\vskip 1.5cm
\centerline{\bf Construction of $YM_{4}$ with an infrared cutoff}
\vskip 2cm
\centerline{Jacques Magnen}
\centerline{Vincent Rivasseau, Roland S\'en\'eor}
\vskip .5cm
\centerline{Centre de physique th\'eorique, CNRS, UPR14}
\centerline{Ecole Polytechnique, 91128 Palaiseau Cedex, France}
\vskip 2.5cm
\centerline{\bf ABSTRACT}
We provide a rigorous construction of the Schwinger functions
of the pure SU(2) Yang-Mills field theory in four dimensions (in the trivial
topological sector)
with a fixed infrared cutoff but no ultraviolet cutoff, in a regularized
axial gauge. The construction exploits the positivity of the
axial gauge at large field. For small fields, a different gauge, more
suited to perturbative computations is used; this gauge and the
corresponding propagator depends on
large background fields of lower momenta.
The crucial point is to control (in a non-perturbative way)
the combined effect of the functional integrals over small field regions
associated to a large background field and of the counterterms which
restore the gauge invariance broken by the cutoff.
We prove that this combined effect is stabilizing if we
use cutoffs of a certain type in
momentum space.
We check the validity of the construction by showing that Slavnov
identities (which express infinitesimal gauge invariance) do hold
non-perturbatively.
\vskip 1.5cm
\line {A 144 01 92 \hfil February 1992}
\vfill\eject
\noindent {\bf I. Introduction and outline}
\medskip
Non-abelian gauge theories form the core of modern high energy physics,
and in the recent years they have been very important in pure mathematics too.
Perhaps the main reason for this success lies in the discovery that these
theories (at the perturbative level) are renormalizable and
asymptotically free. Therefore most physicists are convinced that the
ultraviolet problem in non-abelian gauge theories
is well understood and void of any surprises. However it remains to
substantiate this belief rigorously beyond perturbation theory.
Until now there was only one rigorous program of study of this problem
completed, the one of Balaban [B]. This program defines a sequence of
block-spin transformations for the pure Yang-Mills theory in a finite
volume on the lattice and shows that,
as the lattice spacing tends to 0 and these transformations are iterated many
times, the resulting effective action on the unit lattice remains bounded.
>From this result the existence of an ultraviolet limit for $gauge$
$invariant$ observables
such as ``smoothed Wilson loops" should follow, at least through a compactness
argument using a subsequence of approximations; but the limit is not
necessarily unique. Clearly this is a point which requires further work.
Although very impressive, Balaban's work is not easily accessible,
partly because the use of the lattice regularization is the
source of many technical complications and partly because the results are
scattered over many publications; hence to check the consistency of all the
arguments is very difficult. Also it does not address the problem
of constructing the expectations values of products of the field operators
in a particular gauge (the Schwinger functions), because these are not
gauge invariant observables. It is true that physical quantities should be
gauge invariant. Nevertheless the gauge fixed framework is obviously the
most convenient for perturbative computations, and one can consider in fact
that the ultraviolet problem for the
Yang-Mills $field$ theory is not yet understood until this point is clarified.
A related program in progress but not yet completed is the one of Federbush
[F] for which the above remarks also apply.
In this paper we provide another approach to the same problem, by
constructing the Schwinger functions of the pure $SU(2)$ Yang-Mills field
in the axial gauge, with an infrared cutoff such as a finite volume box. We
give the construction in a single self-contained paper, but it remains
admittedly still very complicated
and technical. It certainly requires some knowledge of constructive
theory; we assume familiarity of the reader with a reference
on the construction of just renormalizable models such as [R]. We
do not repeat most of the arguments which are
already contained there. We do not
claim to provide here the proofs of convergence of our expansion in all
detail. However we think that this paper, which summarizes many years of
efforts and trials on this problem, both provides a detailed outline
of these proofs and remains relatively short and (hopefully!) readable.
Beside these remarks our program has in fact a lot in common with the one
of Balaban. We are indebted to him because his pioneering efforts encouraged
us to attack this problem; we did not take our technical tools directly
in his works, but meeting similar difficulties we think we found often very
similar solutions. We do not use the
lattice cutoff but a momentum cutoff of a certain type
which breaks gauge invariance
and requires gauge restoring counterterms which
stabilize the field at sizes of order $\la^{-1}$.
We think that the role of this stabilizing cutoff
is quite similar to the role played by compactness of integration
over the gauge group in Balaban: it provides us with the initial
information that
the field variables in the Lie algebra are of order
at most $\la^{-1}$. This
information by itself is not enough to start perturbation theory, but we have
found that if we combine it with the positivity of the axial gauge, then the
field variables in the Lie algebra become of order roughly
$\la^{-1/2}$. The fact that the field is of order roughly
$\la^{-1/2}$ is however true only in probability. To exploit
this fact we have to make a division of the phase space for the field
into small field regions and large field region, and expect that
the large field regions are so rare that they cam be resummed and
controlled. This makes
an explicit change of the gauge possible in the small field regions,
and in turn this change of gauge allows the use of
perturbation theory there (remember that
in the axial gauge alone, perturbation
theory is sick). However because
not all couplings between high and low momenta are of
a form which can be dominated in the technical constructive sense,
we have to use a background dependent gauge and a background dependent
propagator for the analysis of the small field perturbative region.
The background field at a given scale and position
is roughly speaking made of all the large fields
of lower frequencies located at this position.
The use of these background dependent gauge and propagator is a source
of technical complications for the cluster expansions of constructive
theory and it is also the source of a new difficulty with the
evaluation of the large field regions;
their functional integral is ``renormalized''
or ``dressed'' by their coupling to higher momenta small field regions.
As could be understood quite intuitively, this coupling results in a
determinant which reflects the difference in normalization between the
gaussian measure with a given background field, and the ordinary
gaussian measure when this background field is 0. One of the main
point of this paper is to prove that this determinant can
be controlled; it turns out that this is true in the
cas of the stabilizing cutoffs that we use.
All these elements are present in
a way or another in what we have understood of Balaban's papers,
and we consider that our solutions of these problems, although technically
very different, reinforce our belief that his solutions are correct.
However there is one point in our approach which we do not see emphasized in
Balaban's work but which seems unavoidable to us: it is
the use of anisotropic lattices for phase space expansions. Indeed the
axial gauge is anisotropic, and large field conditions have to be
adapted to this anisotropy. We hope that some day this point can be
clarified for us.
As a justification that our construction is correct
we show that the Schwinger functions that we construct satisfy the Slavnov
identities which are the remnant of gauge invariance under small gauge
transformations at the level of Schwinger functions.
This is our main result, formulated as a Theorem
at the end of Section VIII.
The drawbacks of our approach is that for the moment it is limited to
the axial gauge (Feynman gauge or similar ones which are Euclidean invariant
and convenient for perturbative computations, and which we use
in the small field regions, cannot be used
directly at the beginning because of their
lack of positivity). Also the stability property
that we require for our ultraviolet cutoff allows many
different cutoffs but certainly
for the moment rules out many others. It would be nice to understand
in a deeper way why some cutoffs stabilize the theory and others do not.
We do not investigate invariance under large gauge transformations and
non-trivial topological effects such as instantons; also we do not try
to lift the infrared cutoff, since this would lead to large values of
the coupling constant, and presumably to so called non-perturbative effects
corresponding to confinement. These problems are for the moment still
out of the realm of our constructive methods, especially because
there is no easy solvable model of these phenomenons around which to expand.
Concerning the axioms of quantum field theory, we cannot study the
complete set of Osterwalder-Schrader's axioms, also because we
do not lift our fixed infrared cutoff. However we think that the main
axiom, the O.S. positivity, could be shown to hold with some additional work
in the following way. It was proved in [L] and [OS] that OS positivity holds in
the lattice gauge theory relatively to the hyperplanes of the lattice.
We could then use as a first cutoff a lattice cutoff, then use the
momentum cutoffs of this paper for slicing and analyzing the theory.
If we start the slicing at a scale quite below the inverse of the
lattice spacing we think that the gauge-restoring counterterms are
close to what they are in the ansatz
of this paper. We think that in this way OS positivity can be proved.
\vfill\eject\medskip
\noindent {\bf II. The starting Ansatz}
\medskip
\noindent {\bf A) The model, notations}
\medskip
We consider the pure Yang-Mills theory with an infrared cutoff, which we never
try to lift. This cutoff may be imposed on the propagator, or we could consider
the theory on a finite volume with some boundary conditions, or on the sphere
$S^4$, the torus $\Lambda = \RR^4 / \ZZ^4$ or an other compact Riemannian
four-dimensional manifold. Naive infrared regularization
breaks gauge invariance, but compactification of space and the choice of a
particular principal bundle with fiber $G$ defines an unbroken group of gauge
transformations. For instance in the case of the torus with the trivial SU(2)
bundle, the gauge transformation are simply the functions $x \to g(x)$
from $\RR^4$ to $G$ which are periodic with period lattice $\ZZ^4$.
The momentum space corresponds to discrete Fourier analysis on the dual
lattice $ {\Lambda}^* = {\ZZ}^4$. Moreover the constant fields or
the zero mode in Fourier space is deleted in all our functional
integrals, hence there is no infrared problem.
For the pure SU(2) Yang-Mills theory the vector potential is a field
$A$ with components
$A_{\mu}^a, \mu =0,1,2,3, a=1, 2, 3$
with Lorentz (greek) indices and Lie algebra (latin) indices (the group is
noted SU(2) and the algebra su(2)).
We have also often to distinguish between the index $\mu =$0,
called the time, and the
three other indices, called the spatial indices, usually
noted $m,n,...$, $m,n,...=1,2,3$. Geometrically $A$ is a connection on the
considered principal bundle; again in the case of the trivial SU(2) bundle
one can consider that each $A_{\mu }$ is simply a function with values in
su(2). Our conventions are those of [IZ], which we recall briefly; later
to simplify the notations we will forget indices most of the time.
We write $ A = \sum_{a=1}^3 A^a t_{a}$, with $t_a =(i\sigma_a /2) $
where the $\sigma$'s are the three
usual hermitian Pauli matrices. With this convention the covariant
derivative is $D_{\mu} = \partial_{\mu} - \lambda [ A_{\mu} ,.]$.
We have ${\rm Tr}\, t_at_b=- {\delta_{ab} \over 2} $.
The field curvature is:
$$ F_{\mu\nu} = (\partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}) - \lambda
[ A_{\mu} , A_{\nu} ] = (\partial \wedge A - \lambda
[ A , A ]) \eqno({\rm II.1})
$$
$\lambda$ being the coupling constant; the second notation is a condensed
one in which indices are omitted (and $\partial \wedge$ is the exterior
derivative).
Remark that in the three dimensional su(2) space, the commutator is a wedge
product: $ [ A_{\mu}^a , A_{\nu}^b ] = \epsilon_{ab}^{\; \; c}
A_{\mu}^aA_{\nu}^b $.
The pure Yang-Mills action is:
$$ - {1 \over 2} \int_{\Lambda} d^4x Tr F_{\mu\nu}F^{\mu\nu} =
{1 \over 4} \int_{\Lambda} d^4x \sum_a F_{\mu\nu}^aF^{\mu\nu a} \eqno({\rm
II.2})
$$(for e.g. Euclidean canonical metric on the flat
torus the raising of ''Lorentz'' indices is trivial so that $F_{\mu \nu
}=F^{\mu \nu }$).
To simplify, we define
a scalar product $$ on space time tensors $A$ and $B$ of the same type
with values in the Lie algebra,
by the convention that a trace is taken
over all correspondent space time indices and
minus a trace over group indices,
so that it is positive definite with a factor 1/2 in component notation.
We also write simply $A^2$ for $$, and with this convention we can
write the action
as ${1 \over 2} \int_{\Lambda} F^2$.
We distinguish between the quadratic, trilinear and quartic pieces
of $F^2$, writing:
$$ F^2 = F_2 + \lambda F_3 + {\lambda}^2 F_4 \eqno({\rm II.3})
$$
This action is invariant under the gauge transformations:
$$ A \to A^{g} \ ; \ (A^g)_{\mu } = g A_{\mu } g^{-1} +
{1 \over \la}\partial _{\mu }g
\cdot g^{-1} \eqno({\rm II.4})
$$
In what follows these gauge transformations are limited to a particular
topological sector, for instance the functions from the compact space to $G$.
It is often useful to consider the infinitesimal gauge transformations
$\gamma $ with values in the Lie algebra, which are tangent to the gauge
transformations, such that $g=e^{\la \gamma }$; the corresponding formula is:
$$ A \to A^{\gamma } \ ; \ (A^{\gamma })_{\mu } = A_{\mu } +
D_{\mu }\gamma \eqno({\rm II.5})
$$
where $D=\partial -\lambda [A,.]$ is the covariant derivative.
Finally for technical reasons it is also useful to introduce infinitesimal
gauge transformations which correspond to expanding to a finite order
in $\gamma $ the exponentials in (II.4).
For instance we are interested in the regime where $A \simeq \lambda ^{-1/2-
\epsilon _1}$ and $\gamma \simeq \lambda ^{-1/2-\epsilon _2}$, where
$\epsilon _1$ and $\epsilon _2$ are very small and we want to keep all
terms not small as $\lambda \to 0$. Then we should define
$$ A^{\gamma ,2}_{\mu } = A_{\mu } + D_{\mu } \gamma + \lambda /2 [\gamma ,
\partial _{\mu }\gamma ] \eqno({\rm II.6})
$$
This ``truncated" gauge transformed configuration $A^{\gamma ,2}$
is a polynomial of second order in $\gamma $ and its derivatives.
We could define further expansions of the gauge transformations;
with these notations,
if $g=e^{\la\gamma }$, we have $A^{g}= A^{\gamma ,\infty}$
and $A^{\gamma }=A^{\gamma ,1}$.
Our starting point is the Yang-Mills theory in the axial gauge.
This gauge is defined by the condition
$$ A_0 = 0 \eqno({\rm II.7})
$$
This is a gauge condition that can be imposed in the sense that for any
field configuration $A$ there is a gauge transformation such that
$A^{g}$ in (II.4) satisfies it; indeed we can take
$$ g = P exp (- \int_{0,\vec x}^{x} A_{\mu } dx^{\mu} ) \eqno({\rm II.8})
$$
where the $P$ means a path ordered exponential (limit of a Trotter product
of exponentials along the path), and the path goes from $\{O,\vec x\}$,
the point on the hyperplane $x_0=0$ to $x$, hence this path is parallel to
the direction 0.
Remark that such an axial gauge condition a
priori is not complete, in the sense that even after
imposing it there remains a subgroup of the gauge group which acts still
on the configurations satisfying (II.7), namely the gauge transformations
independent of $x_0$, the ``time coordinate". We do not fix this remaining
invariance yet. Remark also that (II.7) is not Euclidean invariant, and the
corresponding correlation functions are therefore not Euclidean invariant.
However in principle physical quantities (which are gauge invariant and
Euclidean covariant observables)
can be recovered from the gauge fixed theory. Since these observables
involve composite operators, they have to be renormalized and we do
not provide the corresponding constructions in this paper, although
the task seems accessible to us with our methods.
The main advantage of the axial gauge condition is that it provides
some definite positivity. An other advantage (perhaps related...)
is that there is no Fadeev-Popov determinant in the axial gauge (more
precisely it is a constant absorbed in the normalization),
which is a big simplification.
Indeed in the axial gauge we have
$$ F^{2} = + F_{sp}^{2} \ , \eqno({\rm II.9})
$$
where the spatial part of $F^{2}$
is by definition
$$F_{sp} \equiv - {1 \over 2} \int_{\Lambda} d^4x Tr F_{mn}F^{mn} =
{1 \over 4} \int_{\Lambda} d^4x \sum_a F_{mn}^aF^{mn, a} \ ,
\eqno({\rm II.10})$$
and both pieces in (II.9) are obviously positive.
Also the piece $$ is quadratic, hence
(II.9) looks almost like the usual case of
a positive quadratic measure and a positive interaction. This is not
the case in e.g. the Feynman gauge where the interaction is not
positive in itself but only when combined to the Gaussian measure.
We want to have a well defined functional integral to start with.
The scale of our ultraviolet cutoff is called $M^{\rho }$, and
the ultraviolet limit is when $\rho \to \infty$.
>From standard renormalization group analysis we
learn that in order to get a finite non trivial renormalized
theory at the unit scale of our finite box, we should use a bare coupling
constant which has the usual asymptotic behavior with $\rho$ implied by
asymptotic freedom. Hence a good ansatz for the bare coupling
$\lambda_{\rho}$ should be:
$$ \lambda_{\rho}^{2} = {1 \over -\beta_2 (LogM) \rho + \beta_3 / \beta_2
\log \rho + C} \eqno({\rm II.11})
$$
where $C$ is a large constant,
and $\beta_2$ and $\beta_3$ are the usual first non vanishing coefficients
of the $\beta$ function, whose numerical value is given in standard textbooks
like [IZ]. Then one hopes that the renormalized coupling constant
$\lambda_{ren}$, which should be defined as the last one in a sequence of
effective constants, is finite and arbitrarily small as $C$ becomes
arbitrarily
large (if perturbative renormalization group analysis
turns out to be correct).
Let us define first the tentative effective coupling at scale $i$ as
$$ ( \lambda _{i}^{t} )^{2} \equiv
{1 \over -\beta_2 (\ln M) i + \beta_3 / \beta_2 \ln i + C} \eqno({\rm II.12})
$$
Later (in section V) we will
have performed the necessary expansions
to compute the flow of exact effective coupling constants
$\lambda _{i}$ which will be very close to the tentative ones.
The class of ultraviolet cutoffs we consider is defined as follows.
$\tau$ is a fixed function which is between 0 and 1, is
1 near 0 and decreases at infinity. For
instance we take a one variable $C_0^{\infty}$ function,
monotone decreasing, which is 0 for $ x \ge 2$ and is 1
for $ x \le 1$ (the monotone decreasing and $C_0^{\infty}$ character
are perhaps not essential
but it is important that the slices built out of this cutoff by the
scaling process defined below have
good spatial decay; it is also useful (although perhaps not absolutely
necessary) that they vanish
identically at zero momentum, a property which we call
``good momentum conservation''; this property will be used several
times in our construction when we need to bound contributions
which violate momentum conservation rules. We will usually not provide
the corresponding argument, referring the reader to the last section
of [FMRS1] for an example treated in detail.
Then we define our scaled momentum
cutoff $\ka_{\rh } $ to be:
$$ \ka _{\rho} (p) = \ka ( pM^{-\rho} ) \eqno({\rm II.13})
$$
where $\ka $ is the following function~:
$$ \kappa (p) \equiv 1 \quad {\rm if \ \vert p \vert \le 1}
$$
$$ \kappa (p) \equiv {1+\ta (\vert p\vert ) \over 2}
\quad {\rm if \ 1< \vert p \vert \le 2}
$$
$$\kappa (p) \equiv 1/2 \quad {\rm if \ 2 < \vert p \vert \le 2+ \et ^{-1}}
$$
$$ \kappa (p) \equiv (1/2) \ta (\vert p\vert - 1 - \et^{-1}) \quad
{\rm if \ 2+\et^{-1}< \vert p \vert }
\eqno({\rm II.14})
$$
where $\eta$ is a small constant. This unusual form, shown in Fig. II.1
leads to a stabilizing $A^4$ counterterm whose strength can be made
as large as desired and to a stabilizing
functional integral for large background fields; both effects
are obtained by taking $\eta$ sufficiently small as shown in section III).
We write also
$$\kappa ^{i} = \kappa _{i} -\kappa _{i-1} \ {\rm if} \ i\ge 1;\
\kappa^0 = \kappa _0 \eqno({\rm II.15})
$$
\topinsert
\vskip 12.3cm
\endinsert
\centerline {\bf Figure II.1}
\medskip
The quadratic form $p_0^{2}$ is not invertible when $p_0=0$ and in
order to have
a good propagator we add and subtract $\sum_{i }(\lambda_{i}^{t}) ^{2}
$ to the
action\footnote*{This small term which makes the propagator well defined
is harmless since as shown below the cutoffs that we later
use will generate a term $-c\cdot\lambda
^{4}A^{4}$ in the action which can be used to bound the bad interaction term
$\lambda ^{2}$.}. We warn the reader that below we usually write
$\lambda ^{2}p^{2}$ instead of $\sum_{i }(\lambda_{i}^{t}) ^{2}
p^{2}\kappa ^{i}(p)$ which is quite heavy.
The term added is used to create a well defined
positive quadratic form which is useful
for generating a well defined starting ansatz; also together with the
$p_0^{2}$ term it will be used to prove that the field cannot be much larger
than $\lambda ^{-1/2}$ in probability. The subtracted piece is treated as
an interaction. We define
$$\sum_{i }(\lambda_{i}^{t}) ^{2}
\ =\
\eqno({\rm II.16})$$
For the moment our formal functional integral in the axial gauge without cutoff
is:
$$ e^{(1/2)(-F_{sp}^{2} - + \sum_{i}(\lambda_{i}^{t}) ^{2}
)} d\mu_{0} \eqno({\rm II.17})
$$
where $d\mu_{0} $ is the normalized
Gaussian measure with propagator $C_{0}$.
The support of a Gaussian measure such as (II.17) is made of distributions,
as is well known, and since the multiplication of distributions is
illegal, (II.17) is still formal. To make sense out of it we have
to introduce now a first ``fake" ultraviolet cutoff of the theory, at a scale
$M^{\rho _1}$, with $\rho _1 >> \rho $. This will not be the true cutoff of the
theory but is useful in order to manipulate as soon as
possible well defined quantities.
We could write instead of (II.17) the functional measure of the theory as:
$$ d\mu_{0,\rho _1} (A)
e^{(1/2)(-F_{sp}^{2} - + \sum_{i }(\lambda_{i}^{t}) ^{2}
)}
$$
which is proportional to
$$ d\mu_{axial,\rho _1} (A)
e^{(1/2)(-F_{sp}^{2} + \sum_{i }(\lambda_{i}^{t}) ^{2}
)} \eqno({\rm II.18})
$$
where $d\mu _{0,\rho _1}$ is the Gaussian measure with propagator
$C_{0}(p) \kappa _{\rho _1} (p)$, and the sum over $i$ in (II.18) stops
at $\rho _1$, and the Gaussian measure $d\mu _{axial} $ and
propagators $C_{axial}$ are obtained by joining to $C_{0}$ the quadratic piece
$$:
$$ + \sum_{i }(\lambda_{i}^{t}) ^{2}
=
\eqno({\rm II.19})$$
This formula is still formal, because the positive exponential cannot
be integrated simply with the Gaussian measure $d\mu _{0,\rho _1}$ (this would
give back the ill-defined Lebesgue measure). Fortunately this is also not
the correct starting point
because any continuous ultraviolet cutoff really breaks
gauge invariance and to check ultimately Slavnov identities we have
to introduce gauge-variant counterterms to compensate these gauge
breaking effects of the ultraviolet cutoff. All our construction relies
on the use of the additional positivity given by these counterterms.
However these counterterms
cannot be computed in perturbation theory in the axial gauge (II.18)
because the axial gauge is still incompletely fixed in perturbation
theory. In particular our trick of introducing $\lambda ^{2} p^{2}$
to create a well defined propagator does not allow perturbative computations.
It is only a technical trick to extract easily a small
factor for the large field regions at a later stage where the true
ultraviolet cutoff and the stabilizing counterterms have been introduced.
All our perturbative computations will be done in the small
field region, in which we pass to a particular gauge
well suited for perturbation theory, which we call the $homothetic $
gauge. It is defined exactly as the Feynman or Landau gauge
but with a parameter, called $\la$ in [IZ] and $\ze$ in this paper,
which takes a value close to 3/13\footnote*{More precisely
we pass to a background dependent homothetic gauge as discussed below.}.
This value is chosen so that there is no infinite wave function
renormalization; indeed the one loop wave function renormalization
is proportional to $10/3 + (1-1/\ze)$, hence vanishes for $\ze=3/13$
[IZ]. Taking a value close to 3/13 we can ensure vanishing at any
given order, hence a finite total wave function renormalization either
exactly 0 or as small as we want (if we want an explicit formula for $\ze$).
We want to have an ultraviolet cutoff that
gives us simple gauge-breaking effects, computable in perturbation theory
in the homothetic gauge. An
explicit and still relatively simple cutoff in the axial gauge
will transform in a complicated cutoff in
this homothetic gauge. Therefore we prefer
to impose as our true cutoff a second cutoff which has a simple form
in the homothetic gauge.
It is therefore in the way our true ultraviolet cutoff is defined
that we incorporate the missing piece of information
that we are going to use the homothetic gauge when the field $A$ is small.
This piece of information is critical
because we actually compute the stabilizing $A^{4}$ term generated
by the ultraviolet cutoff by a one-loop
perturbative computation made in the homothetic gauge.
Remark that the stabilizing term which is part of our initial ansatz
is used to stabilize the theory when the field is large
although its value is given by a
perturbative computation, which seems to require small fields. The ultimate
justification of this apparent contradiction
lies in the fact that it allows to construct the model with correct
Slavnov identities; but we can add a further comment.
Stabilization could not be achieved in the
large field regions by artificial means
such as irrelevant operators ($A^{6}$ and so
on) because these operators
would not be enhanced correctly at lower momenta. In
contrast the $A^{4}$ term has a flow governed by the small field perturbative
regions, which keeps it in tune with the increasing coupling constant at lower
scales, and we think that this is
the deep reason why its value, computed in the small field region, can also be
used to stabilize the large field regions.
The formal formula for passing from the axial gauge
to the homothetic gauge is obtained by writing
$$ 1 = \det [K(A)] \int
d\gamma e^{-(\ze/2) (\partial _{\mu }A_{\mu }^{\gamma,\infty })^{2}}
\eqno({\rm II.20})$$
where the determinant is the usual determinant of the Fadeev-Popov operator
$K(A)= \partial _{\mu } D_{\mu }$, with $D_{\mu }$ as in (II.5) (see [IZ]).
This formula in itself cannot contain any new information. But we will use
an approximation to (II.20) which amounts no longer to insert 1 but to insert
cutoffs, hence there is no contradiction.
Remark that we have written (II.20) in terms of an integration variable
$\gamma $ which lies in the Lie algebra rather than in the
Lie group. Indeed it will be easier for us to give a well defined
analogue of this functional integration on a flat Lie algebra
variable, using
standard techniques of constructive field theory such as Gaussian measures
perturbed by polynomial interactions.
First we will modify (II.20) by using an approximate gauge
transformation $A^{\gamma ,2}$ instead of $A^{\gamma ,\infty}$
\footnote*{In fact to have correct renormalization group flows
to third order in later sections we have to be more cautious
and would need something like
$A^{\gamma ,10}$. But the corresponding formulas are just more
complicated and the use of $A^{\gamma ,2}$ should make clear how they
work in a more general case.}.
Also a well defined Gaussian measure with cutoff will be used on $\gamma $
together with a polynomial which ensures that $\gamma $ is small compared
to $\lambda ^{-1}$ (so that small fields after gauge transformations
remain small) but large compared to $\lambda ^{-1/2}$ (so that for small fields
the formula performs its usual job of integrating out gauge degrees of
freedom and changing the gauge at the price of a Fadeev-Popov
determinant).
Then we will introduce the true
ultraviolet cutoff on the
transformed field $A^{\gamma ,2}$, which as we said is
effectively put in the homothetic gauge in the small field region.
The gauge restoring counterterms can be therefore perturbatively computed in
the homothetic gauge as we desired. These counterterms are
also written in terms of $A^{\gamma ,2}$. The combination of cutoff and
counterterms is
balanced so as to restore Slavnov identities.
There is a problem with the use of an ordinary homothetic gauge, which
is that some couplings of high momentum fields to low momentum fields are
not dominable. This and the meaning of dominable is explained in [R],
to which we refer the reader not familiar with this terminology.
This problem can be tackled by using covariant
derivatives with respect to the low momentum field instead of ordinary
derivatives. The price to pay is that the homothetic gauge condition has also
to be written with a covariant derivative in a background field
instead of an ordinary one. This makes formulas more complicated.
The total field is written as the sum of two fields, the one associated
with the large field regions and the one associated with small
field regions. The gauge transformation of the total field is
divided into a gauge transformation on the small field and a rotation
on the large field. The background field at a given scale is then made
of the large field of lower scales. For this reason it is convenient
to introduce the small/large field decomposition before to give the precise
form of the ultraviolet cutoff, although it is possible to proceed also in the
reverse order, but this would require slightly correcting the formulas by
an expansion which suppresses the unwanted small fields from the background.
\medskip
\noindent{\bf B) The small field and large field decomposition}
\medskip
We want to decide, for a sequence of frequencies $M^{i}$, $i=1,...,\rho $
and a sequence of adapted boxes whether the corresponding fields are
smaller or larger than $\lambda ^{-1/2 -\epsilon _1}$. This is done by a first
expansion.
When the field is large, the boxes will be put in the so-called large field
region and the axial gauge positivity together with
the stabilizing counterterms, which restore gauge invariance after
imposition of the cutoff, will provide an associated small factor.
This factor is so small that it can be used to finance the creation of
protection corridors around the large field regions.
In each box of the small field region the sum of the
gradients of the fields
of smaller frequencies localized in the box is small because of the $A^{4}$
term and the protection corridors around the initial large field region.
However for technical reasons it is convenient to increase the strength
of this effect. This is done by a second expansion; the boxes which do not
satisfy the strengthened condition give small factors and are rejected in the
large field region.
At the end of all these tests, in the remaining small field region
where all these conditions are satisfied,
it will be at last possible to perform the change of gauge
which brings us to the homothetic gauge.
We start with the first main test, whether the field $A$ is large or small.
The positivity will come from the axial propagator $C_{axial}$.
This propagator is very anisotropic, hence we need to introduce
a corresponding anisotropic momentum decomposition.
For every value of $i=1,..., \rho_1 $ we introduce an index $\alpha $
with integer values between $N_i$ and $i+1$, where $N_i$ is the integer
part of $i- \vert \ln (\lambda _i^{t})/ \ln M \vert$ (this rule seems
obscure but is introduced because
when $\vert p\vert$ is of order $M^{i}$ we want to decompose $p_0$ between
$\lambda M^{i}$ and $M^{i+1}$). The full set of ordered pairs
$(i,\alpha )$ is called ${\bf P}$,
and the letter $j$ is used for a typical pair $(i,\alpha )$ of {\bf P}.
On this set we introduce an ordering relation, namely we say that
$j'=(i', \al ') < j =( i, \al )$ iff $i'*0$.
\medskip
\noindent {\bf Sketch of proof}
Let us consider the propagator $C_{axial}$ obtained by combining
$e^{-(1/2)}$ to $d\mu_{0}$ as in (II.18). This propagator is
multiplied
in (II.18) by a positive interaction. If we slice the propagator
$C_{axial}(p)$ according to the partition of unity given by the
functions $\ka ^{j}(p)$, we obtain pieces $C_{axial}^{j}(p) \equiv
\ka^{j}(p) C_{axial}(p) $
which satisfy, for any fixed large integer $q$
$$
C_{axial}^{j}(x-y) \le { K_{q}M^{2i} \over \la }
\biggl( { 1 \over 1 + \vert x_{0}-y_{0} \vert M^{\al}} \, \cdot \,
{ 1 \over 1 + \vert \vec x- \vec y \vert M^{i}} \biggr)^{q}
\eqno({\rm II.27})
$$
wher $K_{q}$ is some constant depending on $q$. This bound is
immediate
if we use integration by parts and the bound
$${M^{3i}M^{\al}
\over M^{2\al} + \la^{2} M^{2i}} \le M^{3i}M^{\al}M^{-2r\al }
\la^{-2(1-r)}M^{-2i(1-r)}
\le {M^{2i}\over \la } \ \ {\rm \ if } \ r=1/2
$$
Remark that the
anisotropic nature of $C_{axial}$ leads to different
rates of spatial decay in the zero component and the spatial component
of $x-y$. This is the reason for which we must use rectangular boxes
with a double index. Remark also that the factor $1/\la$ in (II.27)
means, as announced, that the Gaussian measure corresponding to
$C_{axial}$ gives for a field $A^{j}$ a typical size $M^{i} \la
^{-1/2}$, which is large compared to the size $M^{i}$ corresponding to
Gaussian integration with the
propagator of the homothetic gauge, but small compared to the size
$M^{i} \la^{-1}$ where perturbation theory becomes meaningless.
In theory it might be sufficient to take $q$ in (II.27) equal to 4,
so that the propagator is summable, but in practice
we will take it to be large,
e.g. 100, in order to have some margin for the convergence of cluster
expansions.
Using the bound (II.27) we can perform a cluster expansion between the
rectangular boxes of the large field region\footnote*{The
large field frequency splitting is in fact performed on the field, not
on the axial propagator (see (II.23)) so that the covariance is not
diagonal; this complicates slightly the cluster expansion, but the
conclusion is the same.}.
Each factor $E_{\De}$ contains $P_{i}$ fields which are
integrated with respect to $C_{axial}$. As is usual when the spatial
decrease of the propagator is matched to the shape of the boxes in which the
cluster expansion is performed, we obtain a product of local
factorials in the number of the fields in each box [R].
Therefore for each box $\De$ we have a factor
$$
(\la_{i}^{t})^{ \ep _{1} P_{i}} K^{P_{i}}(P_{i}/2) ! \le
e^{-(\la_{i}^{t}) ^{-\ep_{1}/2}}
\eqno({\rm II.28})
$$
if $\la_{i}^{t}$ is small enough
(such that $\sqrt{K}(\la_{i}^{t})^{3\ep_{1}/4} \le 1/e$, recalling
that $P_{i}=(\la_{i}^{t})^{-\ep_{1}/2} $).
This is the small factor announced in Lemma
II.1. However the complete proof of Lemma II.1 is
of course more complicated than this sketch; indeed the functional
integral (II.18) contains the negative factor $-F^{2}_{sp}$ which helps
in reducing the value of the functional integral, but also the positive
factor $\la *$; it is also incomplete because we should
add to it both the counterterms required to restore gauge invariance
and a term coming from the functional integrals over a certain set
of small fields
associated to the large field box $\De$. This last term, a normalizing
determinant announced in the introduction,
comes from the dependence in the background field
of the Gaussian measure used in the small field regions.
The stability estimates
of section VI then prove that the total weight of the positive
factor $\la $, the counterterms $CT$ (see
(II.40)) and the normalizing
determinant coming form the functional integrals over small field regions
associated to any large field box is bounded by 1. This really
achieves the proof of lemma II.1, but we think that to state it here
may help the reader understand the choice of the factors (II.25-26)
and the definition of protection corridors which we introduce later.
\medskip
The morale of Lemma II.1 is that we can associate a small factor not
only to any box of $KLFR$ but also to a lot of neighboring boxes;
first the ordinary neighboring boxes
in the same slices ${\bf D}^{j}$ up to a certain distance,
but also boxes which
are included into or contain a box of $KLFR$ and have neighboring values of
their index $j$. The small factor of Lemma II.1 finances the creation
of all these corridors later in this paper
provided we respect the golden rule that their
width both in space and momentum (index) directions be bounded so
that this small factor in the (II.28) divided into
the total number $p$ of boxes
in the corridors around a single large field box
is still small as $\la \to 0$ (i.e. $\la^{-\ep_{1}/2} >>p$). This rule
is necessary for the cluster and Mayer expansions to converge (see
e.g. [DMR] for a simple example of such a situation).
Let us return to the complete definition of the large field region.
We need further to know in each box of ${\bf D}$ whether the sum of the
gradient of the fields
of lower frequencies localized in the box is large or small.
To gain a small factor we need to create a gap between the scale of
the box and the frequencies tested.
In every box of ${\bf D}$ we write:
$$ 1 = \tau(H_{\Delta }) - \int_0^1 ds H_{\Delta } \tau '((1-s)H_{\Delta })
\ \ ,\eqno({\rm II.29a})$$
where $\tau$ is our reference $C_{0}^{\infty}$ function and
$$H_{\Delta }=
{1\over \Delta }\int _{\Delta }
\biggl( (\lambda_{i}^{t})^{1-(\epsilon _1 / 64)}
M^{-2i}\nabla B(\Delta ,x)\biggr) ^{P_{1,i}} \ \ ,
\eqno({\rm II.29b})$$
where $P_{1,i}$ is an even integer close to
$ (\lambda_{i}^{t}) ^{-\epsilon_1 / 32}$,
$$ B(\Delta ,x) =
\sum _{ \Delta '\in D_{i',\alpha '}, \ r(\Delta )>r(\Delta ')-k(\Delta )}
\chi _{\Delta '}(x) \tilde \kappa _{i',\alpha '}\ast A
\eqno({\rm II.29c})
$$
with $r(\Delta ) = (3i + \alpha )/4$, and if $\Delta \in {\bf D}_{i,\alpha }$:
$$M^{k(\Delta )} = (\lambda_{i}^{t}) ^{-\epsilon _1/16}
\eqno({\rm II.30})
$$
The large field region is now defined as the set ${\bf D}_1$
of boxes in which the
error term of (II.25) is chosen, plus their protection corridors, i.e. the
boxes $\Delta '$ which intersect a box $\Delta $ of ${\bf D}_1$ and
satisfy to $(\lambda _{i}^{t})^{1/16}<
M^{r(\Delta ) -r(\Delta ')} < (\lambda _{i}^{t})^{-1/16}$,
to which we add the set
${\bf D_2}$ of boxes in which the error term in (II.29a) is chosen,
plus a protection corridor around them of the same type but with smaller width,
i.e. the boxes $\Delta '$ which intersect some
$\Delta $ which belongs to $D_{2}$ and satisfy
$$(\lambda ^{t}_{i})^{1/128} < M^{r(\Delta )-r(\Delta ')}
<(\lambda_{i}^{t})^{-1/128}. \eqno({\rm II.31})
$$
Before the final bounds are derived, let us again explain in
anticipation
why the boxes with the error terms have a small factor attached to them.
The reasoning is similar to the sketch of proof of Lemma II.1. This is
because a $\nabla B$ field of scale $i_{2}$ produced at level $i_{1}>i_{2}$
is evaluated by a factor $\la_{i_{2}}
^{-1/2} M^{2i_{2}} \le
\la_{i_{1}}^{-1/2} M^{2i_{1}}M^{-2(i_{1}-i_{2})}$. The local
factorials created by accumulation of many $\nabla B$ factors coming
from the many boxes of scale $i_{1}$ in the same box of scale $i_{2}$
are then compensated by the $M^{-2(i_{1}-i_{2})}$ factors. The rest of the
argument is as in Lemma II.1, the value of $P_{1,i}$ being adapted for
it to work.
Finally we want to prepare the formulas better for the small field change
of gauges. We want that the background field is reduced
to the field of the low
momentum large field regions. Recall that the rationale for introducing this
background field and a modified homothetic gauge in background field, was
that some
interaction terms between high and low momentum field were not dominable; hence
it is necessary to absorb them into the propagator. However fields in the small
field region are always dominable (by the small field condition). Therefore we
do not need to put them in the background. Furthermore it would be bad
to leave
them in the background, because we want to perform a gauge transformation
on the full small field region and to decompose the field into a
gauge-transformed small field plus a rotated background field.
The large field region is called $LFR= \cup_{i,\alpha \in {\bf P}}
L_{i,\alpha }$. Its complement is the small field region $SFR=
\cup_{i,\alpha \in {\bf P}}
S_{i,\alpha }$.
We are going to introduce relations between the rectangular boxes of
$SFR$ and $LFR$. A box $\De \in S_{i, \al}$ is called
$relevant$ if there exists a box $\De ' \in L_{i, \al '}$
such that $\De \subset \De '$. In this case we call the
smallest such box $\De '$ the ancestor of $\De$. If this is not the
case the box $\De$, called irrelevant, is divided into $M^{i+1 - \al}$
boxes of the standard lattice ${\bf D}_{i}$, and we forget about the
corresponding division of frequencies on $p_{0}$. This is justified
because in these regions the frequencies on $p_{0}$ do not have any cutoff
imposed by the presence of large field boxes (recall that there cannot
be any ultraviolet limitation on $p_{0}$ because of our definition of
protection corridors). Since in the small field region an Euclidean invariant
propagator is going to be introduced, there is therefore
no need to keep the decomposition over $\al$ and over rectangular boxes.
>From now on, when we consider a small field region it is therefore
made of relevant rectangular boxes associated to specific ancestors
boxes of large field regions with $same$ index $i$ but lower index
$\al$, and of ordinary $cubes$ of
${\bf D}_{i}$. For these cubes $\De$ we also define a notion of ancestor.
We consider in turn all indices $i' * + \sum_{i }(\lambda_{i}^{t}) ^{2}
)} \eqno({\rm II.35})
$$
and we insert now our analogue of the Fadeev-Popov formula
(II.20).
We want that
the gauge transformations $\gamma $ in (II.20) cover all the small field
regions. As explained in the outline the scale given by the
axial gauge positivity plus the $A^{4}$ counterterm
is $A^{i} \simeq \lambda ^{-1/2}M^{i}$. Since we need
a small margin to gain some
small factor in the large field region, the small field
region was chosen in (II.26) to be of the type $A^{i} < (\lambda _{i}^{t})
^{-1/2-\epsilon _1}M^{i}$. Therefore we use as
a measure over $\gamma $ a quadratic form which
gives to $\ga$ at scale $i$ a typical size
$(\lambda _{i}^t) ^{-(1/2 + \epsilon _2)}$
where $1>> \epsilon _2 > \epsilon _1$. Hence we choose
as propagator
$$ \Gamma _{\rho _2 } (p)=
\sum_{i=1}^{\rho _2 }\Gamma _{\rho _2 } ^{i} (p)
\ ,\ \Gamma _{\rho _2 } ^{i} (p) =
\sum_{i=1}^{\rho _2 } (\lambda _{i}^t) ^{-(1 + 2 \epsilon _2)}
{\kappa ^{i} (p) \over p^{4}} \eqno({\rm II.36})
$$
where $\rho _2 << \rho _1$.
The Gaussian measure on $\gamma $ with covariance $\Gamma _{\rho _2}$
is called $d\nu _{\rho_2 }(\gamma )$. According to the decomposition
(II.15) we can also split in Fourier space $\gamma $ as $\sum_{i=0}^{\rho _2}
\gamma ^{i}(p),\ \gamma ^{i}(p) \equiv \kappa ^{i}(p) \gamma (p)$ (we
do not need the anisotropic indices at this stage because
the homothetic gauge and the corresponding Fadeev-Popov formula that we
are going to introduce are perfectly isotropic).
A Gaussian measure to bound the size of $\ga$ is however
not sufficient; for technical reasons we need to reinforce its
strength by a polynomial of high degree. In order for this polynomial
to behave at small $\ga$ as a small perturbation of a Gaussian measure
so that perturbative analysis remains all right, we take this
polynomial to give a slightly larger size,
$(\lambda _{i}^t) ^{-(1/2 + 2\epsilon _2)}$, to $\ga$, (still
much smaller than $\la^{-1}$). Therefore we define
$$ K_{\rho _2} (A_{s},B_{l}) = \int d\nu_{\rho _2} (\gamma )
e^{-(\ze/2) (\nabla_{B} (A_{s}, B_{l}, \gamma ,2))^{2}} e^{-\sum_i
((\lambda _{i}^t) ^{1/2 + \ep_2 }\gamma^{i}) ^{N}}
\eqno({\rm II.37})$$
where $N$ is some large integer, $\ze$ is the number close to 3/13
defining the homothetic gauge, and
$\nabla_{B}$ is the covariant derivative in the background field,
which is defined in the case of our truncated transformations by
$$ \nabla_{B} (A_{s},B_{l},\gamma ,2) \equiv
\partial _{\mu } (A_{s}^{\gamma ,2 })_{\mu }
- \sum_{j} \la [\kappa _{j} *(B_{l}^{rot\gamma ,2})_{\mu } ,
\kappa ^{j}*(A_{s}^{\gamma ,2})_{\mu } ]
\eqno({\rm II.38})
$$
where the star is a convolution in $x$-space, and for simplification
the Fourier transform
of $\kappa ^{j}$, $\kappa _{j}$... is from now on
also noted $\kappa ^{j}$, $\kappa _{j}$,.... The rotation $A^{rot, n}$
is defined in (II.32a-c).
>From now on we warn the reader that most of the time we use simply
the notation $\la$ instead of $\la _{i}^{t}$ and leave to the reader
to reconstruct the correct value according to the frequency of the
fields concerned; for instance in (II.38), $\la $ should be
understood as $\la_{j}^{t}$. This will simplify the rather complicated
formulas of this section. Similarly we omit from now on the explicit
dependence in $\la _{i}^{t}$ in (II.35-37) etc...
$K_{\rho _2 }(A)$ is well defined since it is a functional integral of
a bounded polynomial interaction with a Gaussian measure
with ultraviolet cutoff.
Then we write instead of (II.18) the functional measure of the theory as:
$$ \sum_{LFR} \int d\mu_{0,\rho _1} (A) d\nu _{\rho _2}(\gamma ) \chi _{LFR}
[K_{\rho _2}(A_{s},B_{l})] ^{-1}
$$
$$e^{(1/2)(-F_{sp}^{2} - + \sum_{i } \lambda ^{2}
)}
e^{-(\ze/2) (\nabla_{B} (A_{s},B_{l},\gamma ,2))^{2}} \eqno({\rm II.39})
$$
We impose now the true cutoff at a scale $M^{\rho }$
with $\rho << \rho _2 <<\rho _1$. To compensate
the gauge breaking effects of this true cutoff will
require some well defined counterterms which are computed in section III.
For the moment these counterterms are written simply as $CT_{\rho }$.
This true cutoff changes the formula (II.39) into
$$ \sum_{LFR}\int d\mu_{0,\rho_1 } (A) d\nu _{\rho_2 }(\gamma )
\chi _{LFR} (A) e^{-\sum_i
(\lambda ^{1/2 +2\epsilon _2}\gamma^{i}) ^{N}}
$$
$$[K_{\rho ,\rho _2} (A_{s},B_{l},\ga)]^{-1} e ^{-(1/2) }
$$
$$e^{(1/2)(-F_{sp}^{2} - + \sum_{i} \la ^{2}
*)}
e^{-(\ze /2) (\nabla_{B} A_{s},B_{l},\gamma ,2)^{2} }
e^{CT_{\rho }(A^{\gamma ,2})} \ . \eqno({\rm II.40})
$$
It remains to explain what is $K_{\rho ,\rho _2}$ in this formula.
It is an analogue of $K_{\rho _2}$ in (II.36) but takes into account
the addition of the true cutoff at scale $\rh$ to the fake
protecting cutoff at scale $\rh_{2}$. However its precise definition
is somewhat complicated and we want to postpone it for a while, but let us
explain at least the guiding idea here.
We want to have a cutoff of the same scale and shape for the propagator
of the $A$ field in the homothetic gauge and the propagator of the
Fadeev-Popov ghosts. Therefore we do not want to use directly
the functional integral $K_{\rho _2}(A)$, which is our analogue of
the Fadeev-Popov
determinant; it would not have the correct cutoff on the ghost field
propagator. We use the fact that this functional integral tends to the
usual Fadeev-Popov determinant in the limit $A \to 0$, and we replace
it by a different functional integral $K_{\rho ,\rho _2}$ in which the
propagator of the ghost field is cut again at scale $\rho $, this time
in the way we want for nice perturbative computations. The reader might
object that therefore we have not inserted really the value 1 as in (II.20),
hence the formula does not correspond really to the axial gauge starting point.
This remark applies also to the imposition of the cutoff on $A^{\gamma ,2}$,
and indeed we explained already that our ansatz contains
more than simply the axial gauge ansatz. However the difference between
$K_{\rho ,\rho _2}$ and $K_{\rho _2}$ will be made of terms with momenta
between $M^{\rho }$ and $M^{\rho _2}$. They will not affect the validity
of Slavnov identities for the final theory, since
their effect on any fixed scale vanishes in the limit
$\rh \to \infty$, hence the replacement
of $K_{\rho _2}$ by $K_{\rho ,\rho _2}$ can be also considered as
part of the definition of our ultraviolet cutoff.
Let us stress that this point is technical; it would be presumably possible to
use the cutoff given by $K_{\rho _2}$ but the computation of the ghost
contribution to the gauge breaking counterterms, in particular the graph $G_4$
in section III would be more difficult. Since the contribution
of this graph is much smaller
than the contributions of $G_1$, $G_2$ or $G_3$ in section III, the
conclusions concerning the stability of the ultraviolet cutoff would
be presumably almost the same. However we prefer to use a complicated
redefinition of the initial cutoff on the $\gamma $
field which is our substitute for the Fadeev-Popov ghosts, in
order to allow in the next section a
simpler perturbative computation of $CT_{\rho }$.
We are going to give later in this section the precise definition
of $K_{\rho ,\rho _2}$. This
definition like most of the ansatz (II.40) is best expressed in terms of
$A^{\gamma ,2}$, the correct variable in (II.40) for a small field.
We explain therefore first the change of
variables which consists in using as a new variable
for the main functional integration $A'= A^{\gamma ,2}$ instead of $A$.
Remark that this change of variables is one to one, namely it is
possible to compute directly the initial field $A$ in terms of $A'$,
since the transformation $A \to A^{\gamma ,2}$ is invertible.
The inversion formula exists (also for higher orders approximations to
true gauge transformations) and is a rational function of $\gamma $.
(There are also good polynomial approximations to the inverse transformation,
namely the transformations $A \to A ^{-\gamma ,n}$).
For instance, if we write $A'= A^{\gamma ,2} = T(\gamma ).A + U(\gamma )$,
with $T.A = A- \lambda [A,\gamma ]$ and $U= \partial \gamma + 1/2[
\gamma ,\partial \gamma ] $, in su(2) space the matrix of $T$ is
$$T_{ab}(\gamma )= \delta _{ab}
-\epsilon _{abc} \lambda \gamma _{c}\eqno({\rm II.41})
$$
Its inverse is
$$T^{-1}_{ab} = {1 \over 1 + \lambda ^{2} \sum_{d}\gamma _{d}^{2}}
(\delta _{ab} + \epsilon _{abc} \lambda \gamma _{c} + \lambda ^{2}\gamma _{a}
\gamma _{b}) = \delta _{ab} + H_{ab} \eqno({\rm II.42})
$$
where $H$ is a small matrix;
the transformation $A \to A'$ is therefore inverted by $A= T^{-1}(A'-U)$.
Furthermore the Jacobian of the change of variables associated to a true gauge
transformation is one, since the linear piece is an inner automorphism.
For the truncated gauge transformation this is no longer exactly true.
For instance for the truncated transformation $A \to A^{\gamma ,2}$
the linear piece is $ A \to TA $,
and the Jacobian is $J(\gamma ) \equiv ( 1+ \lambda ^{2}\gamma ^{2})^{-1}$.
The formal Lebesgue measure changes therefore, if $A' = A^{\gamma ,2}$ as:
$$ \prod_{x} dA (x) \to \prod_{x} dA'(x) \prod _{x}
(1+\lambda ^{2}\gamma ^{2} (x))^{-1} \eqno({\rm II.43})
$$
Since we use Gaussian measures, we have a well defined analogue of this formal
formula. Let us consider again $d\mu _{0, \rho _1}(A)$ which is the
initial normalized Gaussian measure with propagator $C_{0,\rho _1}$ used
to define our functional integral over $A$.
This measure can be recomputed exactly in terms of
$A' = A^{\gamma ,2}$
using as a guide the following formal manipulations:
$A= T^{-1}(A'-U(\gamma ))$
$$ d\mu_{0,\rho _1} (A) = {e^{-(1/2)AC_{0,\rho _1}^{-1}A}dA \over \int
e^{-(1/2)AC_{0,\rho _1}^{-1}A}dA}
$$
$$= {e^{-(1/2)(A'-U)(T^{tr})^{-1}C_{0,\rho _1}^{-1}T^{-1}(A'-U)}J(\gamma )dA'
\over \int e^{-(1/2)(A'-U)(T^{tr})^{-1}C_{0 ,\rho_1}^{-1}T^{-1}(A'-U)}
J(\gamma )dA'}
$$
$$= d\mu _{0, \rho _1} (A')
{G(A',\gamma ) \over \int G(A', \ga) d\mu _{0, \rho _{1} (A')}}
\eqno({\rm II.44})
$$
where $G$ contains correction terms in the difference $H$ between
$T^{-1}$ and $Id$, and terms in $U$:
$$ G(A',\gamma ) \equiv e^{+A'^{t}H^{tr}C_{0,\rho _1}^{-1}T^{-1}A'+
A'^{t}(T^{tr})^{-1}C_{0,\rho _1}^{-1}HA'-A'^{t}H^{tr}C_{0,\rho _1}^{-1}HA'}
$$
$$
e^{+A'^{t}(T^{tr})^{-1}C_{0,\rho _1}^{-1}T^{-1}U+
U^{tr}(T^{tr})^{-1}C_{0,\rho _1}^{-1}T^{-1}A'-
U^{tr}(T^{tr})^{-1}C_{0,\rho _1}^{-1}T^{-1}U}
\eqno({\rm II.45})
$$
Our initial axial field $A$ has only nine scalar components since $A_0$
was identically 0. We want that the special-gauge field $A'$ contains
the usual twelve components. In fact one should have $A'_0\simeq 0^{\gamma ,2}
= \partial _0 \gamma + (\la /2)[\gamma ,\partial_0 \gamma ]$. But since the
change of variables $\gamma \to \partial _0 \gamma $ is not invertible and
we need to keep in our formulas a functional integration over $\gamma $, it
is convenient (although not necessary)
to create the $A'_0$ field ex nihilo by a functional formula which
peaks it automatically around the desired value. This formula is
$$1= L_{0,\rho _1}(\gamma )\int d\mu _{C_{0,\rho _1}}(A'_0)
F(A'_0,\gamma ) \eqno({\rm II.46})
$$
$$
F(A'_0, \gamma ) \equiv e^{-\sum_{\Delta \in {\bf D_{\rho_{1}}} }
\vert \Delta \vert^{-1}
\int_{\Delta } (A'_0 - \partial _0 \gamma -1/2
\lambda [\gamma ,\partial _0 \gamma ])^{N'} } \eqno({\rm II.47})
$$
where $N'$ is some large integer and $L_{0,\rho _1}$ is the inverse
of the integral in (II.46) so that (II.46) is true; it is a slowly
varying function of $\ga$ which can be integrated with the measure
on $\ga$ in (II.37).
In this way, since the frequency $\rho_{2}$ is much smaller than
$\rho_{1}$ the field $A '_{0}$ coincides very accurately at scale
$\rho_{2}$ with the desired expression $\partial _0 \gamma + (\la /2)
[\gamma ,\partial_0 \gamma ]$.
We write
$$ A'_{s} \equiv A_{s}^{\gamma ,2} \quad;\quad B'_{l} \equiv B_{l}^{rot
\gamma ,2} \quad ; (A'_{s})_{0} \equiv A'_{0} \quad; (B'_{l})_{0} \equiv 0
\eqno({\rm II.48})
$$
so that $A' = A'_{s} + B'_{l}$.
We obtain:
$$ \sum_{LFR}\int d\mu_{0,\rho_1 } (A') d\nu _{\rho_2 }(\gamma )
\chi _{LFR} (A) G(A',\gamma )
e^{-\sum_i(\la ^{1/2 +2\epsilon _2}\gamma^{i}) ^{N}}
$$
$$ e ^{-(1/2) }
e^{CT_{\rho }(A')}
$$
$$e^{(1/2)(-F_{sp}^{2}(A) - + \sum_{i} \la ^{2}
)} $$
$$
[K_{\rho ,\rho _2} (A_{s},B_{l},\ga)]^{-1} e^{-(\ze/2)
(\nabla_{B'_{l}}\cdot A'_{s} )^{2} }
L_{0,\rho _1} (\gamma ) F(A'_0, \gamma )
\eqno({\rm II.49})
$$
where now:
$$ \nabla_{B'_l}\cdot A'_{s} = \partial _{\mu } (A'_{s})_{\mu }
- \sum_{j} \la [\kappa _{j} *(B'_{l})_{\mu } ,
\kappa ^{j}*(A'_{s})_{\mu } ]
\eqno({\rm II.50})
$$
The operator
$ \nabla_{B'_l}$ is very important in what
follows, because the covariance of $A'_s$ in the small field regions
where we will perform most of our analysis is built out of it. This operator
has of course a spatial index $\mu $ which is omitted in (II.49-50); we hope
that the scalar product in (II.49) is clear; the r\^ole played by
the factor $\partial _{\mu } A^{\mu }$ in
the Landau, Feynman or homothetic gauge condition
is now played by the factor $( \nabla_{B'_l})_{\mu }(A'_s)_{\mu }$.
In this way we have both a functional integral over a twelve component field
$A'$ and a three component functional integral over $\gamma $. In (II.49)
it is now the old field $A$ which should be considered a function of $A'$
and $\gamma $ through the formula $A(A',\gamma )= T^{-1}(A'-U)$.
Let us turn now to the precise definition of $K_{\rho ,\rho _2}$. It is
given by an integral over a variable which we will call $\gamma '$ to
distinguish it from the variable $\gamma $ in (II.50).
We have first to reexpress $K_{\rho _2}$ in terms of the new variable $A'$.
Instead of using the inverse transformation $T^{-1}$ we will use the
approximate inverse transformation so that we have polynomial error terms.
More precisely we write:
$$ A_{\mu } = (A_{\mu }^{+\gamma ,2})^{-\gamma ,2} +
R_{\mu }(A,\gamma )\eqno({\rm II.51})$$
$$ B_{\mu } = (B^{rot\gamma ,2})^{rot (-\gamma ),2} + R'_{\mu } (B,\gamma )
\eqno({\rm II.52})$$
$$ R_{\mu }(A,\gamma )= \lambda ^{2}\bigl ([[A_{\mu },\gamma ],\gamma ] +
(1/2) [[\partial _{\mu } \gamma ,\gamma ],\gamma ]\bigr)
\eqno({\rm II.53})$$
$$ R'_{\mu }(A,\gamma )= \lambda ^{2}[[A_{\mu },\gamma ],\gamma ]
\eqno({\rm II.54})$$
We have first to express $ K_{\rho _2} (A)$ in terms of $A '$ and $\ga$:
$$ K_{\rho _2} (A',\ga) = \int d\nu_{\rho _2} (\gamma ') e^{-\sum_i
((\lambda _{i}^t) ^{1/2 +\epsilon _2/4}\gamma '^{i}) ^{N}}
$$
$$
e^{-(\ze /2) \biggl(\nabla_{B} \bigl( (A'_s)^{-\gamma ,2}
+R_{\mu }(A_s,\gamma )
, (B'_l)^{-rot \gamma ,2} + R'_{\mu }(B_l,\gamma ),\; \gamma ',\;2\bigr)
\biggr)^{2}} \eqno({\rm II.55})$$
(see (II.37) for definition of our notation $\nabla_{B}(A,B,\ga, 2)$).
We can now compute:
$$ \biggl( (A'_s) _{\mu }^{-\gamma ,2} + R_{\mu }(A,\gamma )
\biggr)^{\gamma ',2}
= (A'_s)_{\mu } + D_{\mu }(A'_s) \cdot
(\gamma ' -\gamma ) + S_{\mu } (A'_s ,\gamma ,\gamma ')\eqno({\rm II.56})$$
$$ S _{\mu } (A', \gamma ,\gamma ') =
+ (\lambda /2)([\partial _{\mu }(\gamma -\gamma ') ,\gamma '] + [\gamma -\gamma
', \partial _{\mu } \gamma ] + O(\lambda ^{2})\eqno({\rm II.57})
$$
$$
O( \lambda^{2}) = R -\lambda [R,\gamma '] - \lambda ^{2}\biggl(
[[A'_{\mu },\gamma ],\gamma '] +
(1/2) [[\gamma , \partial _{\mu }\gamma ], \gamma ']\biggr)
\eqno({\rm II.58})$$
Similarly:
$$ \biggl( (B'_l)^{rot -\gamma ,2} + R' \biggr)^{rot \gamma ',2}
= (B'_l ) ^{rot (\gamma ' -\gamma) ,2} + S' \eqno({\rm II.59})
$$
$$ S' = R' -\lambda [R',\gamma '] - \lambda ^{2} [[B',\gamma ],\gamma ']
\eqno({\rm II.60})
$$
We substitute (II.56-60) into (II.55).
We find:
$$ \nabla_{B} ( (A'_s)^{-\gamma ,2}+R_{\mu }(A_s,\gamma )
, (B'_l)^{-rot \gamma ,2} + R'_{\mu }(B_l,\gamma ),\gamma ',2) =
$$
$$ = \partial _{\mu }
[(A'_s)_{\mu } + D_{\mu } (A'_s) (\gamma '-\gamma ) + S_{\mu }(A'_s,\gamma
,\gamma ')]
$$
$$- \sum_{j} \la [ \kappa_j * ((B'_l)^{rot ( \gamma ' - \gamma),2}
+ S')_{\mu } ,
\kappa ^{j}* ((A'_s)_{\mu } + D_{\mu } (A'_s) (\gamma '-\gamma ) +
S_{\mu }(A'_s,\gamma ,\gamma '))]
$$
$$ = (\nabla_{RB, \gamma '-\gamma })_{\mu }
(( A'_s)_{\mu} + D_{\mu}(A'_s)(\gamma ' - \gamma )) + S" \eqno({\rm II.61})
$$
where the operator $\nabla _{RB, \gamma '-\gamma }$ is defined by
$$ (\nabla_{RB, \gamma '-\gamma })_{\mu } A ' = \partial_{\mu } A ' -
\sum_{j} \la [ \kappa_j * (B'_l)^{rot (\gamma '-\gamma) ,2}_{\mu} ,
\kappa ^{j}* A ' ] \eqno({\rm II.62})
$$
and
$$ S" = \partial S - \sum_{j}\la [ \kappa_j * S' ,
\kappa ^{j}* (A'_s + D (A'_s) (\gamma '-\gamma ) +
$$
$$S(A'_s,\gamma ,\gamma '))]
- \sum_{j} \la [ \kappa_j * ((B'_l)^{rot (\gamma ' - \gamma ) }+S'),
\kappa ^{j}* S(A'_s,\gamma ,\gamma ')] \eqno({\rm II.63})
$$
Therefore
$$ \biggl(
(\nabla_{RB, \gamma '-\gamma })_{\mu }
( A'_s + D(A'_s)(\gamma ' - \gamma ) )_{\mu } +S''
\biggr)^{2} $$
$$ = \biggl( \nabla_{RB,\gamma '-\gamma } \cdot
\bigl( A'_s + D(A'_s)(\gamma ' - \gamma ) \bigr)
\biggr)^{2} + \Si (A', \gamma , \gamma ') \eqno({\rm II.64})
$$
where
$$
\Si (A',\gamma ,\gamma ')
\equiv 2 S''\biggl( \nabla_{RB, \gamma '-\gamma }\cdot
\bigl( A'_s + D(A'_s)(\gamma ' - \gamma ) \bigr) \biggr) + (S")^{2}
\ ,\eqno({\rm II.65})$$
is a small correction term which will be treated as an interaction.
(There are some implicit summations over $\mu $ in the
formulas above).
With these notations:
$$ K_{\rho _2} (A',\ga) = \int d\nu_{\rho _2} (\gamma ') e^{-\sum_i
(\la ^{1/2 +2\epsilon _2}\gamma '^{i}) ^{N}}$$
$$
e^{-(\ze /2) \bigl(\nabla_{RB,\gamma '-\gamma } \cdot
( A'_s + D(A'_s)(\gamma ' - \gamma ) ) \bigr) ^{2}}
e^{-(\ze /2)\Si (A ', \gamma ,\gamma ') }
\eqno({\rm II.66})$$
Let us develop $\nabla_{RB, \gamma '-\gamma } \cdot D$. We have
$$ \nabla_{RB, \gamma '-\gamma } \cdot D(A'_s) =
U(A'_s,B'_l)+V(A'_s,B'_l,\gamma '-\gamma )
\eqno({\rm II.67})$$
$$
U(A'_s,B'_l)
\equiv \partial ^{2} -\lambda \partial [A'_s , . ] -
\lambda \sum_{j} [\kappa _j * B'_l , \kappa ^{j} *\partial .]
\eqno({\rm II.68})$$
$$ V(A'_s, B'_l, \gamma '-\gamma ) = + \lambda ^{2}
\sum_j [\kappa _j * [B'_l, \gamma '-\gamma ],\kappa ^j * \partial . ]
+ \lambda ^{2} \sum_{j}
[\kappa _j * (B'_l )^{rot \gamma '-\gamma } , \kappa ^j * [A'_s,.] ]
\eqno({\rm II.69})
$$
Again we write
$$ \biggl( U+V +\nabla_{RB, \gamma '-\gamma } \cdot A'_s \biggr)^{2} =
\biggl(U +\nabla_{RB, \gamma '-\gamma } \cdot A'_s \biggr)^{2} + W
$$
$$ W \equiv 2 V\cdot \bigl(U + \nabla_{RB, \gamma '-\gamma } \cdot
A'_s \bigr) + V^{2} \eqno({\rm II.70})
$$
(to simplify these formulas we write them like squares instead of
scalar products, and we omit the necessary transpositions of
operators, which are straightforward). Since $V$ is small (with a
factor $\la^{2}$, we can
treat $\Si +W$ in the integral over $\gamma '$ as a complicated
interaction, and we group together the
measure $d\nu _{\rho _2 }(\gamma ')$ with the main quadratic piece
$(\ze/2)< \gamma ' -\gamma ,
U^{tr} U (\gamma ' -\gamma ) >$. Again we write $U^{2}$ for $ U^{tr}
U$, etc...
If the measure $d\nu _{\rho _2 }(\gamma ')$ had been translation
invariant we would have as a main piece
a Gaussian integral over $\ga '-\ga$. This is not exactly the case,
but by our condition $\ep_{2} >> \ep_{1}$ it will be approximately
true in the small field region and the correction terms will be the
ones containing powers of $\Ga^{-1}$, $\Ga$ being the propagator
(II.35) for $d\nu _{\rho _2 }$.
Therefore we define a new Gaussian variable $\gamma"$
which has propagator $( \ze U^{2}+\Gamma ^{-1})^{-
1}$ and which is defined by:
$$ \gamma " = \ga ' -\ga +
{ \ze U (\nabla_{RB, \gamma '-\gamma } \cdot A'_s) + \Ga^{-1}\ga
\over \ze U^{2}+ \Gamma ^{-1} }
\eqno({\rm II.71})$$
We define also
$$ \Om \equiv
{\bigl(\ze U (\nabla_{RB, \gamma '-\gamma } \cdot A'_s) +
\Ga^{-1}\ga \bigr)^{2} \over \ze U^{2}+ \Gamma ^{-1} } -
\ze (\nabla_{RB, \gamma '-\gamma } \cdot A'_s)^{2} - \ga\Ga^{-1}\ga
\eqno({\rm II.72})$$
We see that $\Om$ is small as $\Ga^{-1}$ when $\Ga^{-1} \to 0$. This
is the reason
for which we can treat it as an interaction, and it is here that
enters in a key way the fact that our explicit Fadeev-Popov averaging formula
effectively covers all the small field region, where it
performs therefore correctly its gauge fixing job.
Now we obtain, rewriting everything in terms of $\ga''$
$$ K_{\rho _2} (A', \ga ) = (\det{\ze U^{2}+\Gamma ^{-1}
\over \Gamma ^{-1}})^{-1/2}
$$
$$ \int d\pi _{\rho _2} (\gamma ") e^{-\sum_i
(\la ^{1/2 +2\epsilon _2}\kappa ^{i}
(\gamma '(\gamma ",\gamma ))) ^{N}}
e^{-(\ze /2) (\Si +W+\Om)(A',\gamma ,\gamma '(\gamma ,\gamma "))}
\eqno({\rm II.73})$$
by completing the square.
The determinant $(\det{\ze U^{2}+\Gamma ^{-1}
\over \Gamma ^{-1}})^{1/2} = (\det (1 + \ze \Ga U^{2}))^{1/2} $
which appears in $K_{\rho _2}(A)^{-1}$
is the analogue of the Fadeev-Popov
determinant (up to the constant normalization
$\det \Gamma ^{-1}$). In particular if
we neglect $\Gamma ^{-1}$ (which is small as $\lambda ^{1+2\epsilon_2}$)
and rewrite $\det \vert U\vert$
as a fermionic integral over ghosts, we recover the ordinary ghost-ghost
propagator $p^{2}$ and the complete ghost-ghost-field coupling at least for
fields of lower momentum than the momentum of the ghosts. Indeed in this case
the total field
$A'_s +B'_l$ appears in $U$. This justifies the perturbative
computations of the next section.
Indeed since $\kappa _j* B'_l$ has only low frequencies, we have
$ \lambda [\kappa _j *B'_l ,\kappa ^j*\partial .] \simeq
\lambda \partial [ \kappa _j * B'_l, \kappa ^j *.]$, and therefore
$U(A'_s+B'_l) \simeq \partial _{\mu } \sum_{j}D_{\mu } (
\kappa _j *(A'_s+ B'_l) + (1-\kappa _j)* A'_s) \kappa ^j * $.
We are at last in the position to define $K_{\rho ,\rho _2}$.
We write $\De$ for the ordinary Laplacian and
$$ \ze U^{tr}U + \Ga^{-1} =
(\ze\Delta^{2} +\Gamma ^{-1}) \biggl( 1 -\lambda
{\ze\Delta \over (\ze\Delta^{2} +\Gamma ^{-1})}
(\partial [A'_s , . ] +
\sum_{j} [\kappa _j* B'_l , \kappa ^{j} *\partial .])$$
$$- \lambda {1\over (\ze\Delta^{2} +\Gamma ^{-1})}
( \partial [A'_s , . ] +
\sum_{j} [\kappa _j* B'_l , \kappa ^{j} *\partial .])\ze\Delta
$$
$$+ \lambda ^{2}{\ze\over (\ze\Delta^{2} +\Gamma ^{-1})}
( \partial [A'_s , . ] +
\sum_{j} [\kappa _j* B'_l , \kappa ^{j} *\partial .])^{2}\biggr)
\eqno({\rm II.74})$$
and we change this operator into:
$$ \bigl( \ze U^{tr}U + \Ga^{-1} \bigr)_{\rho } \equiv
(\ze\Delta +\Gamma ^{-1}) \biggl( 1 -\lambda \kappa _{\rho }(p)
{\ze\Delta \over (\ze\Delta^{2} +\Gamma ^{-1})}
( \partial [A'_s , . ] +
\sum_{j} [\kappa _j* B'_l , \kappa ^{j} *\partial .])
$$
$$ - \lambda {\kappa _{\rho }(p)\over (\ze\Delta^{2} +\Gamma ^{-1})}
( \partial [A'_s , . ] +
\sum_{j} [\kappa _j *B'_l , \kappa ^{j} *\partial .])
\ze\Delta $$
$$+ \lambda ^{2}{\ze\kappa _{\rho }^{2}(p)
\over (\ze\Delta^{2} +\Gamma ^{-1})}
( \partial [A'_s , . ] +
\sum_{j} [\kappa _j* B'_l , \kappa ^{j} *\partial .])^{2} \biggr)
\eqno({\rm II.75})$$
We define
$$ K_{\rho , \rho _2} (A', \ga) = (\det{\bigl( \ze U^{tr}U
+ \Ga^{-1} \bigr)_{\rho }
\over \Gamma ^{-1}})^{-1/2}
$$
$$ \int d\pi _{\rho _2} (\gamma ") e^{-\sum_i
((\lambda _{i}^t) ^{1/2 +2\epsilon _2}\kappa ^{i}
(\gamma '(\gamma ",\gamma ))) ^{N}}
e^{-(\ze/2) [\Si+W +\Om ]}
\eqno({\rm II.76})$$
In this formula the remaining functional integral is close to one
since $\Si +W +\Om $ is a small interaction, as explained above.
The main piece is the determinant
which is nothing but the ordinary Fadeev-Popov term.
The important fact about this way to reimpose a cutoff at scale $\rho $
is that in the small field regime and at zero external momenta for $A'$,
the only contribution of $K_{\rho ,\rho _2}$ to the counterterm
$\lambda ^{4}A'^{4}$ (see below) comes from the ordinary Fadeev-Popov
determinant.
At this order we obtain therefore as only contribution (taking out constant
factors, in particular a global power of $\ze$)
$$\biggl( \det \Delta^{2} \biggl( 1 -\lambda \kappa _{\rho }(p)\Delta^{-1}
( \partial [A'_s , . ] +
\sum_{j} [\kappa _j* B'_l , \kappa ^{j} *\partial .])
$$
$$- \lambda \kappa _{\rho }(p) \Delta ^{-2}
( \partial [A'_s , . ] +
\sum_{j} [\kappa _j* B'_l , \kappa ^{j} *\partial .])\Delta
$$
$$+ \lambda ^{2}{\kappa _{\rho }^{2}(p)
\over \Delta^{2}}
( \partial [A'_s , . ] +
\sum_{j} [\kappa _j* B'_l , \kappa ^{j} *\partial .])^{2}\biggr)
\biggr)^{1/2}
$$
$$ = \det \vert \Delta - \la \kappa _{\rho }(p) ( \partial [A'_s , . ] +
\sum_{j} [\kappa _j* B'_l , \kappa ^{j} *\partial .])
\vert
\eqno({\rm II.77})$$
which is the same as the ordinary Fadeev-Popov determinant with a cutoff
on the ghosts propagator of the desired simple form, and a ghost-ghost-field
vertex which is the ordinary coupling to the full field $A'$ at least at zero
momentum external field $A'$. This proves that the gauge
breaking effect associated to this cutoff can be
computed in the way this is done in the next section.
Let us recapitulate our starting point:
$$ \sum_{LFR}\int d\mu_{0,\rho_1 } (A') d\nu _{\rho_2 }(\gamma )
\chi _{LFR} (A) G(A',\gamma )
e^{-\sum_i(\la ^{1/2 +2\epsilon _2}\gamma^{i}) ^{N}}
$$
$$ e ^{-(1/2) }
e^{CT_{\rho }(A')}
$$
$$e^{(1/2)(-F_{sp}^{2}(A) - + \sum_{i} \la ^{2}
)} $$
$$
[K_{\rho ,\rho _2} (A', \ga)]^{-1} e^{-(\ze /2) (\nabla_{B'_{l}} \cdot
A'_{s})^{2} }
L_{0,\rho _1} (\gamma ) F(A'_0, \gamma )
\eqno({\rm II.78})
$$
where $K_{\rho ,\rho _2}$ is defined by (II.76), and we can fix e.g.
$N=100$ in what follows.
This functional integral is now similar in the first orders of perturbation
theory to the
ordinary functional integral with ultraviolet cutoff $\kappa _{\rho }(p)$
both on the field and ghosts propagator.
In these conditions it is easy to compute the counterterms
$CT_{\rho }(A')$ which restore Slavnov identities. We show now how to perform
this task.
\vfill\eject\medskip
\noindent {\bf III. Computation of the counterterms due to the ultraviolet
cutoff}
\medskip
In all this section the collaboration of Joel Feldman is gratefully
acknowledged. The main result on the stability of certain types
of cutoffs was derived with him
around 1986; there is also an exposition of this result
in [S] and [R].
The computation of the gauge variant counterterms which restore Ward identities
is made in terms of the field $A'$. For this computation
we can assume that $A'=A'_s$ and $B'_l = 0$. Furthermore $in$ $this$ $section$
we write $A$ for simplicity instead of $A'$.
Our ultra violet cutoff does not break global SU(2) or
Euclidean invariance (small Euclidean breaking effects nevertheless occur due
to the infrared cutoff; for instance in the case of a torus there exist such
effects due to the lattice
structure of $\Lambda^*$, but they are tied to the unit scale and do not need
counterterms). Therefore the only new relevant or marginal operators that
we should consider are
$ -{\rm Tr}A_{\mu}A_{\mu} $, $ (-{\rm Tr}A_{\mu}A_{\mu})^2 $,
$ (-{\rm Tr}A_{\mu}(-\Delta)A_{\mu})$ and
$ -{\rm Tr}(\partial _{\mu }A_{\mu })^2 $ which we abbreviate respectively as
$A^2$, $A^4$, $A(-\Delta )A$ and $(\partial A)^2$ (recall the convention that
traces are definite negative). This is only true for the SU(2) theory; for
an SU(N) theory there would be a longer list of operators to consider
and the analysis would be more complicated.
In fact our gauge breaking cutoff also disturbs
the magic relation $Z_2Z_4=Z_3^2$ which relates the multiplicative
renormalization of $F_2 , F_3$ and $F_4$ in $F^2$ and expresses the fact that
up to a rescaling of $A$ only the coupling constant $\lambda$ is renormalized
[IZ]. To correct this problem, using the possibility of rescaling $A$, we need
only to introduce a single counterterm, for instance of the type $F_4$.
Therefore the counterterms that we introduce are:
$$ e^{CT} = e^{-a_{\rho} \int_{\Lambda} (A^4/4~!) -b_{\rho}
\int_{\Lambda} (A^2/2~!)
-c_{\rho} \int_{\Lambda} A(-\Delta)A
-d_{\rho} \int_{\Lambda} (\partial A)^2 -e_{\rho} \int_{\Lambda} F_{4} }
\eqno({\rm III.1})
$$
The relevant counterterm $b_{\rho} \int_{\Lambda} (A^2/2) $ must be fine tuned
exactly to have a renormalized mass which is zero. This is the same problem
than fixing the critical
bare mass in infrared $\phi ^{4}_{4}$ [FMRS1],[R] and should be solved
by a fixed point argument as in [R] or
using a full renormalization of the two point function (and a one particle
irreducible analysis) as in [FMRS1].
For the marginal counterterms, an analysis to lowest order in perturbation
theory is in fact enough for our purpose (because of asymptotic freedom,
further orders again should give no contributions to finite scales in the limit
$\rho \to \infty$). We obtain:
\medskip
\noindent {\bf Lemma III.1}
$$ a_{\rho} \simeq a \lambda _{\rho}^4 , \quad
b_{\rho} \simeq b M^{2\rho} \lambda _{\rho}^2 , \quad
c_{\rho} \simeq c \lambda _{\rho}^2 , \quad
d_{\rho} \simeq d \lambda _{\rho}^2 , \quad
e_{\rho} \simeq e \lambda _{\rho}^4 . \eqno({\rm III.2})$$
Furthermore by choosing the cutoff of the form (II.14) with $\eta $ small
enough (depending on the shape of $\tau $), the coefficient $a$
is strictly positive\footnote*{
It is not clear whether a cutoff for which $a$
would be negative (or zero) is strictly
forbidden for a constructive analysis. The answer
may indeed depend on considering irrelevant counterterms
of higher order generated by the cutoff, which may stabilize
the theory. The analysis would certainly be much more
complicated and we will therefore not try to explore this possibility here.}.
\medskip
\noindent {\bf Proof } We recall the Feynman rules for the pure SU(2) gauge
theory in a general gauge with parameter $\ze$ (the case $\ze=1$
corresponds to the Feynman gauge, and $\ze = \infty$ corresponds to
the Landau gauge) [IZ].
The propagators for the Yang-Mills fields and the
ghost fields are respectively:
$$ \de _{ab}\bigl( {\delta _{\mu \nu } \over p^{2}} + (1/\ze-1)
{ p_{\mu}p_{\nu}\over p^{4} } \bigr)\
\quad; \quad { \delta _{ab} \over p^{2} } \eqno({\rm III.3})
$$
The interaction vertices are of three kinds. For simplicity we always
forget to write the overall multiplication factor (of $2\pi $)
and the $\delta $ function
which expresses momentum conservation which equips them.
These three kinds of vertices are then pictured in Fig. III.1.
\vskip 5.5cm
\centerline {\bf Figure III.1}
\medskip
We concentrate on the computation of the $A^{4}$ counterterm, which is the
most interesting, and include also the computation of the $A^{2}$ counterterm.
The other ones are less interesting and left to the reader.
At one loop, which also means at order $\lambda ^{4}$
in perturbation theory, there are 4 graphs which may contribute to the
$A^{4}$ term. They are pictured in Fig. III.2 and called $G_1$, $G_2$, $G_3$
and $G_4$. To compute their contribution to the coefficient $a$,
we may assume by symmetry that in
all four external legs, both the space time and group indices are equal to 1.
\noindent a) Computation of $G_1$
The graph is obtained by applying 4 derivatives ${\partial \over
\partial A_1^1}$ on $(1/2!) (-F^2/4)^2$. The result is
$3 (\partial ^2 F^2/4)^2$ where derivatives are taken with respect
to $A^1_1$. The only non vanishing pieces come from the derivatives acting
on the commutator in $F$, hence $\partial ^2 F^2/4$ gives
$(1/2) (\partial F)^2$.
\eject
\topinsert
\vskip 6.3cm
\endinsert
\centerline {\bf Figure III.2}
\medskip
Moreover we have
$\partial F^c_{\alpha \beta } = \epsilon ^{c1b} [A^b_{ \delta }
\delta _{\alpha 1} -A^b_{\alpha } \delta _{\beta 1}]$,
where $\epsilon $ is the usual antisymmetric tensor.
But remark that if $\alpha =\beta =1$ the term vanishes. Hence when
developing the square $(1/2)(\partial F)^2$ the cross terms vanish.
Therefore this square gives
$(\epsilon ^{c1b})^2(A^b_{\beta })^2 \delta _{\alpha 1}$, $\beta \ne 1$.
There are now two possible Wick contractions,
a sum over three values (2,3 and 4) for $\beta $ and a sum over 2 values
(2 and 3) for $b$. Collecting all factors we obtain a positive coefficient
3.4$(3+3(1/\ze-1)/2 + 5 (1/\ze-1)^{2}/8)$ = $36 + 18 (1/\ze-1)+ 15(1/\ze-1) ^{2}/2$
in front of the integration $\int { d^{4}k \over k^{4}}$
over the loop momentum of $G_1$.
\noindent b) Computation of $G_2$.
We apply 4 derivatives on $(1/3!) (-F^2/4)^{3}$. The result is
$-6(\partial ^2 F^2/4)(\partial F^2/4)^2$ where derivatives are again with
respect to $A_1^1$. The
term in $\partial ^2 F^2/4$ is the same as before, hence gives
$(\epsilon ^{c1b})^2(A^b_{\beta })^2 \delta_{\alpha 1}$, $\beta \ne 1$.
But we have now two trilinear vertices in $\partial F^2/4$ hence terms with
derivative couplings; remark that a partial derivative $\partial _{\mu }$ can
be replaced by $-ik_{\mu }$. The computation of this term leads to two
identical vertices, one which gives $\epsilon ^{1mn} A^n_{\mu }
[\partial _1 A^m_{\mu } -\partial _{\mu } A^m_1]$,
and the other with $m,n, \mu $ respectively replaced by $p,q,\lambda $.
In the Wick contraction schemes we can first contract to form the line between
these two trilinear vertices. Since the two half legs of the remaining vertex
bear the same index $\beta \ne 1$, a tedious computation gives that the only
term compatible with future contractions is
$(\epsilon ^{1mn})^2 (A^m_{\mu })^2 [4k_1^2 +k_{\mu }^2]$.
Using Euclidean symmetry, this is equivalent to
$(\epsilon ^{1mn})^2 (A^m_{\mu })^2[5k_{1}^2]$.
Contracting with the remaining vertex, we have now as before two
possible Wick contractions, a sum over three values (2,3 and 4) for $\beta $
and a sum over 2 values (2 and 3) for $b$. Collecting all factors we
obtain a negative coefficient $-6\cdot 2\cdot 2 [ 15 k_1^2/k^{2} + (1/\ze-1)
(\sum_{\mu \ne 1}k_{1}^{2}k_{\mu}^{2}/k^{4} + 2
(3 \sum_{\mu \ne 1}k_{1}^{2}k_{\mu}^{2} +
\sum_{\mu \ne 1, \mu ' \ne 1} k_{\mu}^{2}k_{\mu
'}^{2})/k^{4} + (1/\ze-1)^{2}
\sum_{\mu \ne 1, \mu ' \ne 1} k_{1}^{2}k_{\mu}^{2}k_{\mu '}^{2}/k^{6})]$
which is
equivalent by Euclidean symmetry to $-90 -45 (1/\ze-1)- 15 (1/\ze-1)^{2} $
in front of the integration
$\int { d^{4}k \over k^{4}}$ over the loop momentum of $G_2$.
\noindent c) Computation of $G_3$
We apply 4 derivatives on $(1/4!) (-F^2/4)^4$. The result is
$+(\partial F^2/4)^4$ where
derivatives are again with respect to $A_1^1$. The term in $\partial F^2/4$
gives the same trilinear vertex as before, hence gives
$\epsilon ^{1mn} A^n_{\mu }[\partial _1 A^m_{\mu } - \partial _{\mu }
A^m_1 ]$, In the Wick contraction schemes we can first choose one particular
leg of vertex 1 to form a first line between two trilinear vertices. To
choose the vertex (2,3 or 4) to which this leg contracts gives a
factor 3. After this contraction has been performed, the line equipped
with two not yet contracted fields gives a
term $(\epsilon ^{1mn})^2 [2k_1^2 (A_{\mu }^m)^2 +k_{\mu }^2
(A_1^m)^2 -3k_1 k_{\mu } A_1^m A_{\mu }^m]$.
Here we can assume $\mu \ne 1$. We can now contract once more to create one
line between the two remaining vertices, and this can be done in all
possible ways, hence gives a different term, which is
$(\epsilon ^{1mn})^2 [4k_1^2 (A_{\mu }^m)^2 +k_{\mu }^2 (A_1^m)^2
-6k_1 k_{\mu } A_1^m A_{\mu }^m +k_{\mu } k_{\lambda } A_{\mu }^m
A_{\lambda }^m]$.
We can assume that $\mu \ne 1$ in the first three terms and that
$\mu =\lambda =1$ is
excluded in the last one. It remains to contract together both
expressions. We have as before two possible Wick contractions, a sum
over three values (2,3 and 4) for $\mu $ and a sum over 2 values (2 and 3)
for $m$. After collecting all factors and taking into account Euclidean
symmetry to convert it into units of
$\int { d^{4}k \over k^{4}}$, we find a final factor
in front of the integration over the loop momentum of $G_3$
$6( 9 +1/4 + 9(1/\ze-1)/2 + 5 (1/\ze-1)^{2}/4 = 55.5 + 27 (1/\ze-1)+ 7.5
(1/\ze-1) ^{2}$.
\noindent d) Computation of $G_4$
We apply 4 derivatives on $(1/4!) (F.P.)^4$, where $F.P.$ means the
Fadeev-Popov term $\partial _{\mu } \bar \eta _{a}
(D_{\mu } \eta )_a$, with $D$ the covariant derivative. The result is
$(\partial _1 \bar\eta _a \epsilon _{ab1} \eta_b)^4$.
The combinatoric is easier. We obtain a factor 6 for the Wick
contractions, a factor 2 for summations over latin indices and a minus
sign corresponding to the fermionic loop, which comes from reordering
correctly the anticommuting fields $\eta$ and $\bar\eta $. Hence the
contribution is $-12\cdot k_1^4$ in front of the integration over the loop
momentum of $G_4$. Applying the same conversion rate, we obtain in units
of $(k^2)^2$ a final combinatoric factor of $-1.5$.
Remark that when the cutoff is 1 we can all add the terms together
and the 4 coefficients add up to 0. This is a particular case of
the famous miracle of renormalizability (at one loop...) of four dimensional
gauge theories.
Let us perform now a similar analysis for the $A^2$ counterterm.
There are three graphs contributing at order $\lambda ^2$, pictured in
Fig. III.3.
The first graph, $G'_{1}$, gives a computation quite similar to that
of $G_{1}$. We have
$\partial \partial(- F^2/4) = -(1/2) (\partial F)^2$. Again
$\partial F^c_{\alpha \beta } = \epsilon ^{c1b} [A^b_{ \be }
\delta _{\alpha 1} -A^b_{\alpha } \delta _{\beta 1}]$
which is non zero only for $\al . \be \ne 1$. The cross terms therefore
again vanish and we find
$(\epsilon ^{c1b})^2(A^b_{\beta })^2 \delta _{\alpha 1}$, $\beta \ne
1$.
There are two values for $b$ and three for $\be$. Hence the
contribution is $-6(1+(1/\ze-1)/4)$ in front of the integration over
the loop momentum (in units of $1/k^{2})$.
\eject
\topinsert
\vskip 5cm
\endinsert
\centerline {\bf Figure III.3}
\medskip
The second graph, $G'_{2}$, is given by
$\partial \partial (1/2)(F^2/4)^2 =
(\epsilon ^{1mn} A^n_{\mu } [\partial _1 A^m_{\mu } -\partial _{\mu }
A^m_{1}])^2$ (which is non zero only for $\mu \ne 1$). The contribution
is $2(9k_1^2/k^{4} +
(1/\ze-1)(k_{\mu}^{2}k_{1}^{2}+k_{\mu}^{2}k_{\mu '}^{2})/k^{6}) =
(9/2 +6(1/\ze-1) /4)$ in front of the integration over the loop momentum.
The last graph, with ghosts, $G'_{3}$, gives $\partial \partial (1/2)(F.P.)^2
=(\partial F.P.)^2 =(\partial _1\bar \eta _a \epsilon _{ab1} \eta _b)^2$.
There is a minus sign due to the fermion loop (two minus signs due
to the rule $\partial_{\mu } \to -ik_{\mu }$ compensate; beware
that there is a sign mistake in the corresponding computation in [R]).
The contribution is therefore
$-2k_1^2 /k^{4} =-(1/2)/k^2$ in front of the integration over the
loop momentum.
The result for the $(A^2/2)$ term in the region where the ultraviolet
cutoff is one is obtained by adding all the terms and is $
-6-6 /4(1/\ze-1)-1/2+9/2+6 /4(1/\ze-1)$=-2
times the loop integration. Remark that this result is independent of
$\ze$.
To complete the Lemma, we want to study the sign of the $A^4$ counterterm.
Let us explain why it is important to us.
Our strategy is to cancel explicitly the $A^4$ and $A^2$ contributions
due to the gauge breaking character of our ultraviolet cutoff by
appropriate counterterms. Remark that strictly speaking, only the $A^2$
contribution diverges as $\rho \to \infty$ and requires a counterterm
(for the $A^4$
term the coefficient of the divergent piece is 0, as computed above).
However this $A^2$ counterterm is positive (since the contribution is
negative, see the -2 above). This is dangerous for stability
estimates. We will use the (finite) $A^4$ counterterm to control this
dangerous $A^2$ term and stabilize the theory. But this requires that we
use an ultraviolet cutoff such that the $A^4$ counterterm is negative,
hence such that the total $A^4$ contribution induced by the cutoff is
positive. As a consequence of our expansion the leading contribution
is the one-loop contribution; we want its sign to be positive.
We show now
that this is possible if we start with a cutoff function
of a particular shape such as (II.14) $\eta$ being a small constant.
This explains
at last the curious definition (II.14) of our ultraviolet cutoff.
Later we will show that this particular shape also leads to a stabilizing
functional integral associated to a large background field.
Let $\ka (p)$ be the ultraviolet cutoff function in momentum space. Up
to now we did not take it into account. Remark that
since there is one cutoff per propagator the cutoff acts differently
on $G_{1}$, $G_{2}$, $G_{3}$ and $G_{4}$. More precisely
using the coefficients computed in the
preceding section, the one loop contribution to the $(A^4/24)$ term is, for a
single cutoff $\kappa _{\rho }(p) =\ka (pM^{-\rho })$ (all our integrals
are infrared regularized and "finite" means finite as $\rho \to\infty$):
$$ \int {d^4p \over p^4}\bigl[
(36 +18 (1/\ze -1) + 7.5 (1/\ze -1) ^{2})\kappa ^2 (pM^{-\rho })
$$
$$ -(90
+45 (1/\ze -1) + 15 (1/\ze -1)^{2}) \kappa^3
(pM^{-\rho })$$
$$+(54 +27 (1/\ze -1) + 7.5 (1/\ze -1) ^{2})\kappa^4 (pM^{-\rho }) \bigr]
=0 \cdot \rho \ + \ {\rm finite \ terms} \eqno({\rm III.4})
$$
where the finite terms are finite functions of the particular shape of
$\ka$ and are therefore difficult to compute in the general case.
However we are going to use a shape such as (II.14) in which there is
a free parameter $\et$ that we can vary, and we will study the finite
terms in the limit $\et \to 0$. In this case it is easy to
analyze the asymptotic behavior of the finite terms in (III.4).
For $\ka_{\rh}$ defined as in (II.13-14), the corresponding
contribution is indeed:
$$ \int_{1\le \vert p\vert M^{-\rh}<2} {d^4p \over p^4} \bigl[
(36 +18 (1/\ze -1) + 7.5 (1/\ze -1) ^{2})\kappa ^2 (pM^{-\rho })
$$
$$ -(90
+45 (1/\ze -1) + 15 (1/\ze -1)^{2}) \kappa^3
(pM^{-\rho })+(54 +27 (1/\ze -1) + 7.5 (1/\ze -1) ^{2})\kappa^4 (pM^{-\rho }) \bigr]$$
$$ + \int_{2+\et^{-1} < \vert p\vert M^{-\rh}\le 3 +\et^{-1}}
{d^4p \over p^4}\bigl[
(36 +18 (1/\ze -1) + 7.5 (1/\ze -1) ^{2})\kappa ^2 (pM^{-\rho })
$$
$$ -(90
+45 (1/\ze -1) + 15 (1/\ze -1)^{2}) \kappa^3
(pM^{-\rho })$$
$$+(54 +27 (1/\ze -1) + 7.5 (1/\ze -1) ^{2})\kappa^4 (pM^{-\rho })
\bigr]
$$
$$
+ \int_{2 \le \vert p\vert M^{-\rh} < 2+\et^{-1}} {d^4p \over p^4} \bigl[
{36 +18 (1/\ze -1) + 7.5 (1/\ze -1) ^{2}\over 4} -$$
$${90
+45 (1/\ze -1) + 15 (1/\ze -1)^{2} \over 8}
+{54 +27 (1/\ze -1) + 7.5 (1/\ze -1) ^{2}\over 16} \bigr] \eqno({\rm III.5})
$$
As a consequence the one loop $(A^4/24)$ contribution behaves as
$$\biggl( {36 +18 (1/\ze -1) + 7.5 (1/\ze -1) ^{2} \over 4} - {90
+45 (1/\ze -1) + 15 (1/\ze -1)^{2} \over 8}$$
$$ +{ 54 +27 (1/\ze -1) + 7.5 (1/\ze -1) ^{2}\over 16}
\biggr) (- \ln \eta ) \ \ + \ {\rm finite \ terms}$$
$$ =
9/8(1 + (1/\ze -1) /2 +5/12 (1/\ze -1) ^{2})\vert \ln \eta \vert \ + \
{\rm finite \ terms} \eqno({\rm III.6})
$$
where "finite terms" now means terms which are uniformly bounded both
as $\rho $ tends to $+\infty$ and $\eta$ tends to 0.
The polynomial $1+(1/\ze -1) /2 + (5/12)(1/\ze -1) ^{2}$ is always
positive and greater than 17/20. Since $(17/20)\cdot (9/8)\ge 1/2$,
taking $\eta$ small enough
(depending on the details of our cutoff, which are responsible for the
particular value of the finite terms) we can always achieve our goal
of a positive total $A^4$ contribution, hence of a negative stabilizing
counterterm, with value at least
$$ e^{ - (1/2)\int_{\La } (A^{4}/24)\vert \ln \eta \vert } \eqno({\rm III.7})
$$
Remark that the coefficient of this stabilizing term can
be made as large as we want, if $\eta$ is small enough.
\vfill\eject\medskip
\noindent{\bf IV. The propagators for large and small fields}
\medskip
Let us recall our starting point:
$$ \sum_{SFR}\int d\mu_{0 } (A') d\nu _{\rho_2 }(\gamma )
\chi _{LFR} (A) G(A',\gamma )
e^{-\sum_i((\lambda _{i}^t) ^{1/2 +2\epsilon _2}\gamma^{i}) ^{100}}
$$
$$ e ^{-(1/2) }
e^{CT_{\rho }(A')}
$$
$$e^{(1/2)(-F_{sp}^{2}(A) - + \sum_{i}(\lambda_{i}^{t}) ^{2}
)}
$$
$$
[K_{\rho ,\rho _2} (A', \ga)]^{-1} e^{-(\ze /2) (\nabla_{B'_{l}} \cdot A'_{s})^{2} }
L_{0,\rho _1} (\gamma ) F(A'_0, \gamma )
\eqno({\rm IV.1})
$$
This starting point is clearly well defined because we have both finite volume
and ultraviolet cutoff on each of the fields involved. Hence the sample
fields are smooth. Furthermore for large fields $A'$ the leading terms are the
$F_{4}$ term and the $(A')^{4}$ term in $CT_{\rho }$,
which are respectively positive and positive
definite. The $\gamma $ integrals are also convergent at large $\gamma $
thanks to the protecting term in $\gamma ^{100}$. Remark however that it is
only for fields of order $\lambda ^{-1}$ that the $(A')^{4}$ term provides
convergence, so this term alone does not confine the field in the true
perturbative region ($A'<<\lambda ^{-1}$). It is only the combination of this
term with the axial gauge positivity which does this.
Our goal in this section is to manipulate the complicated expression (IV.1)
in order to extract the Gaussian pieces which are essential for our analysis
and to combine them with the (fake) measure $d\mu _0$ (which is used mainly
as a substitute for the non-existence of a continuum Lebesgue functional
measure). These essential pieces are all contained in the Yang-Mills
action. We use a rather complicated symmetric way to extract them in order
to preserve positivity as much as possible
(positivity is indeed essential for constructive estimates).
The Yang-Mills action is invariant under exact gauge transformations.
However if we use truncated transformations, i.e. such as $A ' \equiv
A^{\ga, 2}$ the action is not exactly
invariant, but the difference is a complicated polynomial with at least
two powers of $\lambda $:
$$ F_{sp}^{2}( A ) + = F^{2} (A') + M(A,\gamma )\ ,
\quad M(A,\gamma ) = O(\lambda ^{2})
\eqno({\rm IV.2})
$$
Our goal is to perform a multiscale analysis of the theory and we stop
at this point to explain further why we need to pay some special
attention to some vertices in (IV.1) which are called non-dominable.
The main problem when one tries such a multiscale expansion is that
some low momentum fields derived by cluster expansions at a certain
scale have to be bounded using the stability of an effective potential
in the interaction, otherwise (for instance if they are integrated
with respect to the Gaussian measure) they give rise to divergent
factorials which are a remnant of the divergence of perturbation
theory [R]. The interaction vertices created by the various
error terms of section II (or by formula (IV.2))
correspond to factors such that, when
the low momentum fields $A'_{s}$ are bounded using the small field
condition, the low momentum fields $B'_{l}$ are bounded using the
$(B'_{l})^{4}$ counterterm, and the low momentum $\ga$ fields
are bounded using the $\ga^{100}$ term in (IV.1), a small factor
remains. We have therefore to examine the vertices which come from
$F^{2}(A')$ in (IV.2). If we simplify the situation by considering
that we have two fields, $A$ and $B$ where $A$ is high momentum
and $B$ low momentum, the vertices with only one high momentum field
can be eliminated because they violate momentum conservation; the
other vertices which couple $A$ to $B$ have at most two $A$ fields.
When the $B$ field is of the small field type $A'_{s}$, there is never
any domination problem, because the small field condition itself can
be used to dominate the field, and a small factor remains (because
the size of the field at which no small factor remains is $\la ^{-1}$ and the
small field condition acts well before that size).
Hence we conclude that only couplings with low momentum large fields
can be non-dominable. Such fields are called background fields.
(This is the reason for which we use the same
generic letter $B$ (as in background)
both for low momentum fields and for large fields).
Let us consider two such background fields; if they occur in the form of a
commutator, there is no problem because the decoupled effective action
for the $B$
field contains a commutator squared (in $F_{4}(B)$) and the situation is
therefore analogous to that of a positive polynomial coupling
such as $\phi^{4}$ (see e.g. [R]).
If there is a single $B$ field with a partial
derivative acting on it, there is still no problem.
The small factor then comes from the
fact that $B$ is of a much lower frequency than $A$, hence the
derivative gives a small factor compared to the initial scale
of $A$ (also called the localization scale). This small factor
is in turn related to the gap between the frequencies of $A$
and $B$, hence related to the creation of protection corridors
(see Section II.B).
Therefore we can conclude that the only vertices
which are not dominable are the ones with one or two large
$B$ fields coming both from a commutator of the type $[A,B]$.
These are the only remaining possibilities as far as the vertices
of $F^{2}$ are concerned. Since such vertices cannot be treated as
interaction, the only other possibility that remains is to put them
in the measure for $A$ with respect to which the cluster expansion is
performed.
It is very fortunate indeed that this operation gives a Gaussian
measure, albeit a $B$-dependent one; were it not the case we could
not do anything because up to now Gaussian functional
integrals are the only ones
that we know how to perform explicitly.
In fact the corresponding measure on $A$ is just similar to $F_{2}(A)$
(see (II.3)), but with ordinary derivatives replaced by covariant
derivatives in the background field.
Furthermore if we return to (IV.1) we remark that it contains also the
analogue of the Faddeev-Popov determinant (the term $K_{\rho,
\rho_{2}}^{-1}$), which up to small correction terms is equal to the
determinant (II.76). This determinant can be written in the usual
way as an integral over anticommuting ghosts~; this is only a formal
trick which is useful to summarize the rules of perturbation theory.
We realize then that there are two types of vertices coupling the
ghosts to the field, namely the ordinary vertex coupling ghosts
to $A'_{s}$, which is the small field, and new vertices which couple
the ghosts fields of a given frequency to the sum of the large fields
$B'_{l}$ of lower frequencies; these new vertices are the direct
remnant of the fact that we used a gauge condition which depends of
the large background field. If we consider the usual multiscale
analysis of the theory we have to give special attention to the
vertices which couple different scales and are not dominable.
By Pauli principle, low momentum ghosts fields are dominable [R];
their functional integration gives a determinant which can be
evaluated without any factorial effect. Low momentum fields of the
type $A'_{s}$ can be dominated using the small field condition; hence
we conclude that the only non-dominable vertices coming from the
Faddeev-Popov determinant are the ones which contain two high momentum
ghosts and one $B'_{l}$ field. Together with the free measure on the
ghosts which is the Laplacian in (II.76) these vertices form an object
which cannot be expanded in perturbation theory. The corresponding
functional integral compared to the functional integral when the
low momentum field $B'_{l}$ is absent gives a quotient of determinants.
This quotient for a constant background field $B$
is exactly the same as the normalized functional integral over
ghosts of the ordinary Faddeev-Popov determinant of this constant
background field $B$ (indeed the position of the $\partial $ operator in
(II.76) relative to $B$ is then irrelevant since $\partial B \simeq 0$
for a low momentum field).
The conclusion of this discussion is that we have to use background
dependent propagators both for the field $A'$ and for the ghosts.
Only large low momentum fields need to be considered as background.
The normalization of the Gaussian measures with background field
gives a factor which can be associated to the large field regions.
This factor will be called the large field dressing factor. It
must correspond in [Ba9] to the problem of renormalizing the large
field regions.
A nonperturbative evaluation of this factor is crucial in proving
that the total weight of the functional integrals over these large
field regions is small compared to the weight of the small field
regions, hence to complete the rigorous version of the sketchy Lemma II.1.
We start now to implement this program of extracting the desired
Gaussian measures with background fields from the functional integral (IV.1).
We want to
group together the pieces which involve $\nabla_{B'_l}$ in the Yang-Mills
action with the gauge condition in (IV.1) in order to obtain a
Gaussian factor
$$ e^{-(\ze /2) < \nabla_{B'_l} A'_s \cdot \nabla_{B'_l} A'_s >}
\eqno({\rm IV.3})
$$
We write for simplicity $-\Delta_{B'_l} $ instead of
$\nabla_{B'_l} \cdot \nabla_{B'_l}$. This Gaussian piece is exactly
the analogue of the homothetic gauge Gaussian measure on $A'_s$ but with
background field $B'_l$.
The Yang-Mills action is therefore decomposed as:
$$ F_{\mu \nu } (A'_s+ B'_l) = D_{\mu }(B'_l)\cdot (A'_s)_{\nu } -
D_{\nu } (B'_l) \cdot (A'_s)_{\mu }
- \lambda [(A'_s)_{\mu },(A'_s)_{\nu } ] + F_{\mu \nu } (B'_l)
$$
$$ = ( \nabla_{B'_l})_{\mu }(A'_s)_{\nu } -
(\nabla_{B'_l})_{\nu } (A'_s )_{\mu }
- \lambda [(A'_s)_{\mu },(A'_s)_{\nu } ] + F_{\mu \nu } (B'_l) +G_{\mu \nu }
(B'_l, A'_s)
\eqno({\rm IV.4})$$
$$
G_{\mu ,\nu } (A'_s, B'_l) \equiv -\biggl(
\sum_i\lambda [(1-\kappa _i)*( B'_l)_{\mu }, \kappa ^{i} * (A'_s)_{\nu }]
- (\mu \to \nu ) \biggr)
\eqno({\rm IV.5})
$$
We introduce also a protection corridor around $LFR$ and call $ELFR$
(extended large field region) the region $LFR$ plus its corridor.
The region complementary
to $ELFR$, called the core small field region $CSFR$ is contained in $SFR$
and protected from $LFR$ by the corridor (see Fig. IV.1).
The region $ELFR - LFR$ is called the boundary region $BR$.
Returning to the definitions of section II we see that roughly
speaking $A '_s$ lives on $SFR$ and
$B'_l$ leaves on $LFR$; in particular $B'_{l}$ is heavily suppressed
in $CSFR$. This allows us to extract in these regions
the correct pieces that we want to join to $d\mu _0$, namely in $CSFR$
the piece to create the propagator with covariance $(\Delta _{B'_l})^{-1}$
on $A'_s$ and in $LFR$ the piece $**$ to create
the propagator $C_{axial}$.
\vskip 10cm
\centerline {\bf Figure IV.1}
\medskip
In the boundary region $BR$ it is enough to keep the initial measure $d\mu _0$
and to remark that $F^{2}$, which is treated as an interaction, remains
positive. This gives a bad normalization to the boxes of this boundary
(of the order of $\lambda ^{-O(1)}$ per such box), which is compensated
by the excellent small factor for the boxes of $LFR$
(see Lemma II.1) if the width of the
protection corridors is in $\lambda ^{-\epsilon }$
with $\epsilon $ very small. For a simple example of how to treat such
normalization effects we refer e.g. to [DMR].
Now we decompose $F^{2}$ in three pieces
in a way which respects the positivity of each piece:
$$ F^{2}(A') = F^{2}(A')_{CSFR} + F^{2}(A')_{LFR} + F^{2}(A')_{BR}
\eqno({\rm IV.6})
$$
$$ F^{2} (A')_{CSFR} \equiv
\sum_{j} \sum_{\Delta \in CSFR_j}
F_{\mu \nu }(\kappa ^{j})^{1/2} \chi _{\Delta }
(\kappa ^{j})^{1/2} F_{\mu \nu } \eqno({\rm IV.7})
$$
$$ F^{2} (A')_{LFR} \equiv
\sum_{j} \sum_{\Delta \in LFR_j}
F_{\mu \nu }(\kappa ^{j})^{1/2} \chi _{\Delta }
(\kappa ^{j})^{1/2} F_{\mu \nu } \eqno({\rm IV.8})
$$
$$ F^{2} (A')_{BR} \equiv
\sum_{j} \sum_{\Delta \in BR_j}
F_{\mu \nu }(\kappa ^{j})^{1/2} \chi _{\Delta }
(\kappa ^{j})^{1/2} F_{\mu \nu } \eqno({\rm IV.9})
$$
More generally an index like $CSFR$, $LFR$, $BR$ etc, is a short notation
for a decomposition of the type (IV.6-9).
In this decomposition we substitute the value (IV.4) of $F_{\mu \nu }$
in terms of $A'_s$, $B'_l$. furthermore we want to replace the
background field $B'_l$ by a background field $\bar B'_l $ which is
piecewise constant and corresponds to the average of $B'_{l}$ on the
box $\De$ appearing in the decomposition (IV.7-9). The constant
value $\bar B'_l $ in such a box $\De$ is noted $\bar B'_{l,\De}$ when
necessary. A difference
term such as $B'_{l} - {1\over \vert \De \vert} \int_{\De} B'_{l}$
where the box $\De$ is the one appearing in (IV.7-9) is noted
$\de B'_{l}$. Such terms can be treated as interaction and are
dominable, since we can rewrite them as integrals of
gradients applied on $B'_{l}$; using (II.29a-c) these gradients are
bounded.
Therefore we write:
$$ F^{2}(A')_{CSFR} = \biggl( ( \nabla_{\bar B'_l})_{\mu }(A'_s)_{\nu } -
(\nabla_{\bar B'_l})_{\nu } (A'_s )_{\mu } \biggr)^{2}_{CSFR}
+ H (A'_s, B'_l)
\eqno({\rm IV.10})$$
$$ F^{2}(A')_{LFR} = \biggl(
**** \biggr)_{LFR} +
K (A'_s, B'_l) \eqno({\rm IV.11})
$$
where $H$ and $K$, which are localized respectively in $CSFR$
and $LFR$ will be treated as small interactions. One has:
$$ H = \biggl( \ {\rm terms \ with \ at \ least \ one \ }G, \ {\rm
one } \ F_{\mu \nu } (A'_s) \, ,
$$
$$\ {\rm \ one
\ commutator \ } [A'_s,A'_s]{\rm \ or \ one \ difference \ }
\de B'_{l} \biggr) _{CSFR} \eqno({\rm IV.12})
$$
$$ K =
\biggl(F_{sp}(B'_l)\biggr)^{2}_{CLFR}+
\biggl( {\rm terms \ with \ at \ least \ one \ }
( \nabla_{B'_l})_{\mu }(A'_s)_{\nu } -
(\nabla_{B'_l})_{\nu } (A'_s )_{\mu }\, ,
$$
$$ \ {\rm
one } \ G \ {\rm or\ one
\ commutator \ } [A'_s,A'_s] \biggr) _{LFR} \eqno({\rm IV.13})
$$
(We used the fact that $(B'_l)_0 =0$
to replace $ F_{0,\mu } ^{2}(B'_l)$ by $****$).
We want now to extract the fact that the gauge condition in $(\nabla_{B'_l}
\cdot A'_s)^{2}$ is almost equal to the desired one $(\nabla_{\bar B'_l}
\cdot A'_s)^{2}_{CSFR}$.
Again to respect positivity we decompose the gauge condition as:
$$ (\nabla_{B'_l } \cdot A'_s )^{2} = (\nabla_{\bar B'_l } \cdot A'_s
)^{2}_{CSFR} + (\nabla_{B'_l } \cdot A'_s )^{2}_{ELFR} + J
\eqno({\rm IV.14})$$
where $J$ is a term localized in $CSFR$ containing at least one difference
$\de B'_{l}$. Finally we use the fraction of the gauge
condition localized in $CSFR$ to write:
$$ \biggl( ( \nabla_{\bar B'_l})_{\mu }(A'_s)_{\nu } -
(\nabla_{\bar B'_l})_{\nu } (A'_s )_{\mu } \biggr)^{2}_{CSFR}+
\ze \biggl(\nabla_{\bar B'_l } \cdot A'_s \biggr)^{2}_{CSFR} $$
$$=
+ L ( B'_l, A'_s, \gamma )\eqno({\rm IV.15})
$$
In this formula let us explain what are $\De _{B}$ and $L$.
Let us introduce the ``homothetic'' Laplace operator $-\De^{homothetic}
$ which in Fourier space is simply $ p^{2}\de _{\mu\nu}
- (1 - \ze)p_{\mu} p_{\nu}$~; its inverse is $1/p^{2}(\de _{\mu
\nu} - (1 - \ze^{-1})p_{\mu}p_{\nu}/p^{2})$. Then the operator
$\Delta _{B}$ in (IV.15) can be thought of as the analogue
of $-\De^{homothetic}$ but with covariant derivatives in the background
field instead of ordinary
ones. More precisely it is defined by
$$\Delta _{B}=\sum_{j} \sum_{\Delta \in CSFR_j}\Delta _{B,j,\De} $$
$$
\Delta _{B,j,\De} \equiv \biggl( (\nabla_{\bar B'_{l}})_{\si}
(\kappa ^{j})^{1/2} \chi _{\Delta }
(\kappa ^{j})^{1/2}(\nabla_{\bar B'_{l}})_{\si}\de_{\mu\nu} $$
$$ + (\ze-1)
(\nabla_{\bar B'_{l}})_{\mu} (\kappa ^{j})^{1/2} \chi _{\Delta }
(\kappa ^{j})^{1/2}(\nabla_{\bar B'_{l}})_{\nu}\biggr) \eqno({\rm IV.16})$$
This operator is clearly positive but not strictly positive
(in the Appendix, bounds are given in the case of a
constant background field).
Finally $L$ in (IV.15) is a correction term which is treated as an interaction.
This term contains indeed either derivatives acting on $B'_{l}$ or
commutators of the background field
$[A'_{l,\mu},A'_{l,\nu}]$ which are dominable as explained
above. Indeed usually when one
combines the quadratic piece $\sum_{\mu }\sum_{\nu }(\partial _{\mu } A_{\nu
})^{2} - (\partial _{\mu } A_\nu )(\partial _{\nu } A_{\mu }) $ coming from
$F_2$ with the homothetic gauge condition $\ze (\sum_{\mu } \partial _{\mu }
A_{\mu })(\sum_{\nu} \partial _{\nu }A_{\nu })$ one needs an integration by
parts, so that the gauge condition combines with the term with the minus
sign, leaving the term $ \sum_{\mu }\sum_{\nu }(\partial _{\mu } A_{\nu
})^{2} +(\ze -1)\sum_{\mu }\sum_{\nu }(\partial _{\mu } A_{\nu
})(\partial _{\nu } A_{\mu
})$ which corresponds to the homothetic propagator $1/p^{2}(\de _{\mu
\nu} - (1 - \ze^{-1})p_{\mu}p_{\nu}/p^{2})$. In our case
this integration by parts is no longer exact for two reasons. First
the partial derivatives are replaced by covariant derivatives.
However if the background field is constant the reader can check that
at least for su(2) the formula of integration by parts is still
true up to a term
proportional to $[A'_{l,\mu},A'_{l,\nu}][A'_{s,\mu},A'_{s,\nu}]$.
The fact that the field $\bar B'_{l}$ is piecewise constant
then gives an error term containing derivatives of this field.
We want to use this Gaussian measure to perform a multiscale cluster
expansion. The units corresponding to this expansion are roughly
speaking small field
cubes and blocks of large field cubes. The fact that the propagator
corresponding to joining the quadratic form (IV.16) to the ``fake''
measure $d\mu_{0}$ is not translation invariant forces us to use
an expansion more complicated than usual, inspired by random path
expansions used
for propagators with boundary conditions such as Dirichlet, in which
one writes the propagator as a product
of a regular translation invariant operator for which spatial
decay is easy to prove, a ``first hitting time'' to the boundary
and then a messy non-translation invariant piece.
\vfill\eject\noindent{\bf V. The expansion}
\medskip
\noindent{\bf A. The preparation of the propagator}
We want to perform a multiscale cluster expansion, i. e. starting from
the propagator $\De_{B}^{-1}$ we have to distinguish momentum
slices with index $j=\rho, \rho-1, ..., 1$. Recall that by our
convention operators such as $\De_{B}$ are the analogues of minus
Laplacians, so that they are of positive type. (This convention saves
a lot of minus signs). The main problem
is the fact that $\De_{B}^{-1}$ is not translation invariant, due
to the presence of large field regions and their associated background
fields. We shall introduce a
modified version of this propagator $\De_{B}^{-1}$ which is better suited
for a cluster expansion.
The large field region $ELFR$ is first divided into
connected components $E_{1},E_{2},...,E_{n}$, where a connected
component means a maximal set of boxes of $LFR$ belonging to a
connected component (in the ordinary sense!) of $ELFR$. Therefore
two boxes of $E_{i} $ are connected if they are close enough, and
between the $E_{i}$'s there are wide separation corridors.
Our goal is to decompose
the field into an orthogonal sum of fields, $A=A_{0} + \sum_{i=1}^{n}
A_{i}$. The general field $A_{0}$ extends in the full space and has a
good propagator. Each field $A_{i}$ is localized in or near the connected
component $E_{i}= \cup_{j} E_{i}^{j}$, where $E_{i}^{j}$
is the subset of the $i$-th large field region made of its boxes
of scale $j$. Such a field $A_{i}$
has a non translation invariant, hence poorly
decreasing propagator, but this propagator has no longer
any memory of the existence of the other large field regions, so this
formalism is suited for the factorization of these regions. This is
the general outline. Before to proceed, we suggest eventually to read
reference [DMR] for a more detailed account of such a scenario in a
simpler but similar case.
More precisely we define an inductive resolvent expansion. An ordinary
resolvent expansion is of the type
$$
{1 \over \De + \de} = {1 \over \De} - {1 \over \De }\de {1 \over \De + \de }
\eqno({\rm V.1})
$$
In our case we imagine $\De$ to be
a translation invariant propagator suited for a cluster
expansion in the small field region such as
$(-\De^{homothetic})^{-1}$,and
the perturbation $\de$ contains the background field,
hence it is variable. Even inside the small field region we
cannot iterate
formula (V.1) infinitely many times
because the background fields produced in $\de$
could lead to factorials when bounded. Also
in the large field region we must certainly keep this expansion in a
resummed form, since the true Gaussian measure
there, which has propagator $C_{axial}$ is very far
from the small field resion propagator. This is the source of many technical
difficulties.
First because large fields cannot be bounded effectively we must
forget about using a background independent propagator for ${1 \over
\De}$. But derivatives of background fields can be dominated quite
effectively. This suggests that we should
first compare in the small field region the general propagator to
the propagator built with constant background fields.
The interest of using propagators with constant background field is that
they are translation invariant and have obviously good spatial
decrease. But even when the background field is constant these
propagators still have a defect; the Laplacian with background field
can have a zero mode if all spatial components of the background field
are aligned in su(2) space. As a consequence the bounds on the
inverse Laplacian with covariant derivatives are not the same that for
the ordinary Laplacian. We need a further decomposition of the
momentum around the dangerous zero mode which corresponds to
$p_{\mu}=\la (\bar B'_{l,\De})_{\mu}.e$, where the scalar product is in
su(2) and the vector
$e$ is the unit vector of su(2) which is aligned with, say
$(\bar B'_{l,\De})_{1}$.
Since this decomposition is only necessary when all
$(\bar B'_{l,\De})_{\mu}$, $\mu=1,2,3$
are approximately aligned, the particular
choice of $\mu =1$ for $e$ is unimportant.
This decomposition is done in the following
way. Let us consider some
large field box $\De$ of scale $l$, and the corresponding set of boxes
in the small field region which have it as ancestor, with scales $m
>l$.
We redefine only the cutoffs corresponding to the scales $m$ between
$l$ and $l'$ where $M^{l'}$ is the order of magnitude
of the modulus of $\la \bar B'_{l,\De})$. If we introduce the
corresponding sum of slices
$\ka^{l'}_{l}= \sum_{l j+10$.
This term will also deliver a
small factor through momentum conservation corresponding to
integration of $x$ in the box $\De$.
In the case where either one of these two terms is chosen in (V.7)
the expansion step (V.6) is
reiterated on ${1 \over \De_{B}^{0}}$ with $\De$ replaced by $\De ''$.
There remains terms such as
$(\nabla_{\bar B'_{l,\De}})_{\si}
(\kappa ^{j''})^{1/2} \ch_{\De ''} (\ka^{j''})^{1/2}
(\nabla_{\bar B'_{l,\De}})_{\si}$, $j'' \le j+10$, $ \De ''\in ELFR_{j''}$
or $-\la^{2} \nabla_{\si} (\ka^{j''})^{1/2} \ch_{\De ''}
(\ka^{j''})^{1/2}
\nabla_{\si}$, $\De ''\in ELFR_{j''}$.
These error terms couple $\De$ to a box $\De''$ of $ELFR$.
Remark that this coupling arises through a propagator
with constant background field. When any of these terms is chosen
we stop the expansion.
An important additional rule is the following one: when more than
five low momentum background fields have been produced
we stop the expansion and consider that the boxes of the corresponding
string of propagators are attached to the large field region of the
corresponding lower scale; therefore we do not need to consider it as
a part of the small field region any longer. This rule is necessary
even when both ends $x$ and $y$ of our propagators are localized in the small
field region, where we have by (II.29-a-c) a good compact support
restriction on the size of these gradients, because the path of
integration from $x$ to $y$ can cross large field regions where the
gradient of the field is no longer bounded in a $C_{0}^{\infty}$ way
and factorials of accumulation could occur.
If we apply this process symmetrically on $\De$ and $\De '$, i.e. at
both ends of (V.6), we obtain
the covariance in the form
$$
C= {1\over \De_{B}^{0}} = C_{11} + C_{12} + C_{21} + C_{22}
$$
$$C_{11} =\ch_{CSFR} \Ga \De_{B}^{0} \Ga \ch_{CSFR}\ \ ; \ \
C_{12} =\ch _{CSFR}\Ga [ \Ga ' \ch_{CSFR} + \ch_{ELFR}] $$
$$ C_{21} = [\ch _{CSFR} \Ga ' + \ch_{ELFR} ]\Ga \ch_{CSFR} $$
$$
C_{22} = [\ch_{CSFR}\Ga ' + \ch_{ELFR}]{1 \over \De_{B} ^{0}} [\Ga '
\ch_{CSFR} + \ch_{ELFR} ]\eqno({\rm V.8}) $$
where $\Ga$ is some string of propagators each corresponding to a
constant background field
(with insertions of $\nabla B'_{l}$ or of momentum violating terms)
and $\Ga ' = \Ga D_{ELFR}$ where $D_{ELFR}$ is an insertion explicitly
localized in some box of $ELFR$ of scale $j$.
Then we introduce an interpolation parameter $t\in[0,1]$ which at $t=0$
suppresses the coupling pieces $C_{12}$ and $C_{21}$. Hence we write
$C(t) = C_{11} +t C_{12} +t C_{21} + C_{22}$. This is still a positive
operator since $C_{11}$ and $C_{22}$ are positive.
Then we perform a first order Taylor expansion in this parameter.
The interpolating terms contain an explicit $ C_{12}$ or $ C_{21}$
link which connects one or two boxes $\De$,$\De '...$
to one large field box of scale $j$ in the middle
(either in the form of a $D_{ELFR}$ insertion or simply
by a $\ch_{ELFR}$ factor). For this error term we add $\De,...\De '$
to the large field region $ELFR$. The process is then
reiterated on the remaining $ 1 \over \De_{B}^{0}$ factor
with this new definition of $ELFR$, until finally it stops by
exhaustion of all boxes in $CSFR$.
The decoupled term at $t=0$ corresponds to a new
covariance $C^{11}+C^{22}$. If we introduce the simpler
covariance $C^{11} + \bar C^{22}$ with $\bar C^{22}=\ch_{ELFR}
{1 \over \De_{B} }\ch_{ELFR}$, then we can perform the change of
variables $A \to (1+ \ch_{CSFR}\Ga ') A$ and obtain the same
theory with the simplified covariance but a more complicated
interaction. The $C_{11}$ piece links boxes of $CSFR$
through strings of propagators in constant background fields, which
have both good power counting and good spatial decay. The $\bar C^{22}$
lives purely in $ELFR$. However it is not true that at $t=0$
the $CSFR$ and $ELFR$ regions have been factorized. Indeed the field
is now non local, so the interaction still couples both regions. This
coupling however is easy to control since it occurs through the
well controlled $\Ga '$ operator\footnote*{In all this discussion
we have considered
that a $D_{ELFR} $ insertion is equivalent to a $\ch_{ELFR}$
characteristic function. Strictly speaking this is not true~; the
$D_{ELFR}$ insertion contains a $\ch_{ELFR}$ term but followed by a
controlled non local operator $(\ka^{j})^{1/2}$. The necessary
modifications to take this into account are inessential but
painful. They require the use of the corridor $BR$ and some
modifications of the formulas. We do not include them in order not to
distract the reader from the main argument.}.
\medskip
\noindent {\bf B) Decoupling of the different connected components
of the large field regions}
\medskip
We have not yet a satisfying propagator for performing cluster
expansions, because the distant large field regions still interact
together through the $C_{22}$ piece of the propagator in (V.8).
In this subsection we should describe a general method for removing
this interaction. We return to our
decomposition of the large field region into
connected components $E_{1},E_{2},...,E_{n}$ (we recall our rule
that two regions close together are in fact
connected, so that the distance between two large field
regions is at least a fixed number of boxes of the scale considered).
Recall that we want to decompose
the field into an orthogonal sum of fields, $A=A_{0} + \sum_{i=1}^{n}
A_{i}$, each $A_{i}$ being associated with $E_{i}$, with a poorly
decreasing propagator, but this propagator has no longer
any memory of the existence of the other large field regions, so that
these regions factorize. Instead of that we have at the end of the
preceding section
V.A the sum of two fields, one in the small field region and the other
in the large field region, but not factorized over its connected components.
The construction of the fields $A_{i}$ and of their
measure is performed as follows. We introduce for the $i$-th region
$E_{i}$ the operator $\De_{B}^{0,i}$ which is roughly speaking the same
as $\De ^{0}_{B}$ but in which the other regions $E_{j}, j\ne i$
are now treated as small field regions. More precisely the formula
(V.3b) for $\De_{B}^{0}$ is changed into formula (V.3a) for $\De \in
E_{j}, j\ne i$, where the background field $\bar B '_{l}$ is now
introduced also for the boxes of $E_{j}$. Finally we can introduce
also the operator $\De_{B}^{0,\emptyset }$ in which formula (V.3b)
is replaced by (V.3a) for $every $ $\De \in E_{i}$, $i=1,..,n$.
We introduce also $\ch_{i} \equiv \ch _{E_{i}}$ for the characteristic
function of $E_{i}$. In addition to the
background field each insertion $\De_{B}^{0,i}
-\De_{B}^{0,\emptyset } $
contains a characteristic function $\ch_{i}$
and each insertion $\De_{B}^{0,i}
-\De_{B}^{0 } $ contains a characteristic function $\ch_{j}$, $j \ne
i \ $. Let us for a moment forget the background fields and consider
the structure of the expansion according to the localizations.
We start with $\ch_{ELFR} {1 \over \De _{B}^{0}} \ch_{ELFR}$
(see (V.8)). We want to decouple a first large field region, say
$E_{1}$, from the rest. We insert a first resolvent step which is
$$
\ch_{ELFR} {1 \over \De _{B}^{0}} = \ch_{ELFR} [
{1 \over \De _{B}^{0,\emptyset}} +{1 \over \De _{B}^{0,\emptyset}}
( \De_{B}^{0,\emptyset}-\De_{B}^{0}){1 \over \De _{B}^{0}} ]
\eqno({\rm V.9})
$$
Then we decompose the difference $(\De_{B}^{0,\emptyset}-\De_{B}^{0})$
as a sum over insertions of $\ch_{i}$, $i=1,...,n$.
Iterating this formula at infinity we obtain chains of arbitrary
length. In these chains we formally resum every series of insertions
of at least $two$ consecutive identical functions $\ch_{i}$.
This reconstructs the operator ${1 \over \De _{B}^{0,i}}$
sandwiched by characteristic functions $\ch_{i}$ on $both$ sides;
furthermore
each ${1 \over \De_{B}^{0,\emptyset}}$ is sandwiched by $\ch_{i}$
on one side and $\ch_{j} $ on the other side with $i \ne j$.
In order not to use too heavy
notations, let us call $C_{0}$ the kernel for
${1 \over \De_{B}^{0,\emptyset}}$.
Then formally our expansion has the structure:
$$
\ch_{ELFR}{1 \over\Delta^{0} _{B} }\ch_{ELFR} =
\sum_{p\ge 0} \sum_{\scriptstyle i_{0},i_{1},...i_{p+1}}
\ch_{i_{0}} C_{0} \ch_{i_{1}}C_{0}... C_{0} \ch_{i_{p}}
C_{0}\ch_{i_{p+1}} \ .
\eqno({\rm V.10})
$$
This theory is therefore equivalent to a theory with
a substitution rule for the field corresponding to the $C_{22}$
covariance: $ A \to \sum_{i=1}^{n} A_{i}$,
in which the $A_{i}$'s form an independent set of orthogonal
random variables, each $A_{i}$ being distributed with a Gaussian
measure of covariance $\ch_{i}{1 \over \De_{B}^{0,i}}\ch_{i} $, plus
a quadratic interaction of the form
$$
e ^{\sum\limits_{i , j } \sum\limits_{\scriptstyle i_{1},...i_{p},\
i_{1}\ne i,\; i_{p}\ne j
\atop \scriptstyle i_{k}\ne i_{k+1}, \; k=1,...,p-1}
A_{i} C_{0} \ch_{i_{1}}...C_{0} \ch_{i_{p}}C_{0}
A_{j} }
\eqno({\rm V.11})
$$
This interaction and the propagator $\ch_{i}{1 \over \De_{B}^{0,i}}\ch_{i}$
for $A_{i}$ generates precisely the chains (V.10) (see [DMR]).
The fact that we can consider the transition terms $A_{i}... A_{j}$ in (V.11)
as interactions is due to the
fact that they are indeed small because of our rule that two disjoint
regions $E_{i}$, $E_{j}$ are separated by a corridor of some finite
(large) width.
The covariance $C_{0}\equiv {1 \over \De_{B}^{0,\emptyset}}$ can
indeed be now controlled by the same method than in part A) and has
therefore good decrease properties.
In this way the remaining covariances
${1 \over \De_{B}^{0,i}}$ are now factorized over each connected large
field regions. To decouple truly the large field regions there are two
equivalent possibilities. The first is to use (V.11) and to expand
the corresponding quadratic interaction up to infinity. There is no
factorial associated to this expansion, since it is Gaussian. On the
chains developed in this way we can read $algebraically$ the
connections between large field regions. We do not need any
interpolation parameters. Positivity of the Gaussian measure is
therefore automatically respected. The only drawback of this approach
is that one has to be careful that the insertions $\ch_{i}$ in (V.10)
really are a short notation for an insertion of
$\De_{B}^{0,\emptyset}- \De_{B}^{0,i}$, which contains a background
field localized in $E_{i}$. There can be arbitrary accumulations of
such background fields in the same region. If we dominate them naively
(e.g. by the ${e^{-\la^{4}B^{4}}}$ term) we would generate local
factorials and the series corresponding to the
expansion at infinity of the exponential in (V.11) would
not converge. But this is a problem only for
$B$ large and we can use the fact that each insertion is paired
with a new $C_{0} =(\De_{B}^{0,\emptyset})^{-1}$ propagator which
precisely decays exactly in the same way at large $B$.
A second possibility, instead of expanding (V.10-11) to infinity, is
to test inductively the coupling of $E_{1}$ to $E_{2}\cup...\cup
E_{n}$ and to iterate. This generates only one link at a time, hence
prevents the accumulation of background fields. But this process
requires interpolation parameters \`a la Brydges-Battle-Federbush,
and we need to do it symmetrically from both sides of $C_{22}$ in
order to preserve positivity. This is done by applying (V.9)
for $i=1$ on both sides of ${1 \over \De _{B}^{0}}$ as in (V.6).
We give the corresponding result on one side for simplicity:
$$ \ch_{1} {1 \over \De _{B}^{0}} = \ch_{1} \biggl[ {1 \over \De
_{B}^{0,\emptyset}} + {1 \over \De _{B}^{0,1}}
( \De_{B}^{0,\emptyset}-\De_{B}^{0,1}){1 \over \De_{B}^{0,\emptyset}}
$$
$$ + {1 \over \De_{B}^{0,\emptyset}}
( \De_{B}^{0,1}-\De_{B}^{0}) {1 \over \De _{B}^{0}}
+{1 \over \De _{B}^{0,1}} ( \De_{B}^{0,\emptyset}-\De_{B}^{0,1})
{1 \over \De_{B}^{0,\emptyset}}( \De_{B}^{0,i}-\De_{B}^{0})
{1 \over \De _{B}^{0}} \biggr] \ .
\eqno({\rm V.12})$$
Then we multiply the differences $ \De_{B}^{0,1}-\De_{B}^{0}$
by an interpolation parameter $s_{1}$, Taylor expand
to first order, decompose the remainder term as a sum over $\ch_{j}$, $j\ne
1$ and iterating, with $E_{1}$ and $E_{j}$ joined together, and iterate.
In both strategies expansion (V.6) (iterated at most five
times) has to be used on $C_{0}$ to complete the argument, together with
the good spatial decrease (A.29) of the homothetic propagator in a
fixed background field.
Remark that we could have used more complicated formulas which at once perform
the decoupling of the small field regions and the large field regions,
hence gathered subsections A) and B) into a single step, but we think
it is easier to understand this complicated construction in successive stages.
\medskip
\noindent {\bf C) Horizontal decoupling}
\medskip
At this stage we have factorized the connected
large field regions from one another and from the small field region.
For each slice $i=\rh , ...,1$, it remains to perform an ordinary cluster
expansion between all boxes in the small field region
with respect to the Gaussian measures with propagators
$C_{0} $ as prepared in the previous
subsections. This is done \`a la Brydges-Battle-Federbush [R]. A key
requirement for such an expansion is to preserve positivity of the
underlying quadratic form. We have to perform both an ordinary cluster
expansion in the small field region, using the propagator
$C_{11}$ in (V.8), for which expansion (V.6), iterated at most five
times, allows good spatial decrease (see (A.29)). The positivity
requirements are satisfied because of the symmetry of the expansion
step (V.6).
Recall also that in each large field region $E_{i}$ we must combine the measure
$C_{i}= (\De_{B}^{i})^{-1}$
on the field $A_{i}$ with the $p_{0}^{2}$ piece in (IV.11)
to reconstruct a propagator with which we can contract the fields
produced in the large field expansion (II.25), as required in Lemma II.1.
\medskip
\noindent {\bf D) Vertical decoupling}
\medskip
We need to perform an inductive decoupling of the various momentum
slices. There are several facts to consider. First there are
interactions which are dominable, but couple different momentum
slices. A typical interaction of such a type is a term like $[A,A][B,B]$
where $A$ is a high momentum field of slice $i$ and $B$ a low momentum
field of slice $j$. These interactions are treated as regular dominable
interactions of the $\ph^{4}_{4}$ type. This means that a parameter
$t_{\De}$ is introduced for each box $\De$ and an expansion to fifth
order is performed in each of these parameters in the standard
manner explained in [R]. We should not repeat the corresponding
details here.
Some new features however require more explanations. The existence of
rectangular anisotropic boxes (the double valuedness of the
momentum indices $j = (i, \al)$) should not confuse the reader.
As explained in section
II, the small field boxes, in which the propagator
is isotropic, are themselves isotropic, except when they have
a large field ancestor of same index $i$ and lower index $\al$. In
this case we perform the vertical expansion also for the $p_{0}$
slices, which means that a parameter is introduced which tests the
coupling, at fixed $i$, between the $p_{0}$
frequencies larger and smaller than $\al$. This vertical piece of the
expansion should be thought really as an auxiliary one, however,
because we show in section VII that it is superrenormalizable.
The vertical expansion between large field boxes is similar, except
that here like in the horizontal expansion we must first consider that
neighboring boxes in index space, one contained into the other, are
automatically linked. In this way the necessary
vertical expansion to decouple large
field regions one from another will give summable links plus small
factors for the same reason than in the horizontal case, namely because
the vertical distance is at least some large constant.
The background dependence of the Gaussian measure, which
still couples small field regions to large field regions of smaller
frequency, is also a new feature and requires a vertical decoupling.
The main remark here is to consider that in the vertical decoupling,
the parameter $t_{\De}$ is introduced in the Gaussian
measure with propagator $1 \over \De_{B}$ which we constructed in the previous
subsections, but not in the determinant which corresponds to the
normalization of this measure, and also not in the non-dominable part
of the Fadeev-Popov determinant. These determinants (or more
precisely the effective potential extracted from them) are associated to
the normalization of the corresponding large field regions. This is
because if we were to develop these determinants which precisely
contain the non-dominable piece of the interaction, domination
by the $\la^{4}B^{4}$ term in the small field boxes would give a large
product of terms of order 1, not bounded by the single small factor
associated to the large field box; alternatively domination
in the large field box would cost a factorial not summable.
This problem forces us to attribute the
corresponding normalization to the large field regions considered and
to bound it with a special argument in Section VI. For the normalized
background dependent Gaussian measure, there is no such problem,
because the corresponding propagators hit ordinary dominable vertices
which provide the necessary coupling constants, hence
the necessary small factors.
We have to define the $t_{\De}$ interpolation that we introduce on the
explicit propagators created by the previous expansion steps (in
particular the horizontal cluster expansion).
Finally we should explain the modifications introduced by the use
of the slices (V.2) around $\bar B'_{l}$ (the decomposition
with the cutoffs (V.2)). The reader should not
consider that these slices are new momentum
slices but rather that they replace former slices.
In the vertical expansion we introduce therefore
also a dependence in $t_{\De '}$ in the corresponding cutoffs, that is
we write, if the ancestor of $\De '$ is the large field box $\De $:
$$
\ka^{l}_{l'} = \sum_{m=l+1}^{l'}
\ka_{t_{\De '}\bar B'_{l,\De}}^{m}\vert_{t_{\De '}=1}
\eqno({\rm V.13})
$$
where $\ka_{t_{\De '}\bar B'_{l,\De}}^{m}$
restricts $\vert p_{\mu}-\la t_{\De '}(\bar B'_{l,\De})_{\mu}.e\vert$ to be
of order $M^{m}$.
In this way the parameter $t_{\De '}$ interpolates smoothly between the
translated slices at $t=1$ and the ordinary slices at $t=0$.
In the usual way we perform for each box $\De$ a fifth order Taylor
expansion in $t_{\De}$. Each derivative creates at least one
badly localized low
momentum field. The connected objects which have at most four such
low momentum legs, hence which need renormalization are in this way
necessarily decoupled terms at $t_{\De}=0$.
\medskip
\noindent {\bf E) The renormalization and the flow of the effective
coupling constant}
\medskip
To simplify the notations we now call $B(\De)$ or sometimes
simply $B$ the averaged field called previously $B'_{l,\De}$.
After the horizontal and vertical
decoupling has been performed at a given scale, one is in a position
to renormalize the divergent two and four point contributions. This
requires first a Mayer expansion in order to render these
contributions translation-invariant. For the details of such a Mayer
expansion we refer to [R]. Once the divergent contributions are
free from hardcore constraints they can be cancelled by counterterms
of the desired form. These counterterms in turn generate a flow for
the coupling constant of the theory which is
the one of an asymptotically free
theory such as the Gross-Neveu model in two dimensions
and treated in the same way [FMRS2],[R].
In this way the ansatz (II.11) is justified. The effective
constants $\la _{i}$ at scale $i$ are shown
in the standard way to be very close to the simplified effective
couplings given by formula (II.12), since the true flow
corresponds really to a bounded flow for
the constant $C$ in (II.11).
Remark that to control the flow
we are not forced to construct the correct $\be$
function (which would contain renormalons); as explained e.g. in [R]
it is enough to get the
two first terms correctly, plus a uniform bound of the correct order
on the rest, which remains expressed in terms of the effective
couplings.
Remark also that we have to distinguish from the rest the case of
the normalization determinant
corresponding to the background dependent Gaussian
measure and the non-dominable part
of the Fadeev-Popov determinant.
Because these determinant are not expanded
in the vertical expansion but directly associated to
the corresponding large field regions, we do not have a complete
cancellation between the background field counterterms and the
corresponding polymers with two or four background field external
legs. One could believe that the not-cancelled
counterterms together with the
determinants simply form subtracted determinants of the $\det_{4}$ type.
We recall that
$$\det\nolimits_{4}(1+K) \equiv \det (1+K){\rm e}^{Tr
-K+{K^{2}\over 2}-{K^{3}\over 3}+{K^{4}\over 4}} \ .\eqno({\rm V.14})$$
But to our surprise this turned out not to be true. The non-dominable
part of the Fadeev-Popov determinant plus the corresponding
uncancelled counterterms
gives indeed a $\det_{4}$. But because counterterms are gauge
dependent and the background field gauge does not coincide with the
ordinary gauge where the small field flows (hence the counterterms)
are computed, it appears a difference between the normalization
determinant plus the counterterms,
$\det^{-1/2} \Delta^{homothetic} _{B(\Delta )}e^{CT(B)}$ and
$\det_{4}^{-1/2} \Delta^{homothetic} _{B(\Delta )}$. This difference
is computed in detail in the next section, and plays a crucial r\^ole
in the final bound on the large field regions.
The only special difficulty in this case has to do
with the fact that the flow keeps the structure of the theory unchanged.
In the standard textbooks about perturbative renormalization of gauge
theories such as [IZ] one proves that the renormalization group flow keeps
the effective interaction of the theory of the Yang-Mills form but
with a running coupling constant.
This is usually done in perturbation theory using dimensional
regularization. In our case this fact will also be true,
provided we take into account the flow of our gauge-restoring
counterterms in $\la^{4} A^{4}$, because our cutoff
is not gauge invariant and is used at every scale with the same shape to
construct the new cutoff of the effective theory. This has
really to be done
only at the one loop order for the $\la^{4} A^{4}$ term, but
concerning the relevant mass operator,
the value of the mass counterterm has to be fixed exactly by a
fixed point method (as is done e.g. for the critical value of the mass
in infrared $\ph^{4}_{4}$ [R]).
Remark however that if we consider the renormalization group flow of
non-Abelian gauge theories in an ordinary gauge, there is usually both
coupling constant and wave function renormalization. This makes
the discussion of the normalization of the Gaussian measure in
the background field much less transparent. We prefer to use
the homothetic gauge in which there is almost no or no wave function
renormalization (depending on whether one wants
a completely explicit formula for this gauge, or one is satisfied
with the solution of an implicit function theorem). This is a technical trick
which could presumably be circumvented by a more complicated analysis
but it is very convenient in this respect.
\medskip
\noindent {\bf F) The effective potential}
\medskip
It remains now to prepare the theory in order to compute
effectively the normalization of the background dependent Gaussian
measure and of the Fadeev-Popov determinant which have to be
attributed to the large field regions. In order to do this we
use the effective potential
method [BG][MS][dCMSdV]. This means that for a given large field box
we want to introduce specific boundary conditions so that we can compute
explicitly the functional integral over the associated small field
region.
This in turn requires to create some gap between the size of the large
field cube $\De$ which we suppose of index $j=(i,\al)$ and the
frequencies of the fields in the small field
region whose functional integration gives the dressing factor.
We have to recall that the
boxes are rectangular rather than cubic. We have to distinguish two
cases:
\item{-}{if both $p_{0}$ and $\vec p$ are large compared to the
respective scales $M^{\al}$ and $M^{i}$ of the large field box $\De$
(i.e. $\vert \vec p \vert \ge M^{i+100}$
and $\vert p_{0} \vert \ge M^{\al+100}$),
there is no problem
with the boundary conditions. As in the usual effective
potential method, the boundary effects are negligible compared
to the functional integral in the whole volume $\De$.}
\item{-}{if $p_{0}$ or $\vec p$ are small, we can use
a rough bound such as
$$\vert \det (1+K) \vert \le e^{(1/2) {\rm Tr}( K + K^{*} +
K.K^{*}) } \eqno({\rm V.15})$$
on the corresponding determinants; since the
corresponding momentum integrals are unbounded either in three or one
dimensions, the resulting bound in $\la^{2} B^{2}$ is either linearly
divergent or convergent. We can add it to the quadratically divergent
$\la^{2}B^{2}$ term computed in section III, and bound the total by
increasing slightly the coefficient of this term. In Section VI
it is shown that this term is controlled by the effective
potential generated by the dressing factor.}
\medskip
This effective potential method (up to small error
terms) yields in each large field box $\De $ a determinant of the operator
$\Delta^{homothetic} _{B(\Delta )}$, combined with
the non dominable part
of the Fadeev-Popov determinant; we have to
compare it to the normalization of a small field box
in which the Gaussian measure is the one obtained
with the usual homothetic gauge covariance; this is exactly the same
determinant but in which $B(\De)=0$. Similarly the Fadeev-Popov
determinant has to be divided by the Fadeev-Popov determinant at $B(\De)=0$.
The reader should indeed keep in mind that the typical case is (a posteriori,
after all estimates have been performed)
$ELFR = \emptyset$, in which case the covariance of the $A' _{s}$
field (which is then equal to the full $A '$ field)
is simply $1/p^{2}(\de _{\mu
\nu} - (1 - \ze^{-1})p_{\mu}p_{\nu}/p^{2})$,
the homothetic gauge covariance (up to the small ``fake''
term $\lambda ^{2}p^{2}$ which can be
recombined with the corresponding counterterm in (IV.1)).
To this determinant we have to add the effective action
for a constant $B$ field which reduces to the $e^{-F_{4}} =e^{-
\la^{2}[B,B]^{2}}$
term in $e^{- F^{2}}$.
Finally as mentioned in the previous subsection we have to add to this
term the uncancelled pieces of the $B$-dependent counterterms.
Returning to section III, equation (III.1), we remark that the terms
with derivatives of $B$ cancel in the effective potential computation.
We have therefore to consider only the terms in $B^{2}$, $B^{4}$
and $[B,B]^{2}$. The uncancelled piece of the $\la^{4}[B,B]^{2}$ counterterm
is a small correction to the action term in $\la^{2}[B,B]^{2}$, and we
do not need to compute it precisely. But
the $B^{2}$ and $B^{4}$ terms are crucial. Using section III the
reader can check that at one loop the totality of these counterterms is
uncancelled
because the graphs $G_{1},G_{2},G_{3},G_{4},G'_{1},G'_{2}$, and
$G'_{3}$ only contain vertices of the non-dominable type $[A,B]$.
Therefore in the effective potential the full value of these
counterterms has to be included.
The conclusion of this analysis is that we have to
bound the product of these counterterms, of
an $e^{-\la^{2}[B,B]^{2}}$,
and of a quotient of properly normalized determinants
in which the background field is a constant and there are
e.g. periodic boundary conditions on the operator in the determinant;
the proper normalization means simply
that at $B(\De) =0$ this quotient of determinants
is simply 1. The precise form of this factor is written down in the next
section (equation (VI.1)).
To obtain a non-perturbative bound on this object is one of the main
points of this paper and is explained in detail in Section VI.
Again the solution ultimately depends of a correct choice of the ultraviolet
cutoff function.
\vfill\eject \noindent{\bf VI. The main stability estimate for a large field region}
\medskip
In this section we prove that the non-trivial factor
associated to the large field regions which come from the
$B$ dependent gauge fixing and functional integration
in the associated small field regions can be bounded
uniformly in $B$ by 1, if the ultraviolet cutoff is of a certain
stabilizing shape. This result (a kind of non-perturbative stability)
is at the core of our whole analysis.
We have to consider the small field functional integral associated to
the normalization of a unit of the large
field domain such as a fixed cube $\Delta$ of ${\bf D^{i,\alpha}}$.
As explained in the preceding section, we can compute this
normalization in the case of a
constant background field $B(\Delta)=B$ and of e.g. periodic
boundary conditions (the type of boundary conditions being inessential).
This functional integration is (taking into account the action
$F_{\mu\nu}(B)$ which for a constant $B$ reduces to
$(\lambda ^{2}/2)\int_{\Delta } [B,B]^{2}$):
$$ g_{i,\alpha,\Delta }(B)=
{ \det_{i,\alpha,\Delta} (FP) \over
\det_{i, \al, \Delta}(BF)^{1/2} }
e^{ CT_{i,\al , \De} -\lambda ^{2}/2\int_{\Delta } [B,B]^{2}}
\eqno({\rm VI.1})$$
where
$\Delta \in {\bf D^{i,\alpha}}$ is a cube of scale $i,\alpha$,
hence of volume $\vert \De \vert =
M^{-3i-\alpha}$ (at most $\lambda^{-1}M^{-4i}$);
$\det_{i,\alpha , \De }(BF)$
means the determinant with the appropriate cutoffs corresponding to
our background dependent measure on the small field region associated
to $\De$; it has
infrared cutoff of the type $\kappa^{i}(p)\kappa^{\alpha}(p_{0})$
and ultraviolet cutoff given by the exact shape of the small field
domain associated to $\Delta$; this domain in the typical case of an
isolated large field cube corresponds to an integral over momenta with
the ultraviolet cutoff $\kappa^{\rho}(p)$ of scale $\rho$.
As explained in section V, the case
of more complicated large field domains reduce to this case.
Finally $CT_{i,\al ,\De}$ is the
associated set of counterterms (which contains a positive, potentially
dangerous $B^{2}$ counterterm
and a $B^{4}$ counterterm, which thanks to our choice of
ultraviolet cutoff is negative and
stabilizing); the exact value of these counterterms depends on the precise
shape of the ultraviolet cutoff, and in particular of the parameter
$\et$ in (II.14).
Similarly $ \det_{i,\alpha,\Delta} (FP)$
is the background dependent determinant corresponding to
the integration over the ghosts in the small field region associated
to $\De$~; it is equal to an ordinary Fadeev-Popov determinant.
More precisely, if we rewrite the cutoff simply as $\ka (p^{2})$
and take into account our form of the cutoff and the
normalization at $B=0$, we have a twelve by twelve matrix $BF$
in su(2)$\otimes \RR^{4}$ space~(we forget to put the su(2) indices):
$$ BF = \de_{\mu\nu} + {\ka (p) \over p^{2} } \bigl[- (D^{2}\de_{\mu\si} -
(1-\ze)D_{\mu}D_{\si})
(\de_{\si\nu} + (\ze^{-1}-1) {p_{\si} p_{\nu}\over p^{2}}) -p^{2}
\de_{\mu\nu} \bigr]
\eqno({\rm VI.2})$$
where repeated indices are summed, and
$$-D^{2} = -\sum_{\mu} D_{\mu}D_{\mu}
$$
$$= [p^{2} \delta _{ab} +2i p_{\mu } \epsilon
_{abc}\lambda B_{\mu }^{c}+
\lambda^{2}B_{\mu }^{c}B_{\mu }^{c} \delta _{ab}(1-\delta _{ac}) -
\la^{2}
B_{\mu}^{a}B_{\mu}^{b}(1-\de_{ab})],\eqno({\rm VI.3})
$$
Similarly the Fadeev-Popov operator (normalized at $B=0$)
is a three by three matrix in su(2) space:
$$
FP = 1 + {\ka (p) \over p^{2} } \bigl[ -\sum_{\mu}\partial_{\mu}
D_{\mu} - p^{2} \bigr]
\eqno({\rm VI.4})
$$
Using the Euclidean and global SU(2) symmetry and the fact that
$B_{0}=0$ (because of the axial gauge), we can explore
completely the function $g_{i,\al,\Delta }$ by considering a field
$B$ with only two non zero components $B_{1}^{1}=x/\lambda $
and $B^{2}_{2}=y/\lambda $.
Then $\lambda ^{2}B^{2}=x^{2}+y^{2}$ and $\lambda ^{4}
[B,B]^{2}=x^{2}y^{2}$. The function $g_{i,\alpha,\Delta}$
becomes a symmetric function of $x$ and $y$. We are going to prove the
following uniform estimate:
\noindent {\bf Lemma VI.1}
For a sufficiently wide ultraviolet cutoff (in the sense of the parameter
$\eta $ in (II.14) being small) we have:
$$ g_{i,\alpha,\Delta }(x,y) \le 1 \eqno({\rm VI.5})
$$
We are going to compute this function $ g_{i,\alpha,\Delta }$
as the exponential of an action
integrated over $\Delta$, and prove that this action is always
negative. Let us stress that a constant bound such as $O(1)M^{4i}$ on
this action would not be sufficient because the volume of the cube
can be as large as $\lambda^{-1}M^{-4i}$ if $\alpha$ is quite
small compared to $i$, and the small factor gained from the positivity
of the axial gauge for such a cube is in $e^{\lambda^{-\epsilon}}$, $not$ in
$e^{\lambda^{-1}}$.
We write $\ps = \kappa_{i, \al} (p^{2}) /p^{2}$.
It is convenient to define $P_{\mu}$ such that $D_{\mu} =iP_{\mu}$ (in
Fourier space).
With these conventions we have in su(2) space~:
$$
P_{0} = \pmatrix{ p_{0}& 0 & 0 \cr
0 & p_{0} & 0 \cr
0 & 0 & p_{0} \cr }
\quad P_{3} = \pmatrix{ p_{3}& 0 & 0 \cr
0 & p_{3} & 0 \cr
0 & 0 & p_{3} \cr }
$$
$$
P_{1} = \pmatrix{ p_{1}& 0 & 0 \cr
0 & p_{1} & -ix \cr
0 & ix & p_{1} \cr }
\quad P_{2} = \pmatrix{ p_{2}& 0 & iy \cr
0 & p_{2} & 0 \cr
-iy & 0 & p_{2} \cr }
$$
$$
\eqno({\rm VI.6})
$$
The Fadeev-Popov operator $K_{FP}=-\partial \cdot D = \sum_{\mu}p_{\mu}P_{\mu}$ is
$$
K_{FP} = \pmatrix{ p^{2}& 0 & ip_{2}y \cr
0 & p^{2} & -ip_{1}x \cr
-ip_{2}y & ip_{1}x & p^{2} \cr }
$$
$$
\ \eqno({\rm VI.7})
$$
The hermitian matrix $-D^{2}=P^{2}$ is
$$
- D^{2} =P^{2}= \pmatrix{ p^{2}+y^{2}& 0 & 2ip_{2}y \cr
0 & p^{2}+x^{2} & -2ip_{1}x \cr
-2ip_{2}y & 2ip_{1}x & p^{2}+x^{2} +y^{2} \cr }
$$
$$
\ \eqno({\rm VI.8})
$$
The twelve by twelve matrix $BF$ that has to be computed in the general
case $\ze \ne 1$ is decomposed into a four by four matrix of
three by three su(2) blocs~:
$$
BF= \de_{\mu\nu} + \ps \bigl[ (P^{2}\de_{\mu\si} -
(1-\ze)P_{\mu}P_{\si})
(\de_{\si\nu} + (\ze^{-1}-1) {p_{\si} p_{\nu}\over p^{2}})
-p^{2}\de_{\mu\nu} \bigr]
$$
$$= \de_{\mu\nu}(1+\ps U) + \ps {p_{\mu}p_{\nu} \over p^{2}}V
+ \ps W_{\mu\nu} \eqno({\rm VI.9})
$$
where
$$
U=P^{2}-p^{2} = \pmatrix{ y^{2}& 0 & 2ip_{2}y \cr
0 & x^{2} & -2ip_{1}x \cr
-2ip_{2}y & 2ip_{1}x & x^{2} +y^{2} \cr }
$$
$$
\ \eqno({\rm VI.10})
$$
$$
V= \pmatrix{ (1/\ze -1) y^{2} & 0 & (1/\ze -\ze)ip_{2}y \cr
0 &(1/\ze -1) x^{2} &-(1/\ze -\ze)ip_{1}x \cr
-(1/\ze -\ze)ip_{2}y &(1/\ze -\ze)ip_{1}x & (1/\ze -1)( x^{2}+y^{2}) \cr }
$$
$$
\eqno({\rm VI.11})
$$
and
$$
W_{00}=W_{03}=W_{30}=W_{33}=0 \eqno({\rm VI.12a})$$
$$
W_{01}=W_{10}=-(1-\ze)
\pmatrix{ 0 & 0 & 0 \cr
0 &0 &-ip_{0}x \cr
0 &ip_{0}x & 0 \cr }$$
$$
\eqno({\rm VI.12b})
$$
$$
W_{02}=W_{20}=-(1-\ze)
\pmatrix{ 0 & 0 & ip_{0}y\cr
0 &0 & 0 \cr
-ip_{0}y & 0 & 0 \cr }$$
$$
\eqno({\rm VI.12c})
$$
$$
W_{13}=W_{31}=-(1-\ze)
\pmatrix{ 0 & 0 & 0 \cr
0 &0 &-ip_{3}x \cr
0 &ip_{3}x & 0 \cr }$$
$$
\eqno({\rm VI.12d})
$$
$$
W_{23}=W_{32}=-(1-\ze)
\pmatrix{ 0 & 0 & ip_{3}y\cr
0 &0 & 0 \cr
-ip_{3}y &0 & 0 \cr }$$
$$
\eqno({\rm VI.12e})
$$
$$
W_{11}= \pmatrix{ 0 & 0 & 0\cr
xy{(1-\ze )^{2} \over \ze}{p_{1}p_{2}\over p^{2}} &
-x^{2}[{(1-\ze )^{2} \over \ze}{p_{1}^{2}\over
p^{2}}+(1-\ze)] & -ip_{1}x(\ze-1/\ze) \cr
0 &ip_{1}x(\ze -1/\ze) &
-x^{2}[{(1-\ze )^{2} \over \ze}{p_{1}^{2}\over p^{2}}+(1-\ze)]\cr }
$$
$$
\eqno({\rm VI.12f})
$$
$$
W_{12}= \pmatrix{ 0 & 0 & -ip_{1}y(1-\ze)\cr
xy[{(1-\ze )^{2} \over \ze}{p_{2}^{2}\over p^{2}}-(1-\ze)] &
-x^{2}{(1-\ze )^{2} \over \ze}{p_{1}p_{2}\over p^{2}} & -ip_{2}x(1-1/\ze) \cr
ip_{1}y(1-\ze) &ip_{2}x(1-1/\ze) &
-x^{2}{(1-\ze )^{2} \over \ze}{p_{1}p_{2}\over p^{2}}\cr }
$$
$$
\eqno({\rm VI.12g})
$$
$$
W_{21}= \pmatrix{- y^{2}{(1-\ze )^{2} \over \ze}{p_{1}p_{2}\over p^{2}} &
xy[{(1-\ze )^{2} \over \ze}{p_{1}^{2}\over
p^{2}}+(1-\ze)] & ip_{1}y(1-1/\ze)\cr
0 & 0 & ip_{2}x(1-\ze) \cr
- ip_{1}y(1-1/\ze) & -ip_{2}x(1-\ze) &
-y^{2}{(1-\ze )^{2} \over \ze}{p_{1}p_{2}\over p^{2}}\cr }
$$
$$
\eqno({\rm VI.12h})
$$
$$
W_{22}= \pmatrix{-y^{2}[{(1-\ze )^{2} \over \ze}
{p_{2}^{2}\over p^{2}}+(1-\ze)] &
xy{(1-\ze )^{2} \over \ze}{p_{1}p_{2}\over p^{2}} & ip_{2}y(\ze-1/\ze) \cr
0 & 0 & 0\cr
-ip_{2}y(\ze-1/\ze) & 0 &
-y^{2}[{(1-\ze )^{2} \over \ze}{p_{2}^{2}\over p^{2}}+(1-\ze)]\cr }
$$
$$
\eqno({\rm VI.12i})
$$
Similarly we can compute the three by three matrix $FP$~:
$$
FP = 1 + \ps (K_{FP}-p^{2}) = \pmatrix{1 & 0 & i\ps p_{2}y \cr
0 & 1 & -i\ps p_{1}x \cr
-i\ps p_{2}y & i\ps p_{1}x & 1 \cr }$$
$$
\eqno({\rm VI.13})
$$
Remark that all these matrices are homogeneous. We introduce
the variables $u \equiv p^{2}M^{-2i}$, $\theta$ and $\phi$
such that $p_{1}^{2}= u M^{2i}\cos^{2}\theta $ and
$p_{2}^{2}= u M^{2i}\sin^{2} \theta
\cos^{2}\ph $. Then $d^{4}p $ is proportional to
$ M^{4i}udu \sin^{2}\th \sin \ph d\th d \ph$.
We represent the effect of the infrared and ultraviolet cutoffs by
the cutoff $\kappa_{i,\alpha} (p)=
\kappa_{\rho-i}(p^{2}M^{-2i})\equiv \kappa_{k}(u),
k=\rho-i$. Indeed we can limit ourselves to the case where $\al$ takes
its minimum value, in which case there is no particular cutoff on the
value of $p_{0}$; as explained in section V, other cases are similar.
We can suppose that $y \le x$, since the function $g_{i, \al \De}$
is symmetric in $x$ and $y$. We put $v = p^{2}/x^{2} = uM^{2i}/x^{2}$,
$\beta = \kappa_{k}( u)/v = x^{2}M^{-2i} \kappa_{k}( u)/ u $
($\beta$ is a function of $k, u$ and $x$), $\ka = \ka_{k}(u) $
and $y= tx$, $t \in [0,1]$.
Remark that $\vert \De \vert M^{+4i} = M ^{i-\al}$.
With these notations we can compute explicitly the three by three
determinant (VI.13) and in principle the twelve by twelve determinant
(VI.9) and we find (forgetting the commutator $[B,B]^{2}$ in (VI.1),
which
is positive and plays no r\^ole anyway):
$$ g_{i,\alpha,\Delta}(x,y)= g_{k,\al}(x,t)=
\exp\biggl( \vert\Delta\vert M^{4i}$$
$$ \biggl( \biggl[ \int_{0}^{+\infty}
udu\ (2/\pi )\int_{0}^{+\pi } \sin^{2}\theta
d\theta \ (1/2) \int_{0}^{+\pi } \sin\phi d\phi
$$
$$
\ln \vert 1- \be \ka [\cos^{2}\theta + t^{2}
\sin^{2}\theta \cos^{2}\phi ]\vert
-(1/2)\ln \bigl( 1 +\be P(\be, \ka, t, \cos \th, \sin \phi, \ze, 1/\ze )
\bigr)
$$
$$+ \biggl[ \int_{0}^{+\infty} u du (\be/2)[6(1+(1/\ze -1)/4)
- \ka(4 +3(1/\ze -1)/2) ] (1+t^{2}) \biggr]
$$
$$ - \biggl[ \int_{0}^{+\infty} u du ( \be^{2}/24) \bigl[
(36 +18 (1/\ze -1) + 7.5 (1/\ze -1) ^{2})
-\ka (90 +45 (1/\ze -1) + 15 (1/\ze -1)^{2})
$$
$$+ \ka^{2}(54 +27 (1/\ze -1) + 7.5 (1/\ze -1) ^{2}) \bigr]
(1+t^{2})^{2} \biggr)\biggr)
\eqno({\rm VI.14})
$$
where $P$ is a polynomial in all the variables listed,
whose explicit computation requires the
evaluation of the twelve by twelve determinant (VI.9).
In the case of the Feynman gauge $\ze=1$, this determinant simplifies into
a three by three determinant to the fourth power, which is easily
computed, and one finds:
$$ ( 1 +\be P(\be, \ka, t, \cos \th, \sin \phi, \ze, 1/\ze )
= \biggl[ [1 + 2\beta (1-2\ka (\cos ^{2}\theta +
t^{2} \sin^{2}\theta \cos ^{2}\phi )) + \beta ^{2} ]
$$
$$+ \beta t^{2} [2 +\beta (t^{2} + 3-4 \ka (\cos ^{2}\theta + \sin^{2}
\theta \cos^{2}\phi )) + \beta ^{2}(1+t^{2} )]\biggr]^{4} \eqno({\rm VI.15})
$$
Remark that the relative normalization of
the terms and the counterterms in (VI.14) is crucial for what follows.
To check that this relative normalization
is correct the reader should check that the mass term (in $x^{2}$)
which comes from the Fadeev-Popov
determinant in (VI.14) agrees with the mass
counterterm from the same Fadeev-Popov determinant computed in Sect.
III, which is the piece
$e^{ \int_{0}^{+\infty} udu (\be/2) (\ka/2) (1+t^{2})}$ in the total
mass counterterm
$$e^{ \int_{0}^{+\infty} udu (\be/2) [6(1+(1/\ze -1)/4)
- \ka((9/2) +6(1/\ze -1)/4)+ \ka/2] (1+t^{2})} \eqno({\rm VI.16})
$$
as computed in section III. Also the quartic term coming form the
Fadeev-Popov determinant matches the corresponding computation
of the graph $G_{4}$ in section III (recall that the natural
normalization in section III of
the quadratic counterterm was $A^{2}/2$ and of the quartic counterterm
was $A^{4}/24$, see (III.1)).
In the general case we do not attempt the computation of $P$, but we
compute only the integral over $\th$ and $\ph$
of its first order term in $\be$ which is much more
accessible. It can be done by tracing the twelve by twelve matrix
(VI.9) and its square, which is much easier than computing the
determinant.
Equivalently one can perform a graphical computation~;
one finds the same graphs than in section III (remember that they
have the opposite sign of the counterterms in (VI.14)), plus
additional pieces which come from the new vertex $(\ze/2) [(P\cdot A)^{2}
-(p\cdot A)^{2}]$ which is due to our background dependent gauge
fixing. This new vertex is $(\ze/2) (2 \ep _{abc}
B_{\mu}^{b}A_{\mu}^{c}
\partial_{\nu} A_{\nu}^{a} + \ep_{abc}\ep_{a'b'c'}
B_{\mu}^{b}A_{\mu}^{c}B_{\nu}^{b'}A_{\nu}^{c'})$, hence we obtain
additional graphs $G''_{1}$ and $G''_{2}$ analogues to $G'_{1}$
and $G'_{2}$. After a straightforward computation, the contribution of
$G''_{1}$ is $-2 \ze (1+ (1/\ze -1)/4)$ and the contribution of
$G''_{2}$ is $\ka (-2 + (3/2)\ze )$ (beware that in the
computation of $G''_{2}$ one has to add the case with two ``new'' vertices
to the case with one old and one ``new'' vertex). Hence adding the
contribution of the ordinary graphs $G'_{1}$
and $G'_{2}$ we have
$$ \biggl[ (2/\pi )\int_{0}^{+\pi } \sin^{2}\theta
d\theta (1/2) \int_{0}^{+\pi } \sin\phi d\phi
$$
$$
-(1/2)\ln \bigl( 1 +\be P(\be, \ka, t, \cos \th, \sin \phi, \ze, 1/\ze )
\bigr) \biggr]
$$
$$
= (\be/2)(1+t^{2})\biggl(-6(1+(1/\ze -1)/4) -2 \ze (1+ (1/\ze -1)/4)
$$
$$
+ \ka\bigl[(9/2) +(3/2)(1/\ze -1)
-2 +(3/2) \ze \bigr]\biggr) + O(\be ^{2})
\eqno({\rm VI.17})
$$
We will prove Lemma VI.1 using a crude
bound which follows from (VI.17) and the fact that the polynomial
$P$ has a fixed number of (in principle) computable coefficients.
But in Appendix A we provide also an explicit computation of (VI.14)
in the tractable case of the Feynman gauge $\ze =1$ which the reader
might find enlightening.
Using the asymptotic expansion
(VI.17) near $\beta =0$, we find
$$
(2/\pi )\int_{0}^{+\pi } \sin^{2}\theta
d\theta (1/2) \int_{0}^{+\pi } \sin\phi d\phi
$$
$$
\ln \vert 1- \be \ka [\cos^{2}\theta + t^{2}
\sin^{2}\theta \cos^{2}\phi ]\vert
-(1/2)\ln \bigl( 1 +\be P(\be, \ka, t, \cos \th, \sin \phi, \ze, 1/\ze )
\bigr)
$$
$$+ \biggl[ (\be/2)[6(1+(1/\ze -1)/4)
- \ka(4 +3(1/\ze -1)/2) ] (1+t^{2}) \biggr]
$$
$$ \simeq_{\be \to 0} \
(\be /2)(1+t^{2}) [ -3\ze/2 -1/2 -\ka(2-(3/2) \ze)] + O(\be ^{2})
\eqno({\rm VI.18})
$$
If we restrict us to the region $0 \le \ze \le 1$, we have
$$
(\be /2)(1+t^{2})
[ -3\ze/2 -1/2 -\ka(2 -(3/2) \ze )] \le -(\be /4)(1+\ka)
\le -(\be /4) \eqno({\rm VI.19})
$$
Now using the fact that $\ka$, $t$, $\cos \th$,
$\sin \ph$, $\ze$ and $\ze^{-1}$ all
vary in compact intervals (for $\ze$ and $\ze^{-1}$ this is
because we can restrict us to a small interval centered respectively
around 3/13 or 13/3), and the
fact that the logarithms of explicit polynomials in $\be$
such as those of (VI.14) are bounded by a constant
at large $\beta $ (uniform in $\ka$, $t$, $\cos \th$,
$\sin \ph$, $\ze$ and $\ze^{-1}$ by compactness)
it is easy to check the following lemma:
\medskip
\noindent{\bf Lemma VI.2}
If $0 \le \ze \le 1$ there exists two (large...) constants $K_1$ and $K_2$
such that
$$
(2/\pi )\int_{0}^{+\pi } \sin^{2}\theta
d\theta (1/2) \int_{0}^{+\pi } \sin\phi d\phi
$$
$$
\ln \vert 1- \be \ka [\cos^{2}\theta + t^{2}
\sin^{2}\theta \cos^{2}\phi ]\vert
-(1/2)\ln \bigl( 1 +\be P(\be, \ka, t, \cos \th, \sin \phi, \ze, 1/\ze )
\bigr)
$$
$$+ \biggl[ \int u du (\be/2)[6(1+(1/\ze -1)/4)
- \ka(4 +3(1/\ze -1)/2) ] (1+t^{2}) \biggr]
$$
$$ \le K_{2} \beta \quad {\rm if} \ \beta \ge (K_{1})^{-1}
\eqno({\rm VI.20a})
$$
$$ \le -(\be /8) \quad {\rm if} \ \beta \le (K_{1})^{-1}
\eqno({\rm VI.20b})
$$
\medskip
We can then complete the proof of Lemma VI.1.
Indeed we write (using the fact that $\beta = \kappa/v$ and $0 \le \kappa \le
1$ and using (III.7)):
$$ g_k(x) \le \exp\biggl( \vert \De \vert
\biggl( x^{4} \biggl[\int_{v < K_{1}} K_{2} \beta vdv
- \biggl[ \int v dv ( \be^{2}/24) \bigl[
(36 +18 (1/\ze -1)
$$
$$ + 7.5 (1/\ze -1) ^{2})
-\ka (90 +45 (1/\ze -1) + 15 (1/\ze -1)^{2})
$$
$$+ \ka^{2}(54 +27 (1/\ze -1) + 7.5 (1/\ze -1) ^{2}) \bigr]
(1+t^{2})^{2}\biggr] \biggr)\biggr)
$$
$$ \le \exp\biggl( \vert \De \vert
x^{4} \bigl[ K_{1} K_{2}
- \vert \log \et \vert \bigr]\biggr)
\le 1 \eqno({\rm VI.35})
$$
if, again, we choose the parameter $\eta $ in section III sufficiently
small so that $ \vert \log \et \vert \ge K_{1}K_{2}$.
This achieves the proof of the Lemma.
\vfill\eject\medskip
\noindent{\bf VII. The convergence of the expansion: bounds on
error terms}
\medskip
In this section we summarize the reasons for convergence
of our expansion.
We have to give a rough description of the polymers which, at a given
stage of the expansion, have to be summed for the Mayer expansion to
converge. We know that in this type of expansion convergence follows
if in the amplitudes of the polymers there is an adjustable small
constant per box (such as $\la$), and if furthermore we can
resum all the polymers containing a fixed box.
The small constant per box is the easiest part to check. The case
of empty small field boxes is taken into account by the
normalization. A small field box which is not empty must contain some
explicit vertex or error term, which by inspection is small. The large
field boxes are small because of the Lemma II.1 corrected by the computation
of the previous section which justifies the existence of an associated
small factor. But we must check that given some fixed box we can
perform the sum over all the boxes linked to it by previous
horizontal, vertical or Mayer expansions using
the decay of the corresponding links. The
structure of these sums is basically similar to the case considered
in [R] with one most notable exception, the existence of various
lattices with anisotropic shapes. Therefore we focus first on explaining the
convergence of this new feature. More precisely let us explain why
at fixed value of $i$ the vertical expansion in $\al$, the index
labeling the slices for $p_{0}$, is convergent.
Let us consider the small field propagator in a slice of index
$j=\{i,\al\}$. From the point of view of power counting the homothetic
gauge is similar to the Feynman gauge, hence to simplify notations
let us pretend the small field propagator to be simply $1/p^{2}$.
The same propagator in the slice $j$ would then be $
C^{j} = \ka^{j}(p)/p^{2} $. It satisfies the estimate
$$
C^{j}(x-y) \le K_{q}M^{i}M^{\al}
\biggl( { 1 \over 1 + \vert x_{0}-y_{0} \vert M^{\al}} \, \cdot \,
{ 1 \over 1 + \vert \vec x- \vec y \vert M^{i}} \biggr)^{q}
\eqno({\rm VII.1})
$$
for some large integer $q$.
The power counting of the worse vertex (which is a trilinear
vertex with derivative coupling $A^{2}\partial A$ rather than a
quartic $A^{4}$ vertex) integrated in a box
of ${\bf D^{j}}$ is $M^{i} M^{(3/2)(i+\al)}M^{-3i-\al} = M^{- (i-\al)/2}$.
This vertex is equipped with one factor $\la$. However by parity
a single such vertex vanishes. Therefore we have at least two such
vertices, which means the same power counting as a single quartic
vertex (of power counting
$M^{2(i+\al)}M^{-3i-\al} = M^{- (i-\al)}$) with coupling $\la^{2}$.
Since the smallest value of the index $\al$ at $i$ fixed is $\al_{min}$
such that
$M^{- (i-\al_{min})} \le \la ^{-1}$ (see II.21a-b),
we conclude that in the vertical expansion
any contribution of scale $\al $ attached to a box of scale $\al'$,
with $\al_{min } <\al' < \al$ can be resummed
in the box of scale $\al'$ (which contains $M ^{\al -\al' }$
boxes of scale $\al$) using one coupling constant
$\la \le M ^{-(\al -\al' )}$; furthermore
there remains a small factor of size at least $\la$. Finally
there remains a factor at least $M^{- (i-\al)}$ which means that
the sum over $\al$ can be performed and that in fact most of the
sum comes from the case $\al =i$. This confirms the auxiliary nature
of this expansion. We have to perform it because we had
to keep the decomposition
of the isotropic small field propagator into anisotropic scales
in the case of relevant small field boxes
in the sense of section IIb. Otherwise we would not have
the right spatial decay in the
$x_{0}$ direction (this is due to the fact that in the domain
of a large field box the small field
cutoff is limited by the ancestor of the small field boxes). However
we see that the main small field
contribution comes really from the case $i=\al$, so
this decomposition is not very important.
The other types of links are either horizontal between small field
boxes, in which case we have enough spatial decay to resum these links, or
vertical links; in the case of five or more legs ($t_{\De}\ne 0$)
power counting provides the necessary factor $M^{-5\vert i-i'\vert}$
to resum a box of scale $i$ among the $M^{4 \vert i-i'\vert}$ boxes
of scale $i$ contained in a box of scale $i'$. In the case of two and
four point functions, renormalization performs the same task in the
usual way.
We have also a new type of links if we compare
to the multiscale expansion of [R] which are the
``proximity'' links, both in the vertical and horizontal directions
between large field boxes (and the associated ``protection corridors''
introduced previously). All these links extend only
to a bounded distance, independent of $\la$.. It is possible
to resum over such links (which costs only a factor independent of
$\la$) using a piece of the small factors
$O(e^{-\la^{-\ep}} )$ associated to the large field regions (see II.28).
We have to check that the functional integral over $\ga$
can be performed.
In particular, one problem which might worry the reader is that
our use of polynomial approximations to the
true gauge transformations might forbid us from
doing exact computations at all orders in perturbation theory.
Strictly speaking this is correct, but as explained
already we need only to perform, e.g. for the
flow of the coupling constant, one and two loop computations,
and to show that the remainders are of the next order.
Since $A^{\gamma ,2}$ gives vertices small only
as $\lambda ^{1/2}$, it is not, strictly speaking, adapted to this
problem. But we can replace everywhere in section II $A^{\gamma ,2}$
by e.g. $A^{\gamma ,10}$, and still control all the integrals
in the same way, using the fact that the factor $N$
in (II.36-40) can be made arbitrarily large;
the resulting formulas are simply more complicated \footnote*{It is
presumably even possible to use the true gauge transformations
$A^{\gamma ,\infty}$ in our change of gauge formulas (II.36-40)
but it is less clear how to define an adequate analogue of the
protection term in this case.}.
In the small field region the bounds we use are similar to the
case of the infrared $\phi^{4}_{4}$ critical theory (see [FMRS1],[R]);
remark however that for the ``domination'' process,
we have to use the small field
condition $e^{-E_{\De}}$ in (II.25), which costs a factor
$\la^{1/2 + \ep_{1}}$ per field, rather
than the $\la^{4}A^{4}$ term (which would cost one
full $\la$). In this way a vertex such as $\la^{2} [A,A]^{2}$ with
the worst case of three badly localized legs has still
a small factor of order $\la^{1/2 -3 \ep_{1}}$, because it has at
least one well localized leg, and if this well localized leg is a
small field $A '_{s}$, its Gaussian integration does not cost any
fraction of $\la$.
At this point the attentive reader may ask what
happens if this well localized leg is of the large field type.
More generally why do the vertical couplings between high
momentum large field regions and lower
momentum small field regions also
lead to small factors? For instance
a vertex such as $\la[B^{i}, A^{i '}]$ with $i ' < i$,
using the $\la^{4} B^{4}$ term for domination of the $B$ field
seems to eat up the factor $\la$, hence to lead to no small factor.
This is not true because the vertical corridor that we decided
to include can be made much
bigger in the direction of lower frequencies than of higher
frequencies. Indeed there are $M^{4}$ boxes of the next higher
frequency in a typical cubic box, which in practice limit us
to consider corridor of bounded width (independent of $\la$)
in the vertical direction upwards. But there is only one box which contains
a box in the next lower frequency. If we take into account the fact
that we have a small factor of order
$e^{-\la_{i}^{-\ep}}$ in a large field box of scale $i$, we can include
all the boxes which contain it until frequency $i ' = i - K\vert \log
\la \vert $ in the protection corridor ($K$ being
a large constant), and still attribute a small
factor of similar order (with $\ep$ slightly smaller) to all these boxes.
It is then true that the integration of the well localized $B^{i}$
field using e.g. the $\la^{4} B^{4}$ term eats up
the coupling constant, but this is more than compensated by the fact
that if we dominate the $A^{ i '}$ field using the small field
condition we gain a small factor $M^{-K\vert \log
\la \vert }$ which comes from writing $M^{i '} \le M^{i} M^{-K \vert \log
\la \vert}$.
This factor is by itself a very large power of $\la$ so the
corresponding terms are indeed extremely small.
\medskip
\noindent{\bf VIII. Slavnov identities}
\medskip
Slavnov identities can be be used in a perturbative gauge such as the
Feynman or Landau gauge to check the necessary relations which ensure
order by order that up to a rescaling of $A$ (wave function renormalization)
there is only a coupling constant renormalization (perturbative
renormalizability of the model). These identities can be expressed in
terms of the Schwinger functions of the theory, but in this form they
were of course up to now true only in the sense of formal power series.
Our bare ansatz is written for a theory with field $A$
satisfying an axial gauge condition,
hence the Slavnov identities that we have to check are the identities
adapted to this condition. However one can also reexpress these
identities in
terms of the small field $A'$~; one would then prove the usual
identities
in the perturbative homothetic gauge, up to a remainder which is not 0
but
which vanishes to any order in perturbation theory, and which
corresponds to the large field regions in which the field has been
kept in the axial gauge. These identities are those used in the
preceding sections at order 3 in perturbation theory to control
the renormalization group flow of the coupling constant.
To derive the exact form of Slavnov identities in the axial gauge,
one introduces the generating
functional for the theory with gauge condition $A_{0}=0$. Formally we
can write this functional as:
$$ W(J) = < e^{-F^2/4 + J\cdot A} \de (A_{0}) > =< e^{ J\cdot A}
>_{ax}
\eqno({\rm VIII.1})
$$
where the first expectation
value is with respect to the formal Lebesgue measure, and the second
one is the expectation value in the axial gauge constructed by the
limit process described above. The $A$ field can
be thought as the corresponding ${\delta \over \delta J}$ functional
derivation.
Then in (VIII.1) we perform a change of variables $A \to A + D\gamma $;
by (infinitesimal) gauge invariance, there is no first order
dependence in $\ga $, which gives the ``Ward'' or ``Slavnov''
equation:
$$< \biggl(J_{m }^{a}(x) D_{m }^{ab}(x) - [{\partial \over \partial
x^{0}}A_{m}^{a} \cdot D_{m}^{ab}] {\partial \over \partial
x^{0}} \biggr)e^{ J\cdot A} >_{ax}=0 \eqno({\rm VIII.2})$$
Integrating by parts and taking into account the fact that $A_{m}D_{m}
=A_{m}{\partial \over \partial x^{m}}$ we can rewrite this identity
simply as~:
$$ <\biggl(J_{m }^{a}(x) D_{m }^{ab}(x) - [{\partial \over \partial
x^{m}}A_{m}^{b} (x) ] ({\partial \over \partial
x^{0}})^{2} \biggr)e^{ J\cdot A} >_{ax}=0 \eqno({\rm VIII.3})$$
This identity gives rise to a hierarchy of identities with any number
$N$ of external sources. For instance the two point function identity
is obtained by applying one functional derivative
${\delta \over \delta J}(y)$ to (VIII.3) and is simply (exchanging the
names of $x$ and $y$):
$$ < A_{m}^{a}(x) [{\partial \over \partial
y^{n}}A_{n}^{b} (y) ] ({\partial \over \partial
y^{0}})^{2} >_{ax} = \de_{ab} \de (x-y) {\partial \over \partial
y^{m}} \eqno({\rm VIII.4})$$
We should of course understand this identity as applied to two test
functions of $x$ and $y$.
Similarly we can write e.g. an identity involving $N$ point
functions~:
$$ <\sum_{i=1}^{N-1}\biggl(
\prod_{j=1, j\ne i}^{N-1} A_{m_{j}}^{a_{j}}(x_{j})\biggr)
D_{m_{i}}^{a_{i}b} \de(x_{i}-y) -
\prod_{j=1}^{N-1}A_{m_{j}}^{a_{j}}(x_{j})
[{\partial \over \partial
y^{n}}A_{n}^{b} (y) ] ({\partial \over \partial
y^{0}})^{2} >_{ax} = 0 \eqno({\rm VIII.5})$$
For a theory with a fixed infrared-cutoff of a given type, the linear term
in $\ga $ for a gauge transformation $A \to A + D\ga$ receives a
contribution from the presence in (VIII.1) of the cutoff. This leads
to correction terms in the Slavnov identities. For example
equation (VIII.5) takes the form
$$ <\sum_{i=1}^{N-1}\biggl(
\prod_{j=1, j\ne i}^{N-1} A_{m_{j}}^{a_{j}}(x_{j})\biggr)
D_{m_{i}}^{a_{i}b} \de(x_{i}-y) -
\prod_{j=1}^{N-1}A_{m_{j}}^{a_{j}}(x_{j})
[{\partial \over \partial
y^{n}}A_{n}^{b} (y) ] ({\partial \over \partial
y^{0}})^{2} >_{ax} $$
$$ = E_{N} (\{x_{j}\}) \eqno({\rm VIII.6})$$
where $E_{N}$ can be computed for any given infrared cutoff. If this
cutoff is a finite compact box with some kind
of boundary conditions, we see that $E_{N}$ can be
interpreted as a boundary term, because the gauge invariance
is exact inside the box.
The identities (VIII.6) are those that we are going to check in the limit
$\rh \to \infty$. In order to prove them we write first approximate
identities which are satisfied for the theory with cutoffs (both
infrared and ultra-violet) and
gauge restoring counterterms. These identities take the form
of equality between the left hand side of (VIII.6) where $<.>_{ax}$ is replaced
by $<.>_{ax, \rh}$, the normalized functional integral of our
theory with cutoff and a right hand side which is no longer $E_{N}$,
but $E_{N} + \de_{N}(\rh)$
because of the effects $\de_{N}$ of
the gauge transformation $A \to A + D\ga$ on the ultraviolet
cutoff and on the gauge
restoring counterterms. When $\rh \to \infty$, the left hand side,
made of normalized Schwinger functions with cutoff $\rh$,
by definition tends to the same
Schwinger functions without ultraviolet cutoff that we have
constructed. Our expansion
proves that in the right hand side the error term
$\de_{N}(\rh) $ tends to zero, because it is made of
contributions tied to the ultraviolet cutoff.
In conclusion we have obtained the following result
\medskip
\noindent{\bf Theorem}
\noindent
The ultraviolet limit as $\rho \to \infty$ of the Schwinger functions
which are the
moments of the measure (II.77) exists; furthermore
these functions in the ultraviolet limit satisfy the Slavnov
identities (VIII.6).
\medskip
\vfill\eject\medskip
\noindent{\bf IX. Slavnov identities}
\medskip
Slavnov identities can be be used in a perturbative gauge such as the
Feynman or Landau gauge to check the necessary relations which ensure
order by order that up to a rescaling of $A$ (wave function renormalization)
there is only a coupling constant renormalization (perturbative
renormalizability of the model). These identities can be expressed in
terms of the Schwinger functions of the theory, but in this form they
were of course true only in the sense of formal power series.
Our bare ansatz is written for a theory with field $A$
satisfying an axial gauge condition,
hence the Slavnov identities that we have to check are the identities
adapted to this condition. However one can also reexpress these
identities in
terms of the small field $A'$~; one would then prove the usual
identities
in the Perturbative special gauge, up to a remainder which is not 0
but
which vanishes to any order in perturbation theory, and which
corresponds to the large field regions in which the field has been
kept in the axial gauge. These identities are those used in the
preceding chapter at order 3 in perturbation theory to control
the renormalization group flow of the coupling constant.
To derive the exact form of Slavnov identities in the axial gauge,
one introduces the generating
functional for the theory with gauge condition $A_{0}=0$. Formally we
can write this functional as:
$$ W(J) = < e^{-F^2/4 + J\cdot A} \de (A_{0}) > \eqno({\rm IX.1})
$$
where the expectation
value is with respect to the formal Lebesgue measure. The $A$ field can
be thought as the corresponding ${\delta \over \delta J}$ functional
derivation.
Then in (IX.1) we perform a change of variables $A \to A + D\gamma $;
by (infinitesimal) gauge invariance, there is no first order
dependence in $\ga $, which gives the ``Ward'' or ``Slavnov''
equation:
$$ \biggl(J_{m }^{a}(x) D_{m }^{ab}(x) - [{\partial \over \partial
x^{0}}A_{m}^{a} \cdot D_{m}^{ab}] {\partial \over \partial
x^{0}} \biggr) W(J)=0 \eqno({\rm IX.2})$$
Integrating by parts and taking into account the fact that $A_{m}D_{m}
=A_{m}{\partial \over \partial x^{m}}$ we can rewrite this identity
simply as~:
$$ \biggl(J_{m }^{a}(x) D_{m }^{ab}(x) - [{\partial \over \partial
x^{m}}A_{m}^{a} (x) ] ({\partial \over \partial
x^{0}})^{2} \biggr) W(J)=0 \eqno({\rm IX.3})$$
This identity gives rise to a hiererchy of identities with any number
$N$ of external sources. For instance the two point function identity
is obtained by applying one functional derivative
${\delta \over \delta J}(y)$ to (IX.3) and is simply~(exchanging the
names of $x$ and $y$):
$$ < A_{m}^{a}(x) [{\partial \over \partial
y^{n}}A_{n}^{b} (y) ] ({\partial \over \partial
y^{0}})^{2} > = \de_{ab} \de (x-y) {\partial \over \partial
y^{m}} \eqno({\rm IX.4})$$
where the expectation value $<.>$ is with respect to the functional
integral over $A$.
We should of course understand this identity as applied to two test
functions of $x$ and $y$.
Similarly we can write e.g. an identity involving $N$ point
functions~:
$$ <\sum_{i=1}^{N}\biggl(
\prod_{j=1, j\ne i}^{N} A_{m_{j}}^{a_{j}}(x_{j})\biggr)
D_{m_{i}}^{a_{i}b} \de(x_{i}-y) -
\prod_{j=1}^{N}A_{m_{j}}^{a_{j}}(x_{j})
[{\partial \over \partial
y^{n}}A_{n}^{b} (y) ] ({\partial \over \partial
y^{0}})^{2} > = 0 \eqno({\rm IX.5})$$
These identities are those that we are going to check in the limit
$\rh \to \infty$. In order to prove them we write first approximate
identities which are satisfied for the theory with cutoff and
gauge restoring counterterms. These identities take the form
of equality between the left hand side of (IX.5) where $<.>$ is replaced
by $<.>_{\rh}$, the normalized functional intgeral of our cutoff
theory and a right hand side which is not 0, because of the effects of
the gauge transformation $A \to A + D\ga$ on the cutoff and the gauge
restoring counterterms. When $\rh \to \infty$, the left hand side,
made of normalized Schwinger functions with cutoff $\rh$,
by definition tends to the corresponding
Schwinger functions without ultraviolet cutoff that we have
constructed, and we have only to show that
the right hand side tends to zero.
\vfill\eject\noindent{\bf Appendix 1}
\medskip
In this appendix we provide some explicit bounds and computations
of the determinants considered in section VI in the simpler case of the
Feynman gauge $\ze=1$.
We start by a warm up: the case $t=0$. The integral over
$\phi $ is then trivial and we have to compute:
$$ g_{k }(x) = \exp(1/2)\int_{\De} \biggl( x^{4}\int vdv
\biggl[ (4/\pi )\int_{0}^{\pi} \sin^{2}\theta
d\theta
$$
$$
\ln \vert 1- \beta \ka \cos^{2}\theta \vert -2\ln (1 +2\beta -4 \be \ka
\cos ^{2} \theta +\beta ^{2}) \biggr]
$$
$$ -c x^{4} + [ x^{4}\int \beta[6 - 4\ka ] vdv] \biggr)
\eqno({\rm A.1})$$
We perform first at fixed $\beta$ and $\ka $, hence fixed $v$,
the angular integral over $\theta$. In order to simplify slightly
the computation we remark first that we have the rigorous inequality
(since $0 \le \ka \le 1$)~:
$$
-2\ln (1 +2\beta -4 \be \ka
\cos ^{2} \theta +\beta ^{2}) \le -2\ln (1 - 2\beta
\cos 2 \theta +\beta ^{2})
\eqno({\rm A.2})
$$
Using this inequality, we can simply compute:
$$ G(\beta ) = (4/\pi )\int_{0}^{\pi } \sin^{2}\theta d\theta
$$
$$
\ln \vert 1- \beta \ka \cos^{2}\theta \vert -2\ln (1 -2\beta
\cos 2\theta +\beta ^{2}) \eqno({\rm A.3})$$
To study $G$ when $\beta $ varies
we can e.g. differentiate once again, so that the $\theta$
integration can be performed
by elementary contour integrals. Then we integrate the
result. The outcome is:
$$ G(\beta ) = -2(1+\ln 4 ) -4\beta +4/(\beta \ka) +2 \ln( \beta \ka)
+2 \ln {1+ \sqrt{1-\beta \ka }\over 1- \sqrt{1-\beta \ka}}
$$
$$ + {2\over 1+ \sqrt{1-\beta \ka }}
- {2 \over 1- \sqrt{1-\beta \ka}} \ {\rm if}\ \beta \le 1
$$
$$ G(\beta ) = -2(1+\ln (4/\ka) ) -6 \ln \beta + {4 (1-\ka) \over \be \ka}
+2 \ln {1+ \sqrt{1-\beta \ka }\over 1- \sqrt{1-\beta \ka}}
$$
$$ + {2\over 1+ \sqrt{1-\beta \ka }}
- {2 \over 1- \sqrt{1-\beta \ka}} \ {\rm if}\ 1 \le \beta \le 1/\ka
$$
$$ G(\beta ) = -2(1+\ln (4/\ka) ) -6 \ln \beta + {4 (1-\ka) \over \be \ka}
\ {\rm if}\ \beta \ge 1 /\ka
\eqno({\rm A.4})$$
The function $G'$ is always negative (remark that $G'(0)=-4-\ka /2$; this
value is critical for the rest of our analysis). Therefore $G$
is always negative (this can also be checked directly on (A.4));
Moreover for $\beta \le 1$ we have $G(\beta) \le (-4-\ka /2)\be$.
Therefore, taking into account the fact that $\be v = \ka \le 1$,
hence that $\be >1$ implies $v <1$, and the shape
of $\kappa$ chosen in section III:
$$ g_{k }(x) \le \exp\int_{\De} \biggl( x^{4}
\bigl( - c + \int_{\kappa \ge 1/2}
4\beta vdv + \int_{\kappa < 1/2} 6 \beta v dv
$$
$$- \int ( 4 +\ka /2) \beta vdv + \int_{ \be > 1}
(4 + \ka /2)\beta vdv \bigr)\biggr)
$$
$$ \le\exp\int_{\De} \biggl( x^{4} \bigl(-c +9/2
+ \int_{\kappa > 1/2} 4\beta v dv
+ \int_{\kappa < 1/2} 6 \beta v dv
- \int_{\ka =1/2}{\be v \over 4} dv \bigr) \biggr)
$$
$$ \le \exp \int_{\De}\biggl(x^{4} (-c +9/2) + x^{2} M^{2k}
( 8 + 6/2 - (1/8\et)) \biggr)
$$
$$ \le 1 \eqno({\rm A.5})
$$
if we take the parameter $\et$ in (II.14) sufficiently
small, so that the constant $c$ (which diverges like $\vert \log \et
\vert$) is bigger than 9/2 and such that $\et$ is smaller than 1/88;
of course in (A.5) we assumed
a cutoff with the particular shape of Fig. II.1. Many
other shapes will work as well, but it seems that cutoffs which tend
very slowly to
zero spending a lot of time between 1/2 and 0 are ruled out by this method.
Let us return to the general
case $\ze =1$ but $t \ne 0$.
The third order polynomial in $\beta $ in (VI.15) becomes
if we put $\tau =t^{2}$:
$$ [1 + 2\beta (1-2\ka (\cos ^{2}\theta +
\tau \sin^{2}\theta \cos ^{2}\phi ) + \beta ^{2} ]
$$
$$+ \beta \tau [2 +\beta (\tau + 3-4 \ka (\cos ^{2}\theta + \sin^{2}
\theta \cos^{2}\phi )) + \beta ^{2}(1+\tau )]
$$
$$ \ge [1 + 2\beta
(1-2(\cos ^{2}\theta +\tau \sin^{2}\theta \cos ^{2}\phi )) +
\beta ^{2} ] +7\beta \tau /4
\eqno({\rm A.6})$$
If we put $s= \sqrt{\tau}\cos\phi $ and $w=s^{2}$,
we have therefore to bound the integral
$$ H(\beta ) =
(1/2)\int_{-\sqrt{\tau}}^{\sqrt{\tau}} {ds \over \sqrt{\tau} }
\int_{0}^{\pi }(4/\pi ) \sin^{2}\theta
d\theta
$$
$$
\ln \vert 1 - \beta \ka (\cos^{2}\theta + w \sin^{2}\theta ) \vert
-2\ln (1 +2\beta(1+7/8 \tau -2
\cos ^{2}\theta -2w\sin^{2}\theta ) +\beta ^{2}) \eqno({\rm A.7})$$
The singularity in the logarithm is integrable hence the result is obviously
well defined and real. However to compute it we decide
to regularize the singularity in the logarithm by adding $+i\epsilon $
and taking the real part (there are non trivial imaginary parts corresponding
to the way we avoid the singularity but we can just throw them away
since $\ln \vert f\vert = {\bf Re \,} \ln f$). Then we
derive with respect to $\beta $ and changing to the variable
$z=e^{2i\theta }$ we have a rational contour integral to compute. It
is convenient to define $\be ' = \ka \be$. Then one finds:
$$
H (\beta ) = (1/2)\int_{-\sqrt{\tau} }^{\sqrt{\tau} }
{ds \over \sqrt{\tau} }
(1/2i\pi )\oint_{\vert z\vert =1 } {dz \over z}
(2-z -1/z )
$$
$$ \lim_{\epsilon \to 0}{\bf Re \,}
\ln ( 4 -4\beta \ka w -\beta \ka (1-w) (2+z+1/z) +4i\epsilon )
$$
$$-2\ln ( 1 + 2\beta(1+7\tau /8)
-\beta (1-w)(2+z+1/z) -4\beta w +\beta ^{2})
\eqno({\rm A.8})$$
$${d \over d\beta }
H (\beta ) = (1/2)\int_{-\sqrt{\tau} }^{\sqrt{\tau} }
{ds \over \sqrt{\tau} } [\ka A + B] \eqno({\rm A.9})
$$
$$
A = (1/2i\pi )\oint_{\vert z\vert =1 } {dz \over z^{2}}
(2z-z^{2} -1)
$$
$$ \lim_{\epsilon \to 0}{\bf Re \,}
{-4wz -(1-w) (2z +z^{2}+1) \over (4-4\beta ' w)z -\beta ' (1-w)(2z
+z^{2}+1)+ 4i\epsilon z } \eqno({\rm A.10})
$$
$$ B =
(1/2i\pi )\oint_{\vert z\vert =1 } {dz \over z^{2}}
(2z-z^{2} -1)
$$
$$[(-2) { -2w z - (1-w)(z^{2}+1) +
2\beta z +7\tau z/4
\over (1-2\beta w)z - \beta (1-w)( z^{2}+1 ) +\beta ^{2} z +
7\beta \tau z/4} ]
\eqno({\rm A.11})$$
The first integral, $A$, has poles in $z=0$ and
at
$$z_{\pm} = { 2-\beta ' (1+w) +2i\epsilon
\pm 2 \sqrt {1-\beta' -\beta ' w +(\beta') ^{2} w -
\epsilon ^{2} + i\epsilon (2-
\beta' -\beta' w)} \over \beta' (1-w)}\eqno({\rm A.12})$$
The pole at $z=0$ for the first piece of the integrand gives a real
contribution equal to
$$ {-2 \over (\beta ')^{2} (1-w) } (2-\beta ' +\beta ' w)
\eqno({\rm A.13})$$
When $1 <\beta ' <1/w$ the two poles at $z_{\pm}$ are approximately on the
contour of integration and are approximately complex conjugate; if
$\epsilon $ is small positive one of the two poles $z_{\pm}$
is inside the unit circle and the other outside (the one inside depends on the
convention for the square root, but with the most natural convention it
is $z_{+}$ for $2/(1+w) <\beta' <1/w$ and $z_{-}$ for $1<\beta' <1/w$).
However we do not need to take these residues into account since they
become purely imaginary when $\epsilon $ goes to 0, hence when we
take the real part they disappear.
For $\beta '<1$ there is the contribution of one real pole inside
the unit disk, with residue
$$ {4 (1-\beta ') \over (\beta ') ^{2}(1-w)\sqrt{1-\beta' -\beta' w +
(\beta') ^{2}w }} \eqno({\rm A.14})
$$
For $\beta '> 1/w$ there is another pole inside the unit circle with residue:
$$ {4 (\beta' - 1) \over (\beta') ^{2}(1-w)\sqrt{1-\beta' -\beta' w +
(\beta ') ^{2}w }} \eqno({\rm A.15})
$$
Hence
$$ A = {2 \over (\beta') ^{2} (1-w)} [ -2 +\beta' (1-w)
+ { 2 (1 -\beta' ) \over \sqrt{(1 -\beta') (1 - \beta ' w) }} ]
\quad {\rm if} \ \beta' <1 \eqno({\rm A.16})
$$
$$ A = { 2\over (\beta') ^{2} (1-w)} [ -2 +\beta'(1-w) ] \quad
{\rm if} \ 1<\beta' <1 /w \eqno({\rm A.17})
$$
$$ A = {2 \over (\beta') ^{2} (1-w)} [ -2 +\beta' (1-w)
- { 2 (1 -\beta ') \over \sqrt{(1 -\beta') (1 - \beta ' w) }} ]
\quad {\rm if} \ \beta '>1/w \eqno({\rm A.18})
$$
The term $B$ has a pole
at $z=0$ giving the contribution
$$ {2 \over \beta ^{2}(1-w)} (1 - \beta ^{2} -2\beta + 2 \beta w )
\eqno({\rm A.19})$$
Finally there is a pole inside the unit disk at the location
$$
{(1-\be)^{2} +\be \ta ' - \sqrt{(1 -2\be w + \be ^{2}+ \be \ta ')^{2}
-4\be ^{2}(1+w^{2})} \over 2\be (1-w)}
\eqno({\rm A.20})
$$
where $\ta ' \equiv 7\ta /4$, which gives the contribution
$$ -2 { (( 1-\be)^{2} +\beta \tau ')( 1 - \beta^{2 } )
\over \beta ^{2} (1-w) \sqrt{(1-\beta
)^{2}[(1+\beta) ^{2} -4\beta w] + \be \ta '
(2-4\beta w +2\beta ^{2} +\beta \tau ' )}} \eqno({\rm A.21})
$$
Therefore we have
$$ B = {2 \over \beta ^{2} (1-w)} \bigl[ 1 -\beta ^{2} -2\beta +2\beta w
$$
$$- { (( 1 - \beta)^{2}+\be \ta ')( 1 - \beta^{2 } ) \over
\sqrt{(1-\beta
)^{2}[(1+\beta) ^{2} -4\beta w] + \beta \tau '
(2-4\beta w +2\beta ^{2} +\beta \tau ')}} \bigr] \eqno({\rm A.22})
$$
Adding $\ka A$ and $B$ we find:
$$ \ka A+B = {2 \over \beta ^{2} (1-w)} [ 1 -2/\ka -\beta (1- w)
-\beta ^{2}
+ { 2 (1 -\beta \ka ) \over \ka \sqrt{(1 -\beta \ka) (1 - \beta \ka w) }}
$$
$$- { (( 1 - \beta)^{2} + \be \ta ')( 1 - \beta^{2 } ) \over
\sqrt{(1-\beta
)^{2}[(1+\beta) ^{2} -4\beta w] +\beta \tau '
(2-4\beta w +2\beta ^{2} +\beta \tau ' )}}
\quad {\rm if} \ \beta \ka <1
\eqno({\rm A.23})$$
$$ \ka A+B = { 2\over \beta ^{2} (1-w)} [ 1-2/\ka -\beta(1-w) -\beta ^{2}
$$
$$- { (( 1 - \beta)^{2} + \be \ta ')( 1 - \beta^{2 } ) \over
\sqrt{(1-\beta
)^{2}[(1+\beta) ^{2} -4\beta w] +\beta \tau '
(2-4\beta w +2\beta ^{2} +\beta \tau ')}}
] \quad
{\rm if} \ 1<\beta \ka <1 /w
\eqno({\rm A.24})$$
$$ \ka A+B = {2 \over \beta ^{2} (1-w)} [ 1-2/\ka - \beta (1-w) -\beta ^{2}
- { 2 (1 -\beta \ka ) \over \ka \sqrt{(1 -\beta \ka) (1 - \beta \ka w) }}
$$
$$- { (( 1 - \beta)^{2}+ \be \ta ')( 1 - \beta^{2 } ) \over
\sqrt{(1-\beta )^{2}[(1+\beta) ^{2} -4\beta w] +\beta \tau '
(1-4\beta w +2\beta ^{2} +\beta \tau ')}}]
\quad {\rm if} \ \beta \ka >1/w
\eqno({\rm A.25})$$
If we use this explicit computation
to expand the function ${d\over d\beta }H(\beta )$ near
$\beta =0$ we find that it behaves as
$$ (1/2) \int_{-\sqrt{\tau }}^{+\sqrt{\tau }} {ds \over
\sqrt{\tau }}{2 \over 1-w}
\biggl( (\ka /4)(-1 -2w +3w^{2}) -2 +8w -6w^{2}- 2\tau'(1-w) \biggr)
$$
$$= -4 - 3\tau -(\ka /2)(1+\ta ) \eqno({\rm A.26})
$$
using the fact that $w=s^{2}$.
Hence the slope of the function $H$ near $\beta =0$ is more negative
than when $\tau =0$. It is possible to integrate explicitly the
formula for $H$, but we do not give the expression here.
Using the asymptotic expansion
(A.26) near $\beta =0$, the fact that $\tau $
varies in the compact interval [0,1] and the
fact that the derivative of $H$ is bounded by a constant
at large $\beta $ we can of course always
achieve a uniform bound as in section VI.
We want also to derive a bound
showing the strict positivity of $-\De^{homothetic}_{B}$ in a constant
field $B$, unless all the components of $B$ are in the same
direction in su(2) space, and $p$ is then exactly
aligned with the corresponding (unique) vector $\la B$. For this
we use the fact that
$$
\de_{\mu\nu} D^{2} - 10/13 \nabla_{\mu}\nabla_{\nu} \ge 3/13 D^{2}\ ;\
\de_{\mu\nu} p^{2} - 10/13 p_{\mu}p_{\nu} \le 23/13 p^{2}
\eqno({\rm A.27})
$$
in order to show that the normalized operator
$(-\De^{homothetic}_{B})(-\De^{homothetic})^{-1}$ is bounded up to a
factor $(3/23)^{12}$ exactly by the same bound than in the Feynman case of
ordinary Laplacians. In that case we can use the explicit
computations above to establish the necessary bounds. In particular
this proves that the determinant of
$(-\De^{homothetic}_{B})(-\De^{homothetic})^{-1}$ is
bounded away from 0 up to a constant factor by the bound (A.6).
The only zeroes
of the right hand side of (A.6)
$$ \biggl( [1 + 2\beta
(1-2(\cos ^{2}\theta +\tau \sin^{2}\theta \cos ^{2}\phi )) +
\beta ^{2} ] +7\beta \tau /4 \biggr)
\eqno({\rm A.28})$$
occur for $\tau=0$, $\be =1$ and $\th =0$, which correspond to the
announced case of all components of $B$ aligned in $su(2)$ space
(since $\tau=0)$ and $B$ aligned with the momentum $p$ ($\be =1$, $\th=0$).
We can consider that for a fixed $B$ (with approximate alignment of all
components in $su(2)$ space) the zero at $p=0$
of the ordinary Laplace operator
$p^{2}$ is simply translated.
If we use a cutoff function $\ka ^{m}_{B}$ as in (V.2)
with correct scaling around this
translated zero of the operator $(-\De^{homothetic}_{B})$
with $constant$ background, we obtain the correct polynomial
bounds on the spatial decay using integration by parts on the cutoff
function:
$$
\ka ^{m}_{B}*(-\De^{homothetic}_{B})^{-1} (x,y) \le K_{q}M^{2m}
\biggl( {1 \over 1+
M^{m}\vert x-y \vert }\biggr)^{q}
\eqno({\rm A.29})
$$
It is this decrease which is finally used in the horizontal cluster expansion.
\vfill\eject\vskip .5cm
\noindent {\bf Acknowledgements} We thank warmly J. Feldman for his
collaboration at an early stage of this work. V. Rivasseau, who by
himself alone would have abandoned this difficult technical problem,
thanks particularly J. Magnen for his tenacity and for all
his ideas.
\vskip 1cm
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\vfill\eject
**