\vskip 1.5cm
\centerline{\bf Rigorous results on
the ultraviolet limit}
\centerline{\bf of non-Abelian gauge theories}
\vskip 2cm
\centerline{Jacques Magnen}
\vskip .3cm
\centerline{Vincent Rivasseau}
\vskip .3cm
\centerline{Roland S\'en\'eor}
\vskip 1cm
\centerline{Centre de physique th\'eorique, CNRS, UPR14}
\centerline{Ecole Polytechnique, 91128 Palaiseau Cedex, France}
\vskip 2.5cm
\centerline{\bf ABSTRACT}
We report on 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. We review briefly the difficulties
related to Gribov ambiguities, then we give an outline of
our construction.
\vskip 1.5cm
\line {A 147 01 92 \hfil February 1992}
\vfill\eject
\noindent {\bf I. Introduction and outline}
\medskip
Non-Abelian gauge theories are used in high
energy physics because they are the only field theories renormalizable and
asymptotically free. Most physicists are convinced that the
ultraviolet problem for these theories
is well understood. However it is surprisingly hard to
substantiate this belief rigorously beyond perturbation theory,
although rigorous constructions
of simpler asymptotically free theories such as
the Mitter-Weisz or massive Gross-Neveu model in two dimensions have been
available for many years now [FMRS1][GK].
Until now there was only one rigorous program
completed on this problem, the one of Balaban [B].
This program defines a sequence of
block-spin transformations for the 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 for
subsequences.
Although very impressive, Balaban's work is not easily accessible.
Also it does not
construct the expectations values of products of the field operators
in a particular gauge (the Schwinger functions), because these are not
gauge invariant observables. Although physical quantities should be
gauge invariant, the Schwinger functions are obviously very
convenient for perturbative computations.
A related program in progress but not yet completed is the one of Federbush
[F].
In this paper we want to report on our approach to the same problem.
We have constructed the Schwinger functions of the pure $SU(2)$
Yang-Mills field in a regularized axial gauge,
with a finite volume box as infrared cutoff
[MRS]. Our construction remains unfortunately quite complicated and
technical. Therefore in this letter we give an outline of
the method and results. Our conclusion, after many years of efforts, is
the same as Balaban's, namely that the ultraviolet limit of Yang-Mills can
be done constructively, although not easily.
Our approach for the moment is limited to
the axial gauge. Feynman gauges or similar ones which are Euclidean invariant
and convenient for perturbative computations cannot be used directly
because of their
lack of positivity; this point is discussed in detail below.
Also for technical reasons we have to
require stability of our ultraviolet cutoff, which rules out certain types
of cutoffs. As a justification that our construction is correct
we show that the Schwinger functions that we build obey the correct Slavnov
identities (with corrections localized at the boundary of our finite volume,
since this finite volume cutoff breaks gauge invariance).
We cannot lift the infrared volume cutoff,
since this would lead to large values of
the coupling constant, hence to non-perturbative effects
corresponding to confinement. These problems are for the moment still
out of the realm of our methods.
Concerning the axioms of quantum field theory, we cannot study the
complete set of Osterwalder-Schrader's axioms, mainly because we never
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.
Also we do not investigate invariance under large gauge transformations and
non-trivial topological effects such as instantons, which are much
more difficult to control rigorously from the point of view of
mathematical physics.
\medskip
\noindent {\bf B) The model. Stabilizing ultraviolet cutoffs.}
\medskip
We consider the pure Euclidean Yang-Mills theory with an infrared cutoff, which
we never try to lift. This cutoff may be imposed on the propagator,
but we prefer to consider
the theory on a finite hypercube $\Lambda $
with some boundary conditions, which might be periodic, in which case
we recover the torus.
Such a naive infrared regularization
breaks gauge invariance in an explicit way; it creates for instance
terms attached to the boundary of $\Lambda $ in Slavnov identities,
and we shall not try to cancel these terms.
Our construction is limited for simplicity to the pure
SU(2) Yang-Mills theory, but in principle we think that with some
additional work it should apply to any asymptotically free non-Abelian
gauge theories (with matter fields).
For the pure SU(2) Yang-Mills theory the vector potential is a field
$A_{\mu}^a, \mu =1,...,4, a=1, 2, 3$
with Lorentz (greek) indices and Lie algebra (latin) indices.
Our conventions are those of [IZ].
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 $D_{\mu} = \partial_{\mu} - g [ A_{\mu} ,.]$.
We have ${\rm Tr}\, t_at_b=- {\delta_{ab} \over 2} $.
The field curvature is:
$$ F_{\mu\nu} = (\partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}) - g
[ A_{\mu} , A_{\nu} ] = (\partial \wedge A - g
[ A , A ]) \eqno({\rm B.1})
$$
$g $ being the coupling constant.
The pure Yang-Mills action is (for Euclidean canonical metric
$F_{\mu \nu }=F^{\mu \nu }$):
$$ - {1 \over 2} \int_{\Lambda} d^4x Tr F_{\mu\nu}F^{\mu\nu} =
{1 \over 4} \int_{\Lambda} d^4x \sum_a F_{\mu\nu}^aF^{\mu\nu a}
\eqno({\rm B.2})
$$
To simplify, we define
a scalar product $$
by the convention that a trace is taken
over all correspondent space time indices and
minus a trace over group indices,
so that it is positive definite with a factor 1/2 in component notation.
We also write simply $A^2$ for $$, 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 $F^2$, introducing a coupling constant $g$:
$$ F^2 = F_2 + g F_3 + {g }^2 F_4 \eqno({\rm B.3})
$$
This action is invariant under the gauge transformations:
$$ A \to A^{\bf g} \ ; \ (A^{\bf g})_{\mu } = {\bf g} A_{\mu } {\bf g}^{-1} +
g^{-1}\partial _{\mu }{\bf g}
\cdot {\bf g}^{-1} \eqno({\rm B.4})
$$
The linearization of (B.4) is:
$$ A \to A^{\ga } \ ; \ (A^{\ga})_{\mu } = A_{\mu } +
D_{\mu }\ga \eqno({\rm B.5})
$$
where $D=\partial -g [A,.]$ is the covariant derivative
and $\ga$ takes values in the Lie algebra.
As is usual in constructive theory we try to define the limit of the
Schwinger functions with cutoff. They are
the moments of a regularized functional measure. This regularized measure
has to be well-defined and to coincide in some formal
way if the cutoffs are removed with the formal measure of the
Yang-Mills theory:
$$
e^{-(1/2)\int_{\La} F^{2}} \prod_{x\in \La, a , \mu } D A^{a}_{\mu}(x)
\eqno({\rm B.6})
$$
where the infinite product of Lebesgue measures in (B.6) is ill defined.
Due to the invariance (B.4) there is a big number of
flat directions in this formal measure.
If we want to define only gauge invariant observables, like in
Balaban's program one can hope
that they will divide out in the normalization. But we want to build
Schwinger functions which are not gauge invariant, hence we have to
fix the gauge. The nicest choice would certainly be a gauge directly
suited for perturbation theory, in which (B.6) becomes
$$
e^{-(1/2)\bigl(\int^{\La} F^{2} - \la(\partial_{\mu}A_{\mu})^{2}
\bigr) + \partial_{\mu} \bar \et \partial_{\mu} \et
+ g \partial_{\mu} \bar \et [A_{\mu}, \et ] \bigr)}
\prod_{x\in \La, a}
d\et ^{a}(x) d\bar \et^{a}(x)
\prod_{x\in \La, a , \mu } d A^{a}_{\mu}(x)
\eqno({\rm B.7})
$$
where $\bar \et$ and $\et$ are the usual Fadeev-Popov
ghosts.
For $\la=1$ we have the Feynman gauge, for $\la= +\infty$ the Landau
gauge; in the case of $SU(2)$ a particularly convenient choice is
to fit $\la$ to a value close to 3/13 at which the wave function
renormalization cancels exactly; we call this gauge the homothetic gauge.
The first task is to build a well defined analogue of (B.7), by
introducing some ultraviolet cutoff.
The lattice regularization is famous and has many advantages; it is
easy to show that the functional integrals
which take place in the compact SU(2) group are well defined.
Also high temperature expansions show
interesting phenomena such as Wilson's area law etc...
Since the group variable is $e^{g^{-1}A}$, we can consider that the
lattice regularization imposes both an ultraviolet cutoff and an
effective limit of order $O(g^{-1})$ on the size of $A$. In this sense
this cutoff is stabilizing at large field. As explained below, a
stabilizing cutoff is necessary but not sufficient. One has to
understand how the stability is preserved by the multiscale analysis. For
instance in the lattice case one has to use
block spin transformations which preserve the fact that the rescaled
variables at lower momentum remain on a compact group and in Balaban's
works, this is the source of many technical complications.
In order to enlarge the possible cutoffs and also to
clarify the last point, we have searched for other stabilizing
cutoffs in which the stability comes from some explicit
operator which is renormalized in the ordinary way in the multiscale
analysis. Since Gaussian measures for positive quadratic forms are well
defined (e.g. through Minlos's theorem) the standard method in
constructive theory is to absorb the propagators of the fields in such
measures. On these propagators there are many ways
to implement momentum ultraviolet cutoffs (such as Pauli-Villars
etc...). Then one has to show that the remaining interaction is stable
for large fields. Here this is already a difficult task, because the
$F_{3}+F_{4}$ term in (B.3)
is not positive by itself when separated from $F_{2}$. But it is well
known that momentum cutoffs break gauge invariance. We make from this
defect a virtue and remark that this gauge breaking can be cancelled
by the insertion of appropriate gauge variant counterterms in (B.7).
It is only necessary to insert the counterterms corresponding to
relevant and marginal operators, since the others have no effects on
the theory at finite scale when the cutoff is removed.
Using global Euclidean and gauge invariance the important counterterms
to consider in the case of SU(2) reduce to $A^{2}$ and $A^{4}$.
Furthermore it is enough to compute the $A^{4}$ counterterm at one
loop (further contributions being
by asymptotic freedom ``logarithmically irrelevant''). We performed
this task in collaboration with J. Feldman for
ultraviolet cutoffs of compact support in momentum space
and found that for
many shapes of the support function (for instance the shape
show in Fig.1 with $K$ sufficiently large) the $A^{4}$
counterterm has the right sign so that the theory is stabilized at
large field. The detailed analysis can be found in [S1], [R] or [MRS].
It means that the corresponding functional integral with cutoff and
gauge restoring counterterms
is well defined (recall that $F_{4}$ is positive, that $F_{3}$
although not positive is only cubic, and that functional integrals for fermions
with cutoffs can be explicitly defined through their perturbative
expansion, see e.g. [FMRS1][R]).
Since the $A^{4} $ counterterm is at one loop in $g^{4} A^{4}$ it provides
obviously some cutoff on $A$ at size $g^{-1}$, hence it is in this
respect very similar to the lattice cutoff.
\vskip 6cm
\centerline {Fig.1: The ultraviolet cutoff in momentum space}
\medskip
Using such a cutoff at a scale $M^{\rh}$ where $M$ is some integer
we have a reasonable bare ansatz for the theory:
$$
e^{-(1/2)\bigl(\int^{\La} F_{3}+F_{4} + CT(A)
\bigr) + \partial_{\mu} \bar \et \partial_{\mu} \et
+ g \partial_{\mu} \bar \et [A_{\mu}, \et ] \bigr)}
d\mu_{\rh}(A, \bar \et \et)
\eqno({\rm B.8})
$$
where $d\mu$ is a certain Gaussian measure which incorporates the
propagators of the gauge fixed
theory with cutoffs, and $CT$ are the gauge restoring
counterterms which at large $A$ behave as $+c. g^{4}A^{4}$. In (B.8)
we can take the bare coupling as
$$ g _{\rho} = {1 \over \beta_2 (LogM) \rho + \beta_3 / \beta_2
\log \rho + C} \eqno({\rm B.9})
$$
where C is a large constant,
and $\beta_2$ and $\beta_3$ are the usual first non vanishing coefficients
of the $\beta$ function. Then one expects to land
on a renormalized coupling constant
$g _{ren}$ (the last one in a sequence of
effective constants) finite and small if $C$ is large.
\medskip
\noindent {\bf C) The Gribov problem}
\medskip
Once a stabilizing cutoff has been introduced
the bare functional integral is at least well defined.
Unfortunately we have not been able
to remove thew cutoff for an ansatz such as (B.8), namely a bare theory
with a stabilizing cutoff in a perturbative gauge such as the Feynman
or homothetic gauge.
The problem we met under the form of a lack
of positivity is related to the Gribov phenomenon.
To perform the ultraviolet limit in (B.8) we have to perform
on (B.9) a multiscale analysis in the style of [R]
and it is crucial that the
renormalization group flow of $g$ is dominated by the regular
perturbative terms.
For this we need to prove that the interaction terms in (B.9)
at least in a given momentum slice are small when compared
to the Gaussian measure. Equivalently we need
a non perturbative bound in the regions $A \simeq g ^{-1}$
which tells us that the functional weight of these regions is small
compared to the (Gaussian) weight of the region $A \simeq 0$.
Although the $A^4$ term shows that the corresponding functional weight
is bounded, we cannot prove solely with this term that this weight is small.
This problem is related to the Gribov phenomenon.
Gribov discovered [G] that in the Landau gauge there can be different smooth
field configurations which are nevertheless related by a gauge transformation.
A configuration $A_2 \ne A_1$ such that $\partial_{\mu} A_{\mu 1}=
\partial_{\mu} A_{\mu 2 } = 0$ and such that there exists a gauge
transformation $g$ with $A_1=A_2^{{\bf g}}$
($A^{{\bf g}}$ being defined as in (B.4))
is called a Gribov copy of $A_1$.
There are always some configurations
which have copies; this is true even on a compact space and for configurations
with the
same topological properties, hence inside a given topological sector;
it is also
true for any regular gauge [S2], not just the Landau gauge. What we call the
strong Gribov phenomenon
is when there are Gribov copies of the 0 configuration, hence pure gauges
which satisfy the gauge condition. This is possible in an infinite
space when a change of topological sector or weak
decay at infinity is allowed, as shown explicitly
in [G], but this
strong Gribov phenomenon does not occur inside a given topological sector
in a compact space, or under strong decay conditions at infinity if the
space is not compact (see e.g. [R]).
Explicit proofs of absence of the
strong Gribov phenomenon for finite volume show that there is some definite
positivity which lies in the combination of the action $F^2$ and the Feynman
gauge term $(\partial _{\mu } A_{\mu })^{2} $. Unfortunately this positivity
is too weak when the frequencies
considered are much larger than the inverse size of the volume cutoff. In
intuitive terms, this positivity is
tied to boundary conditions: it is useful for the last (physical) momentum
slices in a phase space analysis but is not strong enough to
help at high momenta.
Although in the constructive study of the ultraviolet limit of non-Abelian
gauge theories we can avoid the strong Gribov phenomenon just as we avoid the
infrared problem (i.e. by
appropriate boundary conditions or compactification), we cannot avoid
the existence of copies, in particular in the vicinity of null vectors of the
Fadeev-Popov operator
$$K= -\partial _{\mu }D_{\mu }= -\Delta + g \partial _{\mu }
[A_{\mu },.] \eqno({\rm C.1})
$$
For $g =0$ the Fadeev Popov operator reduces to minus the Laplacian and
is positive definite once the 0-momentum mode (hence translation invariance)
has been deleted. But for non zero $g $ and $A$ it is possible to show
that rescaling $A \to k \cdot A$ one can always have negative
eigenvalues of $K$
for $k$ large enough (which correspond physically to bound states of the ghost
system).
The configurations where $\det K =0$, hence where there exists null
vectors for $K$ are the so-called Gribov
horizons. The regions inside the first Gribov horizon, where $K$ is
positive is called the first Gribov region and so on. In [G] it is shown
that near a Gribov horizon there are typically Gribov copies, one on each side
of the horizon. These copies can be rapidly decreasing at infinity (or smooth
on a compact space) so this ``weak Gribov phenomenon"
problem cannot be eliminated
by an infrared cutoff or by topological restrictions.
The existence of copies
means that the functional measure is not monotonous.
The fact that the Fadeev-Popov
operator is not always positive definite at large fields is also quite
disturbing. Since the determinant of this operator occurs in the functional
measure, we must conclude that the ordinary formula for functional
integration is a signed measure.
It is argued in [H]
that this signed measure, although derived in an incorrect way, is nevertheless
the correct one; essentially the argument is that as we move away from the
gauge orbit of the origin (which, by absence of the strong Gribov phenomenon in
our case, cuts the gauge condition only once) the Gribov copies,
by some continuity argument, should occur in pairs
with equal and opposite values, therefore cancel out
so that the naive prescription with the
signed determinant is equivalent to the correct prescription in which
a single point on each gauge orbit is selected. In any case even if
this conjecture is true it seems difficult to prove it in constructive theory.
An other possibility is to eliminate Gribov copies
at the beginning by use of a better ansatz.
This is the approach
advocated e.g. by Zwanziger [Z],
in which one tries to
define first a correct configuration
space for the functional measure, eliminating Gribov copies by
additional gauge conditions. It seems very reasonable to
consider that the functional integration for the non-Abelian gauge theory
can be written as a positive measure supported on a subdomain
of the first ``Gribov region''. However to characterize this region in
a constructive way seems hard. Therefore we have turned to the
axial gauge which contains some definite positivity and lacks
the Gribov effects.
\medskip
\noindent {\bf D) The regularized axial gauge}
\medskip
The axial gauge is defined by the condition $A_{0}=0$. This gauge
condition (which is not a complete gauge fixing)
can be implemented without paying
the price of a Fadeev-Popov determinant
(more precisely this determinant is a constant
which disappears in the normalization).
In this gauge the action is
$$
e^{- (1/2) ( + F_{sp}^{2} )}
\eqno({\rm D.1})
$$
where $ F_{sp}$ is the sum over spatial indices (excluding the time
component).
It is easy to verify that when coupled to a stabilizing cutoff which
states that $A$ is of order $g^{-1}$ at most, the positivity in
(D.1) implies that $A$ is of order $g^{-1/2}$ in probability. Indeed
if we use $g^{2}< A ,p^{2} A>$ as stabilizing term (it is marginal,
like $A^{4}$, and weaker at large $A$ since it is only quadratic
instead of quartic) we can join it to $ e^{- (1/2) }$
and obtain a covariance of order $g^{-1} $.
For $A \simeq g^{-1/2}$
we are well within the first Gribov region, and $F_{2}$ is much larger
than $F_{3}$ and $F_{4}$.
This means that both perturbation theory and an explicit
change of gauge become possible. However the use of the positivity contained
in the $$ term has again a technical price attached
to it: we must perform the phase space analysis of the theory
in an anisotropic way, with momentum slices which distinguish both on
the value of $p^{2}$ and $p_{0}^{2}$. The corresponding dual cells in
$x$-space are rectangular, elongated in the time direction, and this
is the source of many complications.
\medskip
\noindent {\bf E) The expansion and the normalization of large field regions}
\medskip
A phase space expansion consists in both spatial cluster expansions
and momentum decoupling expansions. The only important restriction to build
these interpolations is to preserve positivity requirements for the
interpolated theory.
In addition to perturbative computations, constructive theory needs
some stability properties of the functional integral at large
fields. When a multiscale analysis is necessary, which is certainly
the case in a renormalizable theory, we need to decouple functional
integrals at different scales and this results in a typical problem
called domination. The low momentum fields produced by the expansion,
if evaluated with the Gaussian measure, would lead to the divergence
of perturbation theory and we must instead bound them by using the
positivity of the effective potential. In our case we cannot use the
perturbative small field effective potential when the low momentum
fields are large. In other
words there are some couplings which are not dominable. Fortunately
the structure of these non dominable terms is simple. They can be
absorbed simply by changing ordinary derivatives of the field
into covariant derivatives corresponding to the background field which
is the sum of all large field of lower momenta. This rule is
a deep consequence of the geometric structure of the theory.
Applying this idea we have to replace derivatives by covariant
derivatives $D$ even in the gauge fixing terms. This reacts on the Fadeev-Popov
determinant. Also the multiscale analysis as to be performed
really around the zeroes of the operator $D^{2}$ rather than the
ordinary Laplacian.
Furthermore we want
the ultraviolet cutoff to be stabilizing in the sense above. The
counterterms being computed by a one loop analysis which is easily
performed for a perturbative gauge like (B.7), we choose to use a
simple form of the cutoff after gauge transformation to the
perturbative gauge. This means that the true cutoff in the axial
gauge is given by an inverse formula applied to a simple
momentum cutoff.
These two effects unfortunately complicate very much
the starting ansatz for our axial gauge theory, and we refer to [MRS]
for the corresponding formulas.
Then the multiscale analysis of the theory proceeds in the usual way except
for the following complications: there is a division of the
functional integral cells corresponding to the phase space analysis
into large field cells and small field cells. The large field cells
$A >> g^{-1/2}$ are suppressed in probability by the positivity of the
axial gauge.
In the small field cells
we perform an explicit change of gauge by a well-defined
analogue of the Fadeev-Popov formula with cutoff, and we obtain a theory with
a perturbative gauge analogous to the homothetic gauge ($\la\simeq 3/13$
in (B.7)), but with covariant derivatives in the background low-momentum
large field instead of ordinary derivatives. The propagator is therefore
not translation invariant, and to perform the usual cluster expansion
analysis and renormalization in each momentum slice requires an
auxiliary expansion, in which one compares
the propagator with a slowly varying background
field to the propagator in a constant background field.
But we have also a last problem, typical of a large
versus small field expansion. To be negligible in the correct sense
the functional weight of the large
field cells has to be not only small, but so small that it
compensates for the difference in the normalization of these cells.
The small field cells have indeed a normalization corresponding to
their background dependent Gaussian measure, whereas the large field
cells have the normalization of the initial axial gauge.
In fact for every
large field cell there is an associated
domain of small field cells for which this large field $B$ is a part of
the background. The difference in normalization of this small
field domain with and without the
background field $B$ gives rise to a determinant which can
and in fact must be associated
to the large field cell (more precisely it gives
a quotient of determinants
if the Fadeev-Popov terms is also taken into account). The
crucial point to check is that the large field condition together with
the positivity of the axial gauge gives a factor so small that it
compensates for the (non-perturbative)
bound that we have on this normalizing ratio
of determinants.
The fact that this ratio of determinants can be explicitly bounded is related
to the fact that its leading behavior when the small field domain
contains many scales is again given by the relevant term in
$B^{2}$, whose coefficient in the determinant can be explicitly
computed; we found that in the case of a stabilizing ultraviolet
cutoff the corresponding bound can
be controlled.
In conclusion the corresponding expansion converges if the
renormalized
coupling at the infrared scale remains small (e.g. when $C$ in (B.9) is big
enough), and the Schwinger functions in the limit satisfy the
(non-perturbative) Slavnov
identities of the axial gauge, with a right hand side depending on the
boundary conditions on the volume cutoff, which represents the
infrared gauge breaking effects. The gauge transformed functions
corresponding to the small field regions
hence to the perturbative gauge also satisfy the corresponding
Slavnov identities at every order in perturbation theory, but not
non-perturbatively, since the large field region give
exponentially small correction terms.
\vskip .5cm
\noindent {\bf Acknowledgements} We thank warmly J. Feldman for his
collaboration at an early stage of this work.
\vskip 1cm
\noindent {\bf REFERENCES}
\vskip 1.2cm
\item{[B]} {T. Balaban, Comm. Math. Phys. 95, 17 (1984),
96, 223 (1984), 98, 17 (1985), 99, 75 (1985), 99, 389 (1985),
102, 277 (1985), 109, 249 (1987), 116, 1 (1988), 119, 243 (1988),
122, 175 (1989), 122, 355 (1989).}
\item{[F]} {P. Federbush, A phase cell approach to Yang-Mills theory
I-VII, Comm. Math. Phys 107, 319 (1986), 110, 293 (1987), 114, 317 (1988);
Ann Inst. Henri Poincar\'e, 47, 17 (1987).}
\item{[FMRS1]} {J. Feldman, J. Magnen, V. Rivasseau and R. S\'en\'eor,
A renormalizable field theory: the massive Gross-Neveu
model in two dimensions, Comm. Math. Phys. 103, 67 (1986).}
\item{[G]} {V. Gribov, Quantization of non-Abelian gauge theories, Nucl.
Phys. B139, 1 (1978).}
\item{[GK]} {K. Gawedzki and A. Kupiainen, Gross-Neveu model through
convergent perturbation expansions, Comm. Math. Phys. 102, 1 (1985).}
\item{[H]} {P. Hirschfeld, Strong evidence that Gribov copying
does not affect
the gauge theory functional integral, Nucl. Phys. B157, 37 (1979).}
\item{[IZ]} {C. Itzykson and J.B. Zuber, Quantum field theory, Mc Graw
and Hill, New-York, 1980.}
\item{[MRS1]} {J. Magnen, V. Rivasseau and R. S\'en\'eor,
Construction of $YM_{4}$ with an infrared cutoff,
preprint Ecole Polytechnique, 1992.}
\item{[R]} {V. Rivasseau, From perturbative to constructive renormalization,
Princeton University Press, 1991.}
\item{[S1]} {R. S\'en\'eor, in Renormalization of Quantum Field Theories
with Non-Linear Field Transformations, P. Breitenlohner, D.
Maison and K. Sibold Eds, Lecture Notes in Physics n$^{o}$ 303, Springer
Verlag, 1988.}
\item{[S2]} {I. Singer, Some remarks on the Gribov Ambiguity, Comm. Math.
Phys. 60, 7 (1978).}
\item{[Z]} {D. Zwanziger, Non-perturbative modification of the Fadeev-Popov
formula and banishment of the naive vacuum, Nucl. Phys. B209, 336
(1982), Action from the Gribov horizon,
Nucl. Phys. B321, 591 (1989), Local and renormalizable action from the Gribov
horizon, Nucl. Phys. B323, 513 (1989).}
\vfill\eject