) is to be understood as usually in the sense of
perturbation theory, i.e. as a relation between $r$-th order terms of
the perturbative expansion. Furthermore we are dealing only with the
coefficient functions for a given number of external
$J,\eta ,\etab$ or $\chi$-fields. The coefficient functions
${\cal L}^{\Lambda,\Lambda_0,r}_{m,2n}$ exist for any positive
value of $\Lambda$ at arbitrary momenta. The same is then obviously
true for the ${\cal W}^{\Lambda,\Lambda_0,r}_{m,2n}$ (see below)
which appear on the l.h.s. of (73). Eliminating the exponential in
(73) we may write this equation as
\[
-<\chi,(\delta_{\chi}W^{\Lambda,\Lambda_0}_{\chi})|_{\chi=0}>
\,-\,
<\chi,\partial R^{\Lambda_0}_{\Lambda}\,J> \,+\,
\]
\eq
+\,ie
\{\,<\etab,r^{\Lambda_0}_{m,\Lambda}\,\chi \,\delta_{\etab}> \,+\,
<\delta_{\eta},\chi \,r^{\Lambda_0}_{m,\Lambda}\,\eta>' \,\}
W^{\Lambda,\Lambda_0}_{(0)} \,=\,
-W^{\Lambda,\Lambda_0}_{(1)}\,.
\eqe
Here $W^{\Lambda,\Lambda_0}_{(1)}$
is defined as follows. We set
\eq
e^{-(W^{\Lambda,\Lambda_0}_O\,+\,f.i.(\chi))}\,:=\,
e^{-L^{\Lambda_0}_0 +<\chi,O^{\Lambda_0}_{\Lambda}>}\,
\,
e^{(1/2\, \,+\,<\etab,S^{\Lambda_0}_{\Lambda}\,
\eta>)} \,.
\eqe
Thus $W_O$ is the generating functional of the connected unamputated
Green functions with insertions of $O$ and we expand
\eq
W^{\Lambda,\Lambda_0}_O\,=\,
W^{\Lambda,\Lambda_0}_{(0)}\,+\,W^{\Lambda,\Lambda_0}_{(1)}\,+\,
O(\chi^2)\,.
\eqe
Therefore $W_{(1)}$ generates those with one $O$-insertion.
In the same way as for the $L^{\Lambda,\Lambda_0}$ we expand
\eq
W^{\Lambda,\Lambda_0}_{(0)}\,=\,
\sum_{r\ge 0} \,e^r \,W^{\Lambda,\Lambda_0,r}\,,\,
W^{\Lambda,\Lambda_0}_{(1)}\,=\,
\sum_{r\ge 0} \,e^r \,W^{\Lambda,\Lambda_0,r}_{(1)}\,,\,
\eqe
\eq
W^{\Lambda,\Lambda_0,r}\,=\,
\sum_{m+2n >0 }\,W^{\Lambda,\Lambda_0,r}_{m,2n}\,,\,
W^{\Lambda,\Lambda_0,r}_{(1)}\,=\,
\sum_{m+2n >0 }\,W^{\Lambda,\Lambda_0,r}_{m,2n,1}
\eqe
and finally
\eq
W^{\Lambda,\Lambda_0,r}_{m,2n,1}
\,=\,
\,\int \,
\frac{d^4q}{(2\pi)^4}
\frac{d^4k_1}{(2\pi)^4}
\ldots
\frac{d^4p_{2n-1}}{(2\pi)^4}
\;({\cal W}^{\Lambda,\Lambda_0,r}_{m,2n,1})(q,k_1,\ldots k_m,p_1,\ldots
p_{2n-1}) \, \times
\eqe
\[
\times \,
\chi(q) \, J(k_1) \ldots J(k_m)
\etab(p_1) \ldots \ldots \eta(p_{2n})\,
,
\]
and similarly for ${\cal W}^{\Lambda,\Lambda_0,r}_{m,2n}$.
For shortness we left out indices.
The ${\cal W}'s$ are assumed to have the
same symmetry and antisymmetry properties as the
${\cal L}'s$.
We know already from ch.3 that the ${\cal W}^{\Lambda,\Lambda_0,r}_{m,2n}$
are finite in the limit $\Lambda \to 0$ for those momentum configurations
for which the ${\cal L}^{\Lambda,\Lambda_0,r}_{m,2n}$ are finite in the
same limit {\sl and} for which no external photon momentum vanishes,
since the external lines are no more amputated in $W$.
\\
With these definitions we get from (75) (remembering (62), (63))
for $r \ge 1$:
\[
(m+1)\,iq_{\rho} D^{-1}_{\rho \mu}(q)
{\cal W}^{\Lambda,\Lambda_0,r}_{m+1,2n}(q,k_1,\ldots k_m,p_1,\ldots
p_{2n-1})_{\mu \mu_1 \ldots j_n} \,+\,
\]
\[
+\,i\, \sum_{a=1}^n \, \{ \, R^{\Lambda_0}_{m,\Lambda}(p_a)
{\cal W}^{\Lambda,\Lambda_0,r-1}_{m,2n}(k_1,\ldots k_m,p_1,\ldots
,p_a+q,\ldots ,p_{2n-1})_{\mu_1 \ldots j_n} \,-\,
\]
\[
-\,
R^{\Lambda_0}_{m,\Lambda}(p_{n+a})
{\cal W}^{\Lambda,\Lambda_0,r-1}_{m,2n}(k_1,\ldots k_m,p_1,\ldots
,p_{n+a}+q,\ldots ,p_{2n-1})_{\mu_1 \ldots j_n} \,\} \,+
\]
\eq
+\, {\cal W}^{\Lambda,\Lambda_0,r}_{m,2n,1}(q,k_1,\ldots k_m,p_1,\ldots
,p_{2n-1})_{\mu_1 \ldots j_n} \,=\,0
\eqe
In the derivation of (81) we used $1+r^{\Lambda_0}_{m,\Lambda}=
R^{\Lambda_0}_{m,\Lambda}$.
(81) for $r=0$ is
realized to be trivially fulfilled.
Since $R^{\Lambda_0}_{m,\Lambda}$
has a well-defined limit for $\Lambda \to 0$,
IR-finiteness of
${\cal W}^{\Lambda,\Lambda_0,r}_{m,2n,1}$ for $\Lambda \to 0$ may
be inferred from that of ${\cal W}^{\Lambda,\Lambda_0,r}_{m,2n}$
(see also Proposition 7). We can pass from unamputated to
amputated quantities.
We define the generating functional of the UV- and IR-
regularized CAG with $O^{\Lambda_0}_{\Lambda}$-insertions
in analogy with (11), (14), (20), (76) as
\eq
e^{1/2\,+
<\etab,S^{\Lambda_0}_{\Lambda}\,\eta>} \,
\exp\{-(L^{\Lambda,\Lambda_0}_{<\chi,O^{\Lambda_0}_{\Lambda}>}
(DJ,S{\eta},{\etab}S)\,+\, f.i.)\}
\eqe
\[
\,:=\,
\exp\{-(L^{\Lambda_0}_{<\chi,O^{\Lambda_0}_{\Lambda}>}
(\delta_J,-\delta_{\eta},\delta_{\etab})\} \,\,
e^{1/2\,+
<\etab,S^{\Lambda_0}_{\Lambda}\,\eta>}
\]
with
\eq
L^{\Lambda_0}_{<\chi,O^{\Lambda_0}_{\Lambda}>} \,:= \,
L^{\Lambda_0} \,-\, <\chi,O^{\Lambda_0}_{\Lambda}>.
\eqe
>From these definitions we will be able to derive a FE for the
$L_{m,2n,1}$ defined as in (78)-(80) with $W \to L$,
${\cal W} \to {\cal L}$. The aim to arrive at such a FE
was the reason for raising $<\chi,O^{\Lambda_0}_{\Lambda}>$ to the exponent
in (73), (76).
The vWI's (81) in terms
of the ${\cal L}_{m,2n},\, {\cal L}_{m,2n,1}$ take the form for
$r\ge 2$:
\[
i(m+1)\,q_{\mu} \, R^{\Lambda_0}_{\Lambda}(q) \,
{\cal L}^{\Lambda,\Lambda_0,r}_{m+1,2n}(q,k_1,\ldots k_m,p_1,\ldots
p_{2n-1})_{\mu \mu_1 \ldots j_n} \,+\,
\]
\[
+\,i\, \sum_{a=1}^n \, \{ \,
((S(-p_a))^{-1}
\,S^{\Lambda_0}_{\Lambda}(-p_a-q))_{i_a i'_a} \,
{\cal L}^{\Lambda,\Lambda_0,r-1}_{m,2n}(k_1,\ldots k_m,p_1,\ldots
,p_a+q,\ldots ,p_{2n-1})_{\mu_1 \ldots i'_a \ldots j_n}
\]
\[
-\, {\cal L}^{\Lambda,\Lambda_0,r-1}_{m,2n}(k_1,\ldots k_m,p_1,\ldots
,p_{n+a}+q,\ldots ,p_{2n-1})_{\mu_1 \ldots j'_a \ldots j_n}
(S^{\Lambda_0}_{\Lambda}(p_{n+a}+q)
\,(S(p_{n+a}))^{-1})_{j'_a j_a} \,
\}
\]
\eq
+\,{\cal L}^{\Lambda,\Lambda_0,r}_{m,2n,1}(q,k_1,\ldots k_m,p_1,\ldots
,p_{2n-1})_{\mu_1 \ldots j_n} \,=\,0 \,.
\eqe
The unviolated WI's, which we want to recover for
$\Lambda \to 0$, $\Lambda_0 \to
\infty$ are obtained from (84) on replacing
$R^{\Lambda_0}_{\Lambda}$
by 1, $S^{\Lambda_0}_{\Lambda}$ by $S$ and
${\cal L}^{\Lambda,\Lambda_0,r}_{m,2n,1}$ by 0.
The first two replacements are true for $\Lambda \to 0$,
$\Lambda_0 \to \infty$
and finite nonvanishing momenta. That the last is also true on taking limits
is a consequence of the FE
for ${\cal L}^{\Lambda,\Lambda_0,r}_{m,2n,1}$
and the boundary conditions at $\Lambda = \Lambda_0$
(see (83),(74)) and at $\Lambda=0$. The latter will follow from
(84) and the r.c. for the ${\cal L}^{\Lambda,\Lambda_0,r}_{m,2n}$.
FE's for Green functions with operator insertions have been studied
extensively in [4] in the $\Phi^4_4$-context. For the present case
they have already been presented in [13]. In the same way as
in Proposition 2 we find
\eq
\exp\{-(L^{\Lambda,\Lambda_0}_{<\chi,O^{\Lambda_0}_{\Lambda}>}
(A,\psib ,\psi) \,+\,f.i.)\} \,=\,
e^{\triangle (\Lambda,\Lambda_0)}\,
\exp\{-(L^{\Lambda_0}_{<\chi,O^{\Lambda_0}_{\Lambda}>}
(A,\psib ,\psi))\}
\eqe
and
\eq
\exp\{-(L^{\Lambda_0}_{<\chi,O^{\Lambda_0}_{\Lambda}>}
(\delta_J,-\delta_{\eta},\delta_{\etab})\} \,\,
e^{1/2\,} \,\,
e^{<\etab,S^{\Lambda_0}_{\Lambda}\,\eta>} \,=\,
\eqe
\[
e^{1/2\,} \,\,
e^{<\etab,S^{\Lambda_0}_{\Lambda}\,\eta>}
\exp\{-(L^{\Lambda,\Lambda_0}_{<\chi,O^{\Lambda_0}_{\Lambda}>}
(A,\psib ,\psi) \,+\,f.i.)\}|_{A=D^{\Lambda_0}_{\Lambda}J,\,
\psi=S^{\Lambda_0}_{\Lambda}\eta,\, \psib=\etab
S^{\Lambda_0}_{\Lambda}}
\]
for the generating functional
of the regularized CAG with $O^{\Lambda_0}_{\Lambda}$-insertions. The FE is
obtained as (22) by taking a $\Lambda$-derivative on both sides
(replace $L$ by$L_{<\chi,O^{\Lambda_0}_{\Lambda}>}$ in (22)).
If we were to apply the $\Lambda$-derivative also to the
$\Lambda$-dependent term $O^{\Lambda_0}_{\Lambda}$ we would obtain
a much more complicated equation than (22), however.
We therefore fix the $\Lambda$-parameter
in $O^{\Lambda_0}_{\Lambda}$ to be equal to $\delta:\;
O^{\Lambda_0}_{\Lambda} \to O^{\Lambda_0}_{\delta}$. We choose
$0<\delta<}
(A,\psib ,\psi) \,+\,f.i.)\} \,=\,
e^{\triangle (\Lambda,\Lambda_0)}\,
\exp\{-(L^{\Lambda_0}_{<\chi,O^{\Lambda_0}_{\delta}>}
(A,\psib ,\psi))\}
\]
Expanding in powers of $e$
and of the external fields $A,\psib,\psi$ we obtain to zeroth
order in $\chi$ the FE (25) and for the first order terms we find
\[
\partial_{\Lambda}\,L^{\delta,\Lambda,\Lambda_0,r}_{m,2n,1} \,=\,
(\partial_{\Lambda}\, \triangle') \,
L^{\delta,\Lambda,\Lambda_0,r}_{m,2n+2,1} \,+\,
(\partial_{\Lambda}\, \triangle'') \,
L^{\delta,\Lambda,\Lambda_0,r}_{m+2,2n,1} \,+\,
\]
\eq
\sum_1 \,\int \, \frac{d^4k_0}{(2\pi)^4} \,\frac{1}{2} \,D_{\mu \nu}(k_0)
\,
(\partial_{\Lambda}\,R(\Lambda,k_0))\,
(\delta_{A_{\mu}(k_0)}\,
L^{\delta,\Lambda,\Lambda_0,r'}_{m',2n',s'}) \,\,
(\delta_{A_{\nu}(-k_0)}\,
L^{\delta,\Lambda,\Lambda_0,r''}_{m'',2n'',s''}) \,-\,
\eqe
\[
\sum_2 \int \frac{d^4p_0}{(2\pi)^4} \,\,S_{i j}(p_0)
\,
(\partial_{\Lambda}\,R_m(\Lambda,p_0))\,
(\delta_{\psi_i(p_0)}\,
L^{\delta,\Lambda,\Lambda_0,r'}_{m',2n',s'}) \,\,
(\delta_{\psib_j(-p_0)}\,
L^{\delta,\Lambda,\Lambda_0,r''}_{m'',2n'',s''}) \,\,,
\]
where $\sum_1$ is over $r'+r''=r,m'+m''=m+2,n'+n''=n,s'+s''=1$
and $\sum_2$ is over $r'+r''=r,m'+m''=m,n'+n''=n+1,s'+s''=1$
and we denote $L_{m,2n,0}=L_{m,2n}$. The equation analogous
to (25) is then
\[
\partial_{\Lambda}\,({\cal L}^{\delta,\Lambda,\Lambda_0,r}_{m,2n,1})
_{\mu_1,\ldots j_n} (q,k_1,\ldots ,p_{2n-1})\,=\,
\]
\[
=\,-{m+1 \choose 2} \,\int \frac{d^4k}{(2\pi)^4} \,\frac{1}{2}
\,D_{\mu \nu}(k)
\,
(\partial_{\Lambda}\,R(\Lambda,k))\,
({\cal L}^{\delta,\Lambda,\Lambda_0,r}_{m+2,2n,1})
_{\mu \nu \mu_1,\ldots j_n} (q,k,-k,k_1,\ldots ,p_{2n-1})\,+\,
\]
\[
+\, (-1)^n (n+1)^2 \,
\int \frac{d^4p}{(2\pi)^4} \,S_{ji}(p)
\, \times
\]
\[
\times \,
(\partial_{\Lambda}\,R_m(\Lambda,p))\,
({\cal L}^{\delta,\Lambda,\Lambda_0,r}_{m,2n+2,1})
_{\mu_1\ldots ii_1\ldots i_nj \ldots j_n} (q,k_1,\ldots
,k_m,-p,p_1,\ldots,p_n,p,\ldots,p_{2n-1})\,+\,
\]
\eq
\sum_1\,m'm''(-1)^{n'n''} [
\,(\partial_{\Lambda}\,R(\Lambda,k'))
\, D_{\mu \nu}(k')
\,
({\cal L}^{\Lambda,\Lambda_0,r'}_{m',2n'})
_{\mu \mu_1\ldots j_{n'}} (k',k_1,\ldots,k_{m'-1},p_1,
\ldots ,p_{n+n'-1})\,\times\,
\eqe
\[
({\cal L}^{\delta,\Lambda,\Lambda_0,r''}_{m'',2n'',1})
_{\nu \mu_{m'}\ldots j_{n}} (q,-k',k_{m'},\ldots
,p_{2n-1})\,]_{SAS}\,+\,
\sum_2\,(-1)^{n'n''+n''} n'n''\,[
\,(\partial_{\Lambda}\,R_m(\Lambda,p'))
\, S_{ji}(p')
\, \times
\]
\[ \times \,
({\cal L}^{\Lambda,\Lambda_0,r'}_{m',2n'})
_{\mu_1\ldots \mu_{m'}i_1 \ldots i_{n'}j j_1\ldots j_{n'}}
(k_1,\ldots,k_{m'},p_1,\ldots, p_{n'},p', \ldots ,p_{n+n'-1})\,\times\,
\]
\[
\times \,
({\cal L}^{\delta,\Lambda,\Lambda_0,r''}_{m'',2n'',1})
_{\mu_{m'+1}\ldots \mu_mii_{n'+1} \ldots i_n j_{n'}\ldots j_n}
(q,k_{m'+1},\ldots ,k_m,-p',p_{n'+1},\ldots, p_n,p_{n+n'},
\ldots ,p_{2n-1})\,]_{SAS}
\]
\[
+\, \sum_2\,(-1)^{n'n''+n''} n'n'' \,[
\,(\partial_{\Lambda}\,R_m(\Lambda,p'))
\, S_{ji}(p')
\, \times
\]
\[
\times \,
({\cal L}^{\delta,\Lambda,\Lambda_0,r'}_{m',2n',1})
_{\mu_1\ldots \mu_{m'}i_1 \ldots i_{n'}j j_1\ldots j_{n'}}
(q,k_1,\ldots,k_{m'},p_1,\ldots, p_{n'},p', \ldots ,p_{n+n'-1})\,\times\,
\]
\[
\times \,
({\cal L}^{\Lambda,\Lambda_0,r''}_{m'',2n''})
_{\mu_{m'+1}\ldots \mu_mii_{n'+1} \ldots i_n j_{n'}\ldots j_n}
(k_{m'+1},\ldots ,k_m,-p',p_{n'+1},\ldots, p_n,p_{n+n'},
\ldots ,p_{2n-1})\,]_{SAS}\,.
\]
$\sum_1,\sum_2$ are defined as in (87) with the exception that we do
not sum over $s',s''$. Otherwise the notation corresponds to that of
(25). \\
To bound the solutions of (88) we have again to look at the b.c.
For $\Lambda= \Lambda_0$ they follow from (83), (74) (with
$\Lambda=\delta$ in (83),(74)). We find at \\
$\underline{\Lambda= \Lambda_0}$ for $r\ge 1$:
\[
({\cal L}^{\delta,\Lambda_0,\Lambda_0,r}_{1,0,1})_{\mu}(q) \,=\,
-iq_{\mu}\,R^{\Lambda_0}_{\delta}(q) \,(q^2 \delta \lambda_r \,+\, \delta
\mu^2_r)
\]
\[
({\cal L}^{\delta,\Lambda_0,\Lambda_0,r}_{3,0,1})_{\nu \rho
\sigma}(q,k_1,k_2) \,=\,
-4iq_{\mu}\,R^{\Lambda_0}_{\delta}(q)\,f_{\mu \nu \rho
\sigma} \,z_{4r} \,, \quad (q=-k_1-k_2-k_3)
\]
\eq
({\cal L}^{\delta,\Lambda_0,\Lambda_0,r}_{0,2,1})_{ij}(q,p)\,=\,
i/\!\!\!q\,[z_{2,r-1}\,-\,R^{\Lambda_0}_{\delta}(q)\, z_{1,r-1} \,+\,
\delta_{1r}\, r^{\Lambda_0}_{\delta}(q)\;] \,+\,
\eqe
\[
+\,iz_{2,r-1}\,(-/\!\!\!p\,r^{\Lambda_0}_{m,\delta}(p) \,+\,
(/\!\!\!q\,+\,/\!\!\!p)\,r^{\Lambda_0}_{m,\delta}(q+p))\,+\,
i\delta m_{r-1} \,(r^{\Lambda_0}_{m,\delta}(p)\,-\,r^{\Lambda_0}_
{m,\delta}(q+p))
\]
\[
({\cal L}^{\delta,\Lambda_0,\Lambda_0,r}_{1,2,1})_{\mu ij}(q,k,p)\,=\,
i(\delta_{r,2} \,+\, z_{1,r-2})\,(r^{\Lambda_0}_{m,\delta}(p)
\,-\,r^{\Lambda_0}_{m,\delta}(q+p)) \,.
\]
Remember $r^{\Lambda_0}_{a,\delta}(p)\,=\,r^{\Lambda_0}_{a,\delta}(-p),
\;\; z_{is}:=0$ for
$s\le 1$ and note that $r^{\Lambda_0}_{m,\delta}(p)=
R_m(\Lambda_0,p)-1$ since $\delta }$. Using
also the invariance of the theory without insertions this implies
\begin{itemize}
\item[(i)] ${\cal L}^{\delta,\delta,\Lambda_0,r}_{m,0,1}
\,\equiv \,0$ for $m$
even (C)
\item[(ii)] $({\cal L}^{\delta,\delta,\Lambda_0,r}_{1,0,1})_{\mu}(0)
\,=\,0$ ( $O(4)$
)
\item[(iii)] $\partial_{\mu}
({\cal L}^{\delta,\delta,\Lambda_0,r}_{1,0,1})_{\nu}(0) \,=\,
c^{\delta}_{1r} \delta_{\mu \nu}$ ( $O(4)$ )
\item[(iv)] $\partial_{\mu} \partial_{\nu}
({\cal L}^{\delta,\delta,\Lambda_0,r}_{1,0,1})_{\rho}(0) \,=\,
0$ ( $O(4)$ )
\eq \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \mbox{(v)}\;\;
\partial_{\mu} \partial_{\nu} \partial_{\rho}
({\cal L}^{\delta,\delta,\Lambda_0,r}_{1,0,1})_{\sigma}(0) \,=\,
c^{\delta}_{2r} f_{\mu \nu \rho \sigma} \;\;
\quad (O(4) \mbox{ and permutation symmetry for }
\mu, \nu, \rho)
\eqe
\item[(vi)] $({\cal L}^{\delta,\delta,\Lambda_0,r}_{0,2,1})_{ij}(0) \,=\,0$ ( $O(4)$
and C )
\item[(vii)] $\partial_{q_{\mu}}
({\cal L}^{\delta,\delta,\Lambda_0,r}_{0,2,1})_{ij}(0) \,=\,c^{\delta}_{3r} \,
(\gamma_{\mu})_{ij} \,=\,
-\,\partial_{p_{\mu}}
({\cal L}^{\delta,\delta,\Lambda_0,r}_{0,2,1})_{ij}(0)
$ ( $O(4)$
and C )
\item[(viii)]
$({\cal L}^{\delta,\delta,\Lambda_0,r}_{3,0,1})_{\mu \nu \rho}
(0) \,=\,
0$ ( $O(4)$ and Bose symmetry )
\item[(ix)]
$\partial_{q_{\mu}}({\cal L}^{\delta,\delta,\Lambda_0,r}_{3,0,1})
_{\nu \rho \sigma}
(0) \,=\, c^{\delta}_{4r} \, f_{\mu \nu \rho \sigma} \,=\,
-\,\partial_{k_{j,\mu}}({\cal L}^{\delta,\delta,\Lambda_0,r}_{3,0,1})
_{\nu \rho \sigma}
(0)$
\item[(x)]
$({\cal L}^{\delta,\delta,\Lambda_0,r}_{1,2,1})_{\mu ij}(0)\,=\,0
\;\;(\,C\,)\,.$
\end{itemize}
In the derivation of (ix) it was not sufficient to use the
symmetries, which also allow for a term
$c'^{\delta}_{4r}(\delta_{\mu \nu}\,\delta_{\rho \sigma} \,+\,
\delta_{\mu \rho }\delta_{\nu \sigma}
\,-\,\delta_{\mu \sigma} \, \delta_{\nu \rho })$.
This term is excluded however using the vWI (84) which gives
\eq
({\cal L}^{\delta,\delta,\Lambda_0,r}_{3,0,1})_{\nu \rho \sigma}
(q,k_1,k_2) \,=\, -4i\,q_{\mu}
R^{\Lambda_0}_{\delta}(q)
({\cal L}^{\delta,\Lambda_0,r}_{4,0})_{\mu \nu \rho \sigma}
(q,k_1,k_2), \;\;
|q|,|k_i| < \Lambda_0\;.
\eqe
We remark that the existence of the derivatives in
(92) for $\delta = 0$ can be inferred from (84).
For $|q|$ between $\delta$ and $2\delta$
the functions $\partial^w R^{\Lambda_0}_{\delta}(q)$
change rapidly and have large derivatives
$\sim O(\delta^{-|w|})$.
The task of finding suitable bounds for the ${\cal
L}^{\delta,\Lambda,\Lambda_0,r}_{m,2n,1}$
is therefore considerably simplified by restricting
to the region $ |q|>2\delta$ (see below). This is compatible
with our way of inductively estimating the solutions of the FE
since $q$ appears as a fixed parameter on both sides of (88).
Such restricted bounds are sufficient for our purposes since
we let $\delta \to 0$ in the end.
In this case we also have to use boundary conditions for the
FE in which the value of $q$ fulfills $|q| >2\delta$.
To be definite we choose some
$q$ with $|q|=3\delta$, and the second momentum argument
appearing in ${\cal L}^{\delta,\delta,\Lambda_0,r}_{m,2n,1}$
is then chosen as $-q$, the others being 0.
Using (93) and (48) as well as (54) we find
\eq
|({\cal L}^{\delta,\delta,\Lambda_0,r}_{3,0,1})_{\mu\nu\rho}(q,-q,0)|\le
\delta^2 \,Plog \delta^{-1}\,,
|\partial_{q_{\mu}}
({\cal
L}^{\delta,\delta,\Lambda_0,r}_{3,0,1})_{\nu\rho\sigma}(q,-q,0)
-(-4i\, z^R_{4r}\, f_{\mu\nu\rho\sigma})| \le
\delta \,Plog \delta^{-1}\,.
\eqe
Using again the vWI (84) we also find from (48) and (54)
\eq
|({\cal L}^{\delta,\delta,\Lambda_0,r}_{1,0,1})_{\mu}(q)|
\le \,\delta^4 \,Plog \delta^{-1}\,, \;
|\partial_{\mu}
({\cal L}^{\delta,\delta,\Lambda_0,r}_{1,0,1})_{\nu}(q)
-(-i\delta \mu^{2R}_r \,\delta_{\mu \nu})| \le
\delta^3 \,Plog \delta^{-1}\,,
\eqe
\[
|\partial_{\mu} \partial_{\nu}
({\cal L}^{\delta,\delta,\Lambda_0,r}_{1,0,1})_{\rho}(q)| \le \,
\delta^2 \,Plog \delta^{-1}\,, \;
|\partial_{\mu} \partial_{\nu} \partial_{\rho}
({\cal L}^{\delta,\delta,\Lambda_0,r}_{1,0,1})_{\sigma}(q) \,-\,
(-6i\delta\lambda^R_r \,f_{\mu \nu \rho \sigma}) |\le \;
\delta \,Plog \delta^{-1}\,.
\]
For $m=0,n=1,r\ge 2$ (84) takes the form
\eq
({\cal L}^{\delta,\delta,\Lambda_0,r}_{0,2,1})_{ij}(q,-q) \,=\,
-iq_{\mu}
({\cal L}^{\delta,\delta,\Lambda_0,r}_{1,2})_{\mu ij}(q,-q) \,+\,
\eqe
\[
(-i) [(\frac{/\!\!\!q+m)}{m}
({\cal L}^{\delta,\Lambda_0,r-1}_{0,2}(0))_{ij} \,-\,
({\cal L}^{\delta,\Lambda_0,r-1}_{0,2})(-q)\,
\frac{m}{/\!\!\! q+m})_{ij} ] \,,
\]
where we used the fact that all $R^{\Lambda_0}_{a,\delta}$'s
equal 1 for the chosen momentum arguments.
Using the continuity of ${\cal L}_{1,2}$
and the continuous differentiability of
${\cal L}_{0,2}$
we obtain from (96)
\eq
|({\cal L}^{\delta,\delta,\Lambda_0,r}_{0,2,1})_{ij}(q,-q)| \le \,
\delta^2 Plog \delta^{-1}\,,
\eqe
\[
|\partial_{q_{\mu}}
({\cal L}^{\delta,\delta,\Lambda_0,r}_{0,2,1})_{ij}(q,-q)
-\{-iz^R_{1,r-1}-i[-z^R_{2,r-1} +\frac{2}{m}\delta m_{r-1}^R]
\}(\gamma_{\mu})_{ij}| \le \,
\delta Plog \delta^{-1}\,.
\]
For $r=1$ one realizes on going back to (81)
that ${\cal W}^{\delta,\delta,\Lambda_0,1}_{0,2,1}(q,p)=0$
for $|q|,|p| \le \Lambda_0/2$
which implies ${\cal L}^{\delta,\delta,\Lambda_0,1}_{0,2,1}(q,p)=0$
for $|q|,|p| \le \Lambda_0/2$. All other
${\cal L}^{\delta,\delta,\Lambda_0,1}_{m,2n,1}$
also vanish by inspection of the vWI's (trivially).
Similar considerations finally show
\eq
|({\cal L}^{\delta,\delta,\Lambda_0,r}_{1,2,1})_{\mu ij}(q,-q,0)|
\le \,\delta \, Plog \delta^{-1}\,.
\eqe
(93),(94),(95),(98) now tell us
that all terms of dimension $\le 4$ which do not vanish a priori by
symmetry, are explicitly given in terms of the renormalization
constants from (48) -
up to corrections bounded by $\delta \, Plog \delta^{-1}$.
The aforementioned restriction on the values of $q$
is implemented by adapting
the norms (34), with the aid of which we estimated the
solutions of the FE, to the present situation:
\\
For a system a functions $f_{\mu_1,\ldots,i_1,\ldots,j_1,\ldots}
(q,k_1,\ldots,p_1,\ldots,p_{2n-1})$ depending on $q$
and on photon momenta $k_1,\ldots,k_m$
and fermion and antifermion momenta $p_1,\ldots,p_{2n}$ we define
\[
||\partial^{|w|}f ||^{\delta}_{(a,b)}
\]
as in (34) with the additional restriction that we only
take the $\sup$ over the momenta fulfilling
$|q|>2 \delta $.
\\
We also need an adaptation of Def.1-5 in sect.3.2.
The following changes are necessary:
$Q$ is now the set $\{q,k_1,\ldots,p_1,\ldots,p_{2n}\}$.
In Def.1 we replace (ii) by: (ii') $m$ odd if $n=0$.
In Def.3 one of the sets $E_{\nu}$ now contains $q$.
This set is then counted as if it were $E_{\nu}\setminus\{q\}$
in all subsequent definitions, and we find immediately
that Lemmas 4, 5 may be restated for the new situation without
change. Now we prove:\\
\noindent
{\bf Proposition 7}:
Let $B>0$ be any fixed constant, $|w| \le 4$, $0< \delta < 2\delta$, $\delta \le \Lambda \le \Lambda_0 < \infty$.
All subsequent estimates are uniform in $\delta$ and $\Lambda_0$.
For the r.c. (48) together with the following restriction
\eq
z_{1,r}^R = z_{2,r}^R - \frac{2}{m} \delta m^R_{r}, \; r\ge 1,
\eqe
in particular for the r.c.(49),
we obtain the following bounds for $\Lambda \ge 1$
\eq
\|\partial^{|w|} {\cal L}_{m,2n,1}^{\delta,\Lambda, \Lambda_0 ,r}
\|^{\delta}_{(2\Lambda,B)} \le \,
(\frac{\Lambda}{\Lambda_0} \,Plog \Lambda_0
\,+\, \delta \,Plog\delta^{-1})\,
\Lambda^{4-m-3n-|w|} \,.
\eqe
The constants in $Plog$ depend on $m,n,r,|w|,B$.
\\
Now let $\delta \le \Lambda \le 1$ and all momenta $q,k_1, \ldots ,p_{2n-1}$
be bounded by $B$. We find for nonexceptional momentum configurations $Q$
( or for $\{q,-q\}$ if $m=1,n=0$ )
\eq
|\partial^w {\cal L}_{m,2n,1}^{\delta,\Lambda, \Lambda_0 ,r}
(q,k_1, \ldots ,p_{2n-1})| \le \, \Lambda_0^{-1} \, Plog\Lambda_0
\,+\, \delta\,Plog \delta^{-1}\,.
\eqe
The constants in $Plog$ now also depend on $\eta$ (45),
but (101) holds uniformly in $U_{\varepsilon}(\{q, \ldots ,
p_{2n} \})$ (see Proposition 6).
\\
Furthermore we find in the same sense as in Proposition 6
for exceptional momentum sets $Q \,=\,\{q,k_1, \ldots ,
p_{2n} \}$
\eq
|\partial^w {\cal L}_{m,2n,1}^{\delta,\Lambda, \Lambda_0 ,r}
(\hat{q},\hat{k}_1, \ldots ,\hat{p}_{2n-1})|
\le \,\Lambda^{-2g_1(Q)-|w|}\,(\Lambda^{-1}_0 \,Plog\Lambda_0
\,+\, \delta\,Plog \delta^{-1})
\eqe
uniformly in $U_{\varepsilon}(Q)$.
The statement analogous to (b2) in Proposition 6 is the following:
Let
$Q\,=\,\{q,k_1, \ldots ,p_{2n} \}$ be such that
$Q= Q_a \cup Q_b$ and $q\in Q_a$,
where $Q_b$ is of the form $Q_b = \{q_1^{(b)} ,-q_1^{(b)}, \ldots ,
q_l^{(b)},
-q_l^{(b)} \}$. And let $Q_a$ be such that for any
$E \subset Q$ with $\sum_E q_i =0$ either $Q_a \subset E$
or $E \cap Q_a = \emptyset$.
Write $Q_a = \{q,q_1, \ldots ,q_s \}$; we denote by $Q_{a_1}$ any
strict subset of $Q_a$
and by $\partial^w_{a_1}$
any sequence of $w$ derivatives w.r.t. to $q_i \in Q_{a_1}$.
Then we claim
\eq
|\partial^w_{a_1} {\cal L}^{\delta,\Lambda , \Lambda_0 ,r}_{m,2n,1}
(Q_b, \hat{Q}_{a_1})| \le
\, \Lambda^{-2g_1(Q)}
\, (\Lambda^{-1}_0 \, Plog \Lambda_0 \,+\,
\delta \,Plog \delta^{-1} ) \,,
\eqe
where the statement is uniform in $U_{\varepsilon}(Q_a)$ (see Proposition 6). \\
For $m+3n+|w| \le 4$ we obtain the bounds
\eq
|\partial^w {\cal L}_{3,0,1}^{\delta,\Lambda, \Lambda_0 ,r}
(q,k_1,k_2)|
\le \,\Lambda^{2-|w|} \, Plog\Lambda^{-1} \,,\;
|\partial^w {\cal L}_{1,0,1}^{\delta,\Lambda, \Lambda_0 ,r}
(q)|
\le \,\Lambda^{4-|w|} \, Plog\Lambda^{-1} \,,\;
\eqe
\eq
|\partial^w {\cal L}_{1,2,1}^{\delta,\Lambda, \Lambda_0 ,r}
(q,k,p)|
\le \,\Lambda^{1-|w|} \,Plog\Lambda^{-1}\,,\;
|\partial^w {\cal L}_{0,2,1}^{\delta,\Lambda, \Lambda_0 ,r}
(q,p)|
\le \, \Lambda^{2-|w|} \,Plog\Lambda^{-1}\,.
\eqe
(104), (105) are uniform in $\delta, \; \Lambda_0$.
As in Prop.6,(c) those statements for which the r.h.s. vanishes
for $\Lambda \to 0$ hold uniformly only for momenta bounded
by $O(1) \Lambda$, otherwise they hold for momenta $\le B$.
\noindent
{\sl Proof}: The proof is in many aspects similar to that
of Propositions 3 and 6. We will concentrate on those aspects which
are new. The two contributions appearing on the r.h.s. of
(100)-(103) enter through the boundary conditions at
large and small $\Lambda$. We use the standard inductive scheme.
The statements for $r=1$ are immediately verified, since all
${\cal L}_{m,2n,1}^{\delta,\Lambda, \Lambda_0 ,1}$
vanish. For $r>1$ we go down in $m+2n$ and for given $m+2n$
down in $|w|$. We start with
$m+3n+|w| \ge 5$. At $\Lambda\,=\,\Lambda_0$ the bound (100)
is verified using (89)-(91). Using the induction assumption and the
bounds from Prop.3 on the r.h.s. of the FE (88) we also verify
(100) down to $\Lambda=1$. \\
Now we may integrate further down to $\Lambda \ge \delta$.
At $\Lambda=1$ (101) to (105) for $m+3n+|w| \ge 5$ are true
since (100) has been verified for $\Lambda=1$.
We start verifying (102) by estimating the r.h.s. of the
FE with the aid of the induction hypothesis and Prop.6.
The proof proceeds then as that of (53),
Prop.6. As there we also need the statements (104) to
lower order in the respective estimates. Having verfied
(102) we may also prove (103) and (101), (104) and (105).\\
Coming now to $m+3n+|w| \le 4$ we have as usually to integrate
the FE from the lower end, here from $\Lambda=\delta$ upwards,
with the momenta fixed at some renormalization point. It follows
from (92)-(98), (99) that all $\partial^w
{\cal L}^{\delta,\delta,\Lambda_0,r}_{m,2n,1}$ with $m+3n+|w| \le 4$
fulfill the estimates (101) to (105) for the momentum
arguments imposed in (94)-(96).
Integrating then the FE from
$\delta$ to $\Lambda > \delta$ at these arguments and
using the induction hypothesis on the r.h.s. we verify
(104), (105) also for $\Lambda > \delta$.
The next step is then
to go from the renormalization points to arbitrary momenta
$q,p,k_1,k_2,k$ ($\le B$) via the Schl\"omilch formula (41),
starting from $m+3n=4$ and treating then $m+3n=3,|w|=1,0$
and $m=1,|w|=3,2,1$. Using the induction hypothesis
allows then to verify all statements (100) to (105) for abitrary
momenta bounded by $B$. \qed
\\
An immediate consequence of Proposition 7 and (84) is now\\
\noindent
{\bf Proposition 8}:
(Restoration of the Ward Identities)\\
Sending the UV-cutoff $\Lambda_0$ to $\infty$ and
$\delta $ to $0$ (in arbitrary order) the connected amputated Green
functions ${\cal L}^r_{m,2n}$ fulfill the standard QED Ward
identities. That means - for momenta for which they are well-defined
- they satisfy the equations (for $r>1$)
\[
(m+1) q_{\mu} ({\cal L}^{r}_{m+1,2n})_{\mu \mu_1 \ldots j_n}
(q,k_1, \ldots , p_{2n-1})=
\]
\eq
-\sum_{a=1}^n [ ((S(-p_a))^{-1}\, S(-p_a-q) \,
{\cal L}^{r-1}_{m,2n}(k_1,\ldots ,p_a+q, \ldots ,p_{2n-1}))_{\mu_1,
\ldots ,j_n} -
\eqe
\[
( {\cal L}^{r-1}_{m,2n}(k_1,\ldots ,p_{n+a}+q, \ldots ,p_{2n-1})
S(p_{n+a}+q) \, (S(p_{n+a} )^{-1})_{\mu_1,
\ldots ,j_n} ].
\]
\section{ On Massless QED }
\vskip.5cm \noindent
In this paper we have treated Euclidean QED with massive fermions.
In view of physical reality one should also find a way to pass
to the Minkowski metric which we intend to do. The case of massless
fermions is less important from this point of view, still there are
massless fermions in the standard model (and maybe in nature). So
we shortly indicate the modifications necessary in this case
without giving a proof.
In massless QED we have to regularize the fermion propagator
by $R(\Lambda,p)$ instead of $R_m(\Lambda,p)$, since $R_m$ is
not an infrared regulator. This change induces a deterioration
in the IR estimates (see (39)).\\
The definition of the index $g$ (Def.4) has to be changed as follows:
\begin{itemize}
\item[(i)]
Momentum sets $Q$ consisting of two fermion momenta only are no more
admissible.
\item[(ii)]
Assume a momentum subset $E_{\nu}$ (43) or $Q$ itself can be
subdivided into two
subsets $E_{\nu 1},\, E_{\nu 2}$ such that the sum over the momenta
in both vanishes and such
that both contain an odd number of momenta from
$\{ p_1,\ldots,p_{2n} \}$.
In this case $Q$ is called exceptional and the set $E_{\nu}$
contributes $3/2$ to $g_Z(Q)\;$ (as before) if it consists of two
single momenta, it contributes $2$ if one subset $E_{\nu i}$
consists of more than one momentum, and it contributes $5/2$
otherwise.
\end{itemize}
These changes are then sufficient to prove
$g(Q'_F) +g(Q''_F) +1/2 \, \le \, g(Q)$, if $p' =0$ instead of Lemma
4 (d). We need this sharpened inequality to prove Proposition 6
in the massless case. The improved index $g_1$ is of no use any more
since Lemma 5 (c),(d1)
are no more sufficient to bound the ${\cal L}^{ \Lambda}_{m,2n}(Q)$
by $\Lambda ^{-2g_1} Plog \Lambda^{-1}$.
The new index $g$ is then such that ${\cal L}^{ \Lambda, \Lambda_0}_{4,0}$,
${\cal L}^{ \Lambda,\Lambda_0}_{1,2}$
and $ \partial_{\mu} {\cal L}^{ \Lambda, \Lambda_0}_{0,2}$,
$ \partial_{\mu}\partial_{\nu} {\cal L}^{ \Lambda, \Lambda_0}_{2,0}$
are allowed to be logarithmically divergent for $\Lambda \to 0$ at zero
momentum, when estimated with the aid of the FE.
All these terms therefore have to be renormalized at
nonexceptional momenta whereas the undifferentiated two point functions
have to be renormalized at zero momentum.
Renormalization at nonexceptional momenta induces notational complications.
Regard e.g.
${\cal L}_{1,2}$ (leaving out upper indices). We find from symmetry
considerations
\[
({\cal L}_{1,2})_{\mu,i,j} =
(\gamma_{\mu})_{ij} \, l_1(k,p)\, + \, k_{\mu} /\!\!\!p_{ij} \,l_2(k,p)
+ \, p_{\mu} /\!\!\!k_{ij} \,l_3(k,p)\,
+ \, k_{\mu} /\!\!\!k_{ij} \,l_4(k,p)\,
+ \, p_{\mu} /\!\!\!p_{ij} \,l_5(k,p)\,
\]
(to be compared with (31)),
where the $l_i$ depend on $k,p$ only through $O(4)$-invariant
combinations. The function $l_1$ is then fixed by a r.c. at some
nonexceptional momentum configuration $\{k,p,-k-p\}$, whereas
$l_{2,3,4,5}$ are to be calculated from terms with $m+3n+|w| \ge 5$.
To solve for $l_2$ choose e.g. $k,p$ nonexceptional such that
$p=(0,p_2,0,0),\; k=(0,0,k_3,0)$. Then
\[
-4p_2 \, l_2(k,p)\,= \, tr(\gamma_2 \partial_{k_1} ({\cal
L}_{1,2})_1(k,p)) \, ,
\]
and similar expressions for $l_{3,4,5}$.
Arbitrary nonexceptional momenta can now be reached on application
of the Schl\"omilch formula.
Observing these changes and imposing r.c.
as described above it is then straightforward to rewrite
Proposition 6. In part (a) the degree of smoothness is generally reduced
by 1 ($C^2(\R^4) \, \to \, C^1(\R^4),\; C^1(\R^4) \, \to \,
C^0(\R^4))$,
whereas
${\cal L}^{ \Lambda, \Lambda_0}_{4,0}$,
${\cal L}^{ \Lambda,\Lambda_0}_{1,2}$
may diverge logarithmically at exceptional momenta for $\Lambda \to 0$.
In (b3) we have to replace $g_1$ by the new $g$. In (c) we claim
\[
|\partial^w {\cal L}^{\Lambda ,\Lambda_0 ,r}_{2,0}(k)|
\le
\Lambda^{2-|w|} Plog \Lambda^{-1} ,
\]
\[
|\partial^w {\cal L}^{\Lambda ,\Lambda_0 ,r}_{0,2}(p)| \le
\Lambda^{1-|w|} Plog \Lambda^{-1} ,
\]
in the same sense as in Proposition 6. \\
These changes in the IR behaviour and the corresponding modifications
of the r.c. have to be taken into account in ch.IV,
i.e. in the relations between the
${\cal L}^{ \Lambda}_{m,2n}$ and${\cal L}^{\Lambda}_{m,2n,1}$
for $\Lambda = 0 \; (92),\ldots$. These then also have to be exploited at
the new renormalization points. This does not change the statement on
the restoration of the WI's at nonexceptional momenta. But the proof
of the statement corresponding to Proposition 7 becomes more
complicated. This is due to the fact that on the r.h.s. of
(89) all $r_{m,\delta}^{\Lambda_0}$ are to be replaced by
$r_{\delta}^{\Lambda_0}$ which have large derivatives for
small $\delta$. A restriction as $|q|>2\delta$ is no more sufficient
to exclude their appearance
since the $r_{\delta}^{\Lambda_0}$ also carry arguments $p$ which
in turn appear as integration variables on the r.h.s. of the FE.
Therefore we have to make a new induction hypothesis.
The first step is again to adapt the norms (34) to the new situation
by the following
definition:\\
For a system a functions $f_{\mu_1,\ldots,i_1,\ldots,j_1,\ldots}
(q,k_1,\ldots,p_1,\ldots,p_{2n-1})$ depending on $q$
and on photon momenta $k_1,\ldots,k_m$
and fermion and antifermion momenta $p_1,\ldots,p_{2n}$ we define
\[
||\partial^{|w|}f ||^{\delta}_{(a,b)}
\]
as in (34) with the additional restriction that we only
take the $\sup$ over the momenta fulfilling
\eq
|q|>2\delta \mbox{ and also: } \,
|p_i|,\, |q+p_i+S_i| \; > 2\delta \;\mbox{ or }\;
|p_i|,\, |q+p_i+S_i| \; < \delta \,,\; i=1,\ldots,n\,,
\eqe
where $S_i$ denotes any (possibly empty) subsum over momenta from
$\{q,\ldots,\ldots p_{2n}\} \,\setminus \{q,p_i\}$
which contains as many fermion as antifermion momenta
(for $n=0$ only $q$ is restricted, as in the massive case).
\\
The bounds of Prop.7 now hold again if the conditions (106) are satisfied.
If they are not satisfied we claim weaker bounds to hold which
are obtained from those of Prop.7 on multiplication by
$\delta^{-|w|}$.
Since the volume of those regions in $p_i$-space
where (107) is violated is $O(\delta^4)$
the factor $\delta^{-|w|}$ is compensated by the integration volume
on performing the momentum integral on the
r.h.s. of the FE, as long as we restrict to $|w| \le 4$
(which is sufficient for us).
The restoration of the WI's is then obtained letting $\delta \to 0$,
$\Lambda_0 \to \infty$
as before.
\\[2cm]
\noindent
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\end{document}