\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 possibiliti