_-} \over ip_0-e(\p)} \eqno({\rm I.1}) $$where $$

_-=\p\cdot\x-p_0\tau \eqno({\rm I.2}) $$ and $$ e(\p)={\p^2\over2m}-\mu. \eqno({\rm I.3}) $$ The variable $\xi=(\tau,\x,\sigma)$ consists of time, space and spin components and the $(d+1)$-momentum $p=(p_0,\p)$. The interaction is given by $$ {\cal V}={\la\over 2} \int \prod_{i=1}^4 d\xi_i\ \ V\!\left(\xi_1,\xi_2,\xi_3,\xi_4\right) \bar\psi(\xi_1)\bar\psi(\xi_2)\psi(\xi_4)\psi(\xi_3)\eqno({\rm I.4}) $$ where $\la$ is the coupling constant, the kernel $V\!\left(\xi_1,\xi_2,\xi_3,\xi_4\right)$ is translation invariant with $V\!\left(0,\xi_2,\xi_3,\xi_4\right)$ integrable and $$ \int d\xi=\sum_{\sigma\in\{\uparrow,\downarrow\}} \int_\bbbr d\tau\int_{\bbbr^d} d\x.\eqno({\rm I.5}) $$ The partition function is $$ Z=\int e^{-{\cal V}(\psi,\bar\psi)}d\mu_{C}(\psi,\bar\psi) \eqno({\rm I.6})$$ so that ${\cal G}(0,0)=0.$ The fact that $\cal V$ is non-local (although short range) is the source of a few inessential complications. Therefore in this paper we restrict ourselves to the case of a $\de$ function to focus the reader's attention on essential aspects. Therefore from now on, we will consider $$ {\cal V}_{\La}={\la\over 2} \int_{\La} d\x d\ta \ \bar\psi_{\uparrow}(\x, \ta)\bar\psi_{\downarrow} (\x, \ta)\psi_{\uparrow}(\x, \ta)\psi_{\downarrow}(\x, \ta) \eqno({I.4.b}) $$ The Euclidean Green's functions with $2p$ points $$ G_p(\xi_1,\bar\xi_1,\dots,\xi_p,\bar\xi_p)=\prod_{i=1}^p {\delta^2\hfill\over\delta\psi^e(\xi_i)\delta\bar\psi^e(\bar\xi_i)}{\cal G} \eqno({\rm I.7})$$ generated by the effective potential are the connected Green's functions amputated by the free propagator. By definition, ${\cal G}$ exists when the norm $$ \norm G_p\norm=\max_{j}\ \sup_{\xi_j}\ \int\prod_{i\ne j}d\xi_i\ |G_p(\xi_1,\cdots,\xi_{2p})| \eqno({\rm I.8}) $$ of each of its moments, $G_p,\ p\ge 1$, is finite. Intuitively, $\norm G_p\norm$ is the supremum in momentum space of $G_p$. In fact, the supremum in momentum space was used as the standard norm on vertices in [FT2]. Our goal is to give a rigorous proof that the standard model for a weakly interacting system of electrons and phonons has a superconducting ground state at sufficiently low temperature. Perturbation theory and, in particular, the renormalization of the two point function was controlled in [FT1]. A renormalization group flow for the four point function was defined and analyzed in [FT2]. Two additional ingredients are required to complete this program. First we need an infinite volume expansion that controls non-perturbatively this renormalization group flow. This first ingredient, ``constructive stability'' was provided in two space dimensions (d=2) in [FMRT1]. The other physically interesting case, namely $d=3$, is much harder. In this paper we prove by a completely different method that one of the key results of [FMRT1] extends to $d=3$: the radius of convergence of the single scale theory is uniformly bounded below. However up to now, we have not been able to extend this result to the multiscale theory. Therefore ``constructive stability'' in $d=3$ remains an open problem. The second ingredient is the control of the reduced BCS model for Cooper pairs via another expansion (of the $1/N$ type). One has to prove existence of the BCS gap, of the associated spontaneous $U(1)$ symmetry breaking, and of the masslessness of the associated Goldstone boson. This program is well under way. For references on this part of the program and on the general strategy see [FMRT3-5]. In the remainder of this paper we specialize to $d=3$. Our method extends however without difficulty to any $d\ge 2$ (see Appendix I). As in [FT1,2], the model is sliced into energy regimes by decomposing momentum space into shells around the Fermi surface. The $j^{\rm th}$ slice has covariance $$ C^{(j)}(\xi,\bar\xi)=\delta_{\sigma,\bar\sigma}\int {d^{4}p\over(2\pi)^{4}} {e^{i

_-}
\over ip_0-e(\p)}f_j(p),\eqno({\rm I.9})
$$ where $$ f_j(p)=f\left(M^{-2j}\left(p_0^2+e(\p)^2\right)\right)
\eqno({\rm I.10})$$
effectively forces $|ip_0-e(\p)|\sim M^j$. The function $f\in
C^\infty_0([1,M^{4}])$. The parameter $M$ is strictly bigger than one so
that the scales near the Fermi surface have $j$ near $-\infty$. The
model is defined in finite volume and at fixed scale by the following
lemma:
\medskip
\noindent{\bf Lemma I.1}{\it \ \
$$ {\cal G}_\Lambda^{(j)}(\psi^e,\bar\psi^e) =\log\,{1\over
Z_\Lambda^{(j)}}\int e^{-{\cal
V}_\Lambda(\psi+\psi^e,\bar\psi+\bar\psi^e)}
d\mu_{C^{(j)}}(\psi,\bar\psi)
\eqno({\rm I.11})
$$ where $$ {\cal V}_\Lambda={\lambda\over 2}
\int_{\Lambda^4} \prod_{i=1}^4 d\xi_i\ \
V\!\left(\xi_1,\xi_2,\xi_3,\xi_4\right)
\bar\psi(\xi_1)\bar\psi(\xi_2)\psi(\xi_4)\psi(\xi_3)
\eqno({\rm I.12})$$
and
$$ Z_\Lambda^{(j)}=\int e^{-{\cal V}_\Lambda(\psi,\bar\psi)}
d\mu_{C^{(j)}}(\psi,\bar\psi)
\eqno({\rm I.13})$$
is
analytic in $\lambda$ in a neighborhood of the origin that includes
the disk of radius $\const\left(M^{2j}|\Lambda|\right)^{-1}.$}
\medskip
The full proof (not very difficult) is given in [FMRT1, Lemma 1].
Remark however
that the radius of convergence depends on volume and scale in a
patently unsatisfactory way.
Recall that $G^{(j,\Lambda)}_p$
is the $2p$-point Green's function generated by
${\cal G}^{(j)}_\Lambda$. By the Lemma above the Taylor series
$$
G^{(j,\Lambda)}_p=\sum_{n=0}^\infty g_n(p,j,\Lambda)\lambda^n
\eqno({\rm I.14})$$
has a
strictly positive, though possibly $j$ and $\Lambda$ dependent, radius
of convergence. The main result of this paper is
\medskip
\noindent{\bf Theorem I}{\it \ \
There exists a $\ \const $, independent of $j$ and
$\Lambda$, such that $$
\norm g_n(p,j,\Lambda)\norm \le\const^{n+p}
M^{(4-2p)j}\norm V\norm^n
\eqno({\rm I.15})$$
where
$$
\norm g_n(p,j,\Lambda)\norm=
\max_{k}\ \sup_{\xi_k}\ \int\prod_{i\ne k}d\xi_i\
|g_n(p,j,\Lambda)(\xi_1,\cdots,\xi_{2p})|.
\eqno({\rm I.16})$$ Furthermore the limits
$$
g_n(p,j)=\lim_{\Lambda\rightarrow \bbbr^3}g_n(p,j,\Lambda)
\eqno({\rm I.17})$$
exist and the infinite volume Green's functions at scale $j$
$$
G^{(j)}_p=\sum_{n=0}^\infty g_n(p,j)\lambda^n
\eqno({\rm I.18})$$
are analytic in
$|\lambda| _-}
\over ip_0-e(\p)}f_j(p)\,\eta_m(\p)
$$
and of the fields
$$
\psi^{j}=\sum_{m=1}^{M^{-(d-1)j}}\psi^{(j,m)},\hskip.25truein
\bar\psi^{j}=\sum_{m=1}^{M^{-(d-1)j}}\bar\psi^{(j,m)}.
$$
The standard power counting bound for an individual graph is still
easy when there are sectors. First, one selects a spanning
tree for the graph. To each line not in the tree there is a
corresponding momentum loop, obtained by joining its ends through a
path in the tree. This construction produces a complete set of
independent loops. Ignoring unimportant constants, each propagator is
bounded by its supremum $M^{-j}$. The volume of integration for each
loop is now $M^{(d+1)j}$. A priori, there is one sector sum with
$M^{-(d-1)j}$ terms for each line. But, by conservation of momentum,
there is only one sector sum per loop. Thus, if there are $n$ vertices
and $e=2p$ external lines, the supremum in momentum space of the graph is
bounded by $$\eqalign{
\prod_{\rm lines}M^{-j}\prod_{\rm loops}M^{(d+1)j}M^{-(d-1)j}
&=M^{-j(4n-2p)/2}M^{2j\left[(4n-2p)/2-(n-1)\right]}\cr &=M^{{1\over
2}(4-2p)j}.
}$$
In the course of a non-perturbative construction, estimates cannot be
made graph by graph because there are too many of them. Rather, the
perturbation series must be blocked and the blocks estimated as units.
The blocks are estimated using the exclusion principle to implement
strong cancellations between the roughly $n!^2$ graphs of order $n$.
However, once the series is blocked, momentum loops can't be defined
and the argument leading to the estimate above cannot be made.
Conservation of momentum has to be implemented at each vertex, rather
than through loops. Even though the volume cutoff $\Lambda$ breaks
exact conservation of momentum, many of the $M^{-2\ell(d-1)j}$ terms
in the sector sums for a general $2\ell$-legged vertex must be zero.
In [FMRT1] the following lemma is proved:
\noindent{\bf Lemma 2}
{\it Fix $\ m\in {\bf Z}^{d+1}\ $ and $\ \ell\ \ge\ 2\ $.
Then, the number of $2\ell$-tuples
$$
\left\{S_1,\ \cdots\,S_{2\ell}\right\}
$$
of sectors for which there exist $\ \k_i\in \bbbr^d,\ i=1,\ \cdots,\
2\ell\ $ satisfying
$$
\k_i^\prime\in S_i,\ \ |\k_i-\k_i^\prime|\le {\rm const}\ M^j
\ , \ \ \ i=1,\ \cdots,\ 2\ell\
$$
and
$$
|\k_1+\ \cdots\ +\k_{2\ell}|\le{\rm const}\left(1+|m|\right)M^j
$$
is bounded by
$$
\const^\ell(1+|m|)^dM^{-(2\ell-1)(d-1)j}M^j
\left\{1+|j|\delta_{d,2}\delta_{\ell,2}\right\}.
$$
In particular, for a four legged vertex, the number of $4$-tuples is
at most
$$
\const(1+|n|)^dM^{(-3d+4)j}\left\{1+|j|\delta_{d,2}\right\}.
$$
Here, $\ \k^\prime = {\k\over |\k|}\ $ denotes the projection of $\
\k\ $ onto the Fermi surface.}
\vskip.25truein
This lemma although instructive, is neither for $d=2$ nor for $d>2$
powerful enough for the non-perturbative construction. However for
$d=2$ the number of active sector 4-tuples at a vertex is of
order $\ |j| M^{-2j}\ $; ordinary power counting at a vertex
can accomodate at most $M^{-2j}$, hence there is only a ``logarithmic''
power counting deficit of order $|j|$ at each vertex. For $d=3$
the number of active sector 4-tuples at a vertex is oforder $\ M^{-5j}\ $,
and ordinary power counting at a vertex
can accomodate only $\ M^{-4j}\ $ so that the deficit
at each vertex is a full power $M^{-j}$. This deficit is worse and
worse as $d$ increases beyond 3. In three dimension
this difficulty can be geometrically thought
as the twist or torsion which occurs in the figure
made by four momenta of unit length adding up to 0 (this figure is no longer
a planar rhombus as in two dimensions).
To circumvent the extra logarithm at $d=2$, in [FMRT1] we divided the Fermi
surface into sectors of length $M^{j/2}$ rather than $M^j$. Taking
into account the anisotropic spatial decay associated to these new
``rectangular'' sectors beats the logarithmic deficit. If we apply the
same idea for $d=3$, the number
of active 4-tuples sectors at a vertex becomes of order $\ M^{-5j/2}\ $
but the power counting in the manner of [FMRT1] gives $\ M^{-2j}\ $,
so that there still remains a deficit of order $M^{-j/2}$ per vertex.
Making the sectors still longer does not work because the anisotropic
decay no longer holds since the Fermi surface does not look flat for
lengths bigger than $M^{-j/2}$.
Remark that this difficulty
is presumably linked to the graph $G_{2}$, which as we recall is in
$M^{-j} \la^{2}$ in a cube of size $M^{-j}$.
The conclusion is that even the single slice theory in $d \ge 3$
is a non-trivial theory that contains a rather non-trivial
renormalization, like $\ph^{4}_{3}$. It is this renormalization
that can be analyzed by means of the auxiliary
scales used in this paper.
\medskip
\noindent{\bf Appendix III}
\medskip
In this appendix we discuss briefly the present situation for the
full (multi-scale) theory.
The most straightforward
strategy in the case of multi-scale Fermionic models
consists in writing the effect of inductive integration
of higher scales in the form of an effective action for the single scale
model, and to check some inductive estimate for this effective action.
This is the road followed in [FMRT1]. However when
combined with the Hadamard estimate
on determinants, this approach runs
into a major difficulty: even with a quartic bare action
the effective action contains irrelevant
operators of degree 6 and more in the fermionic fields.
The Hadamard inequality when applied to such operators does not
lead to a convergent answer (it develops divergent factorials).
The obvious thing to do seems to treat the theory as a bosonic one.
For bosonic theories we know well
that one cannot simply exponentiate all the effective
action, as in Wilson's original program, but
that one must keep the irrelevant effects coming
from higher scales in the form of a polymer gas with hardcore constraints
[B][R]. A typical stability estimate in this strategy is to prove that
the ``completely convergent'' multi-scale polymers
(i.e. those who do not contain any two or four point subcontributions)
form a geometric convergent series. At the moment our best estimates
with the Hadamard method indicate that the sum over such
``completely convergent'' polymer, extending over
at most $N$ different scales, converges
provided the (bare) coupling constant $\la$ is small, $and$
the number of scales is limited to $N \le \ep / \la$
($\ep$ being a small number). This
is encouraging, but tantalizing. Indeed although this last estimate is
much better than what a sector analysis would provide
(namely $N \le K|\log \la|$), it is still not sufficient.
For the construction of the BCS3 theory,
where the BCS gap cuts the renormalization group
flow at a scale $\De_{BCS} \simeq M^{-N} \simeq e^{-O(1)/\la}$, we would need
$N \simeq K /\la$ (where $K$ is not particularly small). For
the construction of
non-superconducting Fermi liquids as in [FKLT],
the renormalization group flows for ever and we would
need $N$ independent of $\la$. Therefore
at the moment we consider that the question of the non-perturbative
stability of these theories remains open, but we hope that the Hadamard
method, clearly much better than the sector method, is a step
towards the final solution.
\vskip 1cm
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%taken by R. Fernandez, Universit\'e
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\end