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many-electron system, renormalization, Feynman diagrams, Anderson model
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\begin{document}
\include{epsf}
{
\thispagestyle{empty}
\baselineskip=16pt
$$ $$
\vskip 3cm
\centerline{\gross Resummation of Feynman Diagrams } 
\smallskip
\centerline{\gross and the Inversion of Matrices  }
\bigskip
\bigskip
\centerline{by}
\bigskip
\bigskip
\centerline{{\mittel Detlef Lehmann}\footnote{ e-mail: 
 lehmann@math.tu-berlin.de}}
\centerline{\mittel Technische Universit\"at Berlin}
\centerline{\mittel Fachbereich Mathematik Ma 7-2} 
\centerline{\mittel Sta{\ss}e des 17. Juni 136}
\centerline{\mittel D-10623 Berlin, GERMANY}
\vskip 4.5cm 
\noindent{\bf Abstract:} In many field theoretical models one has to
resum two- and four-legged subdia\-grams in order to determine their 
behaviour. In this article we present a novel formalism which does this 
in a nice way. It is based on the central limit theorem of probability 
 and an inversion formula for matrices which is obtained 
 by repeated application of the Feshbach projection method. We discuss 
applications to the Anderson model, to the many-electron system and 
 to the $\vp^4$-model. In particular, for the many-electron system 
 with attractive delta-interaction,   
we find that the expectation value of the Hubbard-Stratonovich field 
for small momentum $q$ has a delta-function singularity instead of 
 the commonly expected $1/q^2$-type singularity. 



\bigskip
\bigskip
%\centerline{\it (preliminary version)}
\vfill
\eject  }
%
%
% E I N L E I T U N G 
%
%
\pagenumbering{arabic}
\baselineskip=16pt

\section{Introduction}
\bigskip
\indent
The computation of correlation functions in field theoretical models is a difficult 
 problem. In this article we present a novel approach which applies to models 
where a two point function can be written as 
$$ S(x,y)=\int [P+Q]_{x,y}^{-1}\> d\mu(Q)\>. \eqno (1.1) $$
Here $P$ is some operator diagonal in momentum space, typically determined by the 
unperturbed Hamiltonian, and $Q$ is diagonal in coordinate space. 
The functional integral is taken with respect to some probability measure 
 $d\mu(Q)$ and goes over the matrix elements of $Q$. $[\;\cdot\;]_{x,y}^{-1}$ 
denotes the $x,y$-entry of the matrix $[P+Q]^{-1}$. Our starting point 
is always a model in finite volume and with positive lattice spacing in 
which case the operator $P+Q$ and the functional integral in (1.1) 
  becomes 
huge- but finite-dimensional. In the end we take the infinite volume limit 
 and, if wanted, the continuum limit. 
\par
Our treatment is based on the following identity which is obtained by
repeated application of the Feshbach formula (Lemma 3.2 below). 
 It  is proven in Theorem 3.3. 
 Let $B=(B_{kp})_{k,p\in\M}\in \mathbb C^{N\times N}$, $\M$ some 
 index set, $|\M|=N$ and let 
$$G(k):=\left[ B^{-1}\right]_{kk}  \eqno (1.2)$$
Then one has
$$G(k)={1\over\ds B_{kk}+\sum_{r=2}^N (-1)^{r+1}
   \!\!\!\!\!\!\!\!\!
    \sum_{p_2\cdots p_{r}\in\M\setminus\{k\}\atop p_i\ne p_j}
  \!\!\!\!\!\!\!  
  B_{kp_2}G_k(p_2)B_{p_2 p_3}\cdots B_{p_{r-1} p_r} 
  G_{kp_2\cdots p_{r-1}}(p_r)B_{p_r k}  }\eqno (1.3)$$
where 
 $ G_{k_1\cdots k_j}(p)=\left[ (B_{st})_{s,t\in \M\setminus 
    \{k_1\cdots k_j\}}\right]^{-1}_{pp}$ 
is the $p,p$ entry of the inverse of the matrix which is obtained
from $B$ by deleting the rows and columns labelled by  
  $k_1,\cdots,k_j$. In Section 2  we apply this formula to a matrix of 
 the form $B=$ self adjoint + $i\vep\,{I\!d}$, which, 
  for $\vep\ne 0$, has 
 the property that all submatrices $ (B_{st})_{s,t\in \M\setminus 
    \{k_1\cdots k_j\}}$ are invertible. 
\par
There is also a formula for the off-diagonal inverse matrix 
 elements. It reads 
$$\left[B^{-1}\right]_{kp}=-G(k) B_{kp} G_k(p)+
  \sum_{r=3}^N(-1)^{r+1} \!\!\!\!\!\!\!  
 \!\!\! \sum_{t_3\cdots t_r\in\M\setminus\{k,p\}\atop t_i\ne t_j}
  \!\!\!\!\!\! G(k) B_{kt_3} G_k(t_3) B_{t_3t_4} 
  \cdots B_{t_rp}G_{kt_3\cdots t_r}(p)  \eqno (1.4)$$
These formulae also hold in the case where the matrix $B$ has a 
block structure $B_{kp}=(B_{k\sigma,p\tau})$ where, say, $\sigma,\tau\in 
 \{\up,\down\}$ are some spin variables. 
   In that case the $B_{kp}$ are small matrices, 
the $G_{k_1\cdots k_j}(p)$ are matrices of the same size 
 and the 1/$\cdot$ in (1.3) 
means inversion of matrices, see Theorem 3.3 below. 
\par
 The paper is organized as follows. 
In the next section we demonstrate the method by applying  it 
  to the averaged Green function of the
Anderson model. The Schwinger-Dyson equation for that model reads 
 $G^{-1}=G_0^{-1}+\Sigma(G_0)$  where  $\Sigma(G_0)$ is the sum of 
 all  two-legged one-particle irreducible diagrams. 
Application of (1.3) 
 leads to an integral equation $G^{-1}=G_0^{-1}+I(G)$ where 
$I(G)$ is the sum of all two-legged graphs without two-legged 
 subgraphs. The latter equation has two advantages. First, 
 $\Sigma$ is the sum of one-particle irreducible diagrams, but 
 these diagrams may very well have two-legged subdiagrams and 
usually these are the diagrams which produce anomalously large 
 contributions. 
And second, the propagator for $I(G)$ is the interacting two 
 point function $G$, which, for the Anderson model, is more regular 
than 
the free two point function $G_0$ which is the propagator for the 
 diagrams contributing to $\Sigma(G_0)$. More precisely, 
 the series for  
 $I(G)$ can be expected to be asymptotic, that is, its 
 lowest order contributions are a good approximation if the 
 coupling is small, but, usually, the series for $\Sigma(G_0)$ 
 is not asymptotic.
\par
For the many-electron system and for the $\vp^4$ model 
repeated application of (1.3,4) amounts to a resummation of two- and 
four-legged subgraphs. This is discussed in Section 4. In particular, 
for the many-electron system with attractive delta-interaction we find, 
as one expects, the formation of a BCS gap, but we are also able to 
obtain information on the expectation 
 value of the Hubbard-Stratonovich 
field $\la |\phi_q|^2\ra$ which is related to a certain four point 
function (see (4.11,12) and (4.3,4)).  
This function has been investigated by several authors [FMRT,CFS,B]. 
 One 
expects a singularity at $q=0$. The usual argument, based on a Taylor  
 expansion of the effective potential around its minimum  
predicts a $1/q^2$ behaviour. Our analysis, given in Section 4.1, 
 results in  $\la |\phi_q|^2\ra 
 ={|\Delta|^2\over\lambda}\> \delta(q)+{regular}$ where $\Delta$ is the BCS 
 gap and the regular term is bounded for small $q$. In 
 particular, we 
 find that the existence of a gap and a macroscopic 
 $\la |\phi_0|^2\ra=\beta L^d\,{|\Delta|^2\over\lambda}$ enforce 
 each other ($L^d$ volume, $1/\beta=T$ temperature). The proof 
 of the inversion formula is given in Section 3. 
\bigskip
\bigskip
%
%
% S E C T I O N   I I
%
%
\section{Application to the Anderson Model}
\bigskip
\indent
Let coordinate space be a lattice of finite volume with periodic 
 boundary conditions, lattice spacing $1/M$ and volume 
 $[0,L]^d$: 
$$\ts \Gamma=\left\{ x={1\over M}(n_1,\cdots,n_d)\;|\;
 0\le n_i\le ML-1\right\}=
 \left( {1\over M}\Bbb Z\right)^d/(L\Bbb Z)^d  \eqno (2.1)$$
Momentum space is given by 
$$\ts \M:=\Gamma^\sharp=\left\{k={2\pi\over L}(m_1,\cdots,m_d)\;|\; 
    0\le m_i\le ML-1\right\}
 =\left( {2\pi\over L}\Bbb Z\right)^d/(2\pi M \Bbb Z)^d \eqno (2.2)$$
We consider the averaged Green function of the Anderson model given
by 
$$\la G\ra(x,x'):=\int \left[ -\Delta-z+\lambda V
     \right]_{x,x'}^{-1} 
  dP(V) \eqno (2.3)$$
where the random potential is Gaussian,  
$$dP(V)= \pro_{x\in\Gamma} \ts e^{-{V_x^2\over 2}} {dV_x\over
       \sqrt{2\pi}}.  \eqno(2.4)$$
Here $z=E+i\vep$ and $\Delta$ is the discrete Laplacian, 
$$\left[ -\Delta-z+\lambda V
     \right]_{x,x'}= -M^2\sum_{i=1}^d 
   \left( \delta_{x',x+ e_i/M} 
  +\delta_{x',x- e_i/M}-2\delta_{x',x}\right) -z \,\delta_{x,x'}
  +\lambda V_x \,
  \delta_{x,x'} \eqno (2.5)$$
By taking the Fourier transform, one has 
\setcounter{equation}{5}
\label{sec2}
\beqn
 \la G\ra(x,x')&=&{\ts {1\over M^d L^d}} \sum_{k\in\M} 
  e^{ik(x'-x)} \la G\ra (k) \label{2.6} \\
 \la G\ra (k)&=& \int_{\Bbb R^{N^d}} \ts
   \left[ a_k\delta_{k,p}+{\lambda\over \sqrt{N^d}} v_{k-p}
  \right]^{-1}_{k,k}\;dP(v) \label{2.7}
\eeqn
where $N^d=(ML)^d=|\Gamma|=|\M|$, $dP(v)$ is given by (2.10) or (2.11) 
 below, depending on whether $N^d$ is even or odd, and 
$$a_k=4M^2\sum_{i=1}^d \sin^2\left[\ts {k_i\over 2M}\right] 
  -E-i\vep \eqno (2.8)$$
 The rigorous control of $\la G\ra (k)$ for small 
 disorder $\lambda$ and energies inside the spectrum of the unperturbed 
 Laplacian, $E\in [0,4M^2]$, in  which case $a_k$ has a root if $\vep\to 0$, 
 is still an open problem [AG,K,MPR,P,W]. It is expected 
 that $\lim_{\vep\searrow 0}\lim_{L\to\infty} 
 \la G\ra (k)=1/(a_k-\sigma_k)$ where Im$\sigma=O(\lambda^2)$. 

The integration variables $v_{q}$ in (2.7) are given by the discrete 
Fourier transform of the $V_x$. In particular, 
observe that, if $F$ denotes the unitary matrix of discrete 
 Fourier transform, the variables  
$$v_{q} \equiv(FV)_{q}={\ts {1\over \sqrt{N^d}}}\sum_{x\in\Gamma} 
   e^{-iqx} V_x={\ts \left({M\over L}\right)^{d\over2}  \;
  {1\over M^d}} \sum_{x\in\Gamma} 
   e^{-iqx} V_x \equiv  {\ts \left({M\over L}\right)^{d\over2}}
  \, \hat V_q  \eqno(2.9)$$
would not have a limit if $V_x$ would be deterministic and cutoffs are 
removed, since the $\hat V_q$ are the quantities which have a limit
in that case. But since the $V_x$ are integration variables, we 
choose a unitary transform to keep the integration measure
invariant. Observe also that $v_q$ is complex, 
 $v_q=u_q+iw_q$. Since $V_x$ is real, $u_{-q}=u_q$ and 
 $w_{-q}=-w_q$. In order to transform $dP(V)$ to momentum space, 
we have to choose a set $\M^+\subset\M$ 
  such that either $q\in \M^+$ or $-q\in \M^+$. If $N$ is odd, 
  the only momentum with  
 $q=-q$ or $w_q=0$ is $q=0$. In that case $dP(V)$ becomes 
$$ dP(v)=e^{-{u_{0}^2\over2}}
 {\ts 
  {du_{0}\over \sqrt{2\pi}}} \pro_{q\in \M^+} 
  e^{-(u_q^2+w_q^2)} 
 {\ts {du_q dw_q\over \pi}} \eqno (2.10) $$
For even $N$ we get 
$$ dP(v) =e^{-{1\over2}(u_{0}^2+u_{q_0}^2)}
 {\ts 
  {du_{0}du_{q_0}\over {2\pi}}} \pro_{q\in \M^+} 
  e^{-(u_q^2+w_q^2)} 
 {\ts {du_q dw_q\over \pi}}   \eqno (2.11) $$
where $q_0={2\pi m\over L}$ is the unique nonzero momentum  for 
which  ${2\pi\over L}m=2\pi M 
 (1,\cdots,1)- {2\pi\over L}m$. 

\vspace{0.6cm}
Now we apply the inversion formula (1.3) to the inverse matrix 
 element in 
$$\la G\ra (k)= \int_{\Bbb R^{N^d}} \ts
   \left[ a_k\delta_{k,p}+{\lambda\over \sqrt{N^{d}}} v_{k-p}
  \right]^{-1}_{k,k}\;dP(v) \eqno (2.7) $$
We start with the `two loop approximation',  
 which we define by retaining only the $r=2$ term in the 
 denominator of the right hand side of (1.3),
$$G(k)\approx 
  {1\over B_{kk}-\sum_{p\in\M\setminus\{k\}} 
    B_{kp} G_k(p) B_{pk} }  \eqno(2.12)$$
Thus, let 
$$\ts G(k):=\left[ a_k\delta_{k,p}+{\lambda\over \sqrt{N^{d}}} v_{k-p}
  \right]^{-1}_{k,k}=G(k;v,z) \eqno (2.13)$$
In the infinite volume limit the spacing $2\pi/L$ of the momenta becomes 
arbitrary small. Hence, in computing an inverse matrix element, 
it should not matter whether a single column and row labelled by some 
  momentum $t$ is absent or not. 
 In other words, in the infinite volume limit one should have 
$$ \phantom{ {\;\;\;{\rm for}\;\;L\to\infty} }
    G_t(p)=G(p)\;\;\;{\rm for}\;\;L\to\infty  \eqno (2.14) $$
and similarly $G_{t_1\cdots t_j}(p)=G(p)$ as long as $j$ is independent of the 
volume. We remark however that if the matrix has a block structure, say 
 $B=(B_{k\sigma,p\tau})$ with $\sigma,\tau\in\{\up,\down\}$ some spin variables,
 this structure has to be respected. That is, for a given momentum $k$ 
 all rows and columns labelled by $k\!\up,\;k\!\down$ 
   have to be eliminated, since otherwise (2.14) may not be 
 true. 
\par
Thus the two loop approximation gives 
$$ G(k)={1\over a_k+{\lambda\over \sqrt{N^d}} v_0 
  -{\lambda^2\over N^d}\sum_{p\ne k} v_{k-p}\,G(p)\, v_{p-k} }
   \eqno (2.15)$$
For large $L$, we can disregard the ${\lambda\over \sqrt{N^d}} v_0$ 
 term. Introducing $\sigma_k=\sigma_k(v,z)$ according to 
$$G(k)=:{1\over a_k-\sigma_k}\;,  \eqno (2.16)$$
we get 
$$\sigma_k={\ts {\lambda^2\over N^d}}\sum_{p\ne k} {|v_{k-p}|^2 
  \over a_p-\sigma_p }
 \approx {\ts {\lambda^2\over N^d}}\sum_{p} {|v_{k-p}|^2 
  \over a_p-\sigma_p } \eqno (2.17)$$
and arrive at 
$$\la G\ra (k)=\int {1\over \ds a_k-{\ts {\lambda^2\over N^d}} 
  \sum_{p}^{\phantom{I}} {|v_{k-p}|^2\over a_p- 
  {\ts {\lambda^2\over N^d}} 
  \sum_{t} {|v_{p-t}|^2\over a_{t}-{\lambda^2\over N^d}\Sigma\cdots}  }  }
  \;dP(v)  \eqno (2.18)$$
Now consider the infinite volume limit $L\to\infty$ or $N=ML\to\infty$. 
   By the central limit theorem of probability 
 ${1\over \sqrt{N^d}}\sum_q\left(|v_q|^2-\la |v_q|^2\ra\right)$ is, 
as a sum of independent 
 random variables, normal distributed. Note that only 
 a prefactor of $1/\sqrt{N^d}$ is required for that property. 
In particular, 
if $F$ is some bounded function independent of $N$, 
 sums which come with a prefactor 
 of $1/N^d$ like 
  ${1\over N^d}\sum_q c_q |v_q|^2$ can be substituted by
their expectation value,
$$\lim_{N\to\infty}\int F\Bigl({\ts{1\over N^d}}
  \sum_kc_k |v_k|^2\Bigr)dP(v)=F\Bigl(\lim_{N\to\infty}{\ts{1\over N^d}}
  \sum_kc_k \la |v_k|^2\ra\Bigr) \eqno (2.19)$$
Therefore, in the two loop approximation, one obtains in the infinite 
 volume limit
$$\la G\ra (k)= {1\over \ds a_k-{\ts {\lambda^2\over N^d}} 
  \sum_{p}^{\phantom{I}} {\la |v_{k-p}|^2\ra \over a_p- 
  {\ts {\lambda^2\over N^d}} 
  \sum_{t} {\la |v_{p-t}|^2\ra \over a_{t}
  -{\lambda^2\over N^d}\Sigma \cdots}  }  }
  \;=:\; {1\over a_k-\la \sigma_k \ra }  \eqno (2.20)$$ 
where the quantity $\la \sigma_k \ra$ satisfies the integral 
 equation 
$$\la \sigma_k \ra= {\ts {\lambda^2\over N^d}}\sum_{p}{
   {\la |v_{k-p}|^2 \ra \over a_p-\la \sigma_p\ra }\;  }
  \buildrel L\to\infty\over \to \;
   {\ts {\lambda^2\over M^d}}\int_{[0,2\pi M]^d}  
  { {d^dp\over (2\pi)^d}} \,
   {\la |v_{k-p}|^2\ra  \over a_p-\la\sigma_p\ra }  \eqno (2.21)$$
For a Gaussian distribution $\la |v_q|^2\ra=1$ for all $q$ 
such that $\la \sigma_k\ra =\la\sigma\ra$ becomes 
 independent of $k$. Thus we end up with 
$$\la G\ra(k)={1\over 4M^2\sum_{i=1}^d\sin^2\left[ {k_i\over 2M}
  \right] -E-i\vep-\la\sigma\ra} \eqno (2.22)$$
where $\la\sigma\ra$ is a solution of
\setcounter{equation}{22}
\beqn
  \la\sigma\ra&=&
    {\ts {\lambda^2\over M^d}}\int_{[0,2\pi M]^d}
   { {d^dp\over (2\pi)^d}} 
\, {1\over 4M^2\sum_{i=1}^d
  \sin^2\left[{p_i\over 2M}\right]-z-\la\sigma\ra}\nonumber \\
 &=&\lambda^2\int_{[0,2\pi]^d}
   { {d^dp\over (2\pi)^d}} 
\, {1\over 4M^2\sum_{i=1}^d
  \sin^2\left[{p_i\over 2}\right]-z-\la\sigma\ra}\>.  
\eeqn
This equation is of course well known and one deduces from it that
it generates a small imaginary part Im$\,\sigma=O(\lambda^2)$ if 
the energy $E$ is within the spectrum of $-\Delta$. 
       
\bigskip
We now add the higher loop terms (the terms for $r>2$ in the  
 denominator of (1.3)) to our discussion and give an 
interpretation in terms of Feynman diagrams. To make the volume
factors more explicit, asume that the lattice spacing in 
 coordinate space is $1/M=1$ such that $N=L$. 
\par
For the Anderson model, Feynman graphs may be obtained by brutally 
expanding 
\beqn
  \lefteqn{\int 
   \bigl[a_k\delta_{k,p}+{\ts{\lambda\over \sqrt L^{d}} v_{k-p}}
  \bigr]^{-1}_{k,k}\;dP =\sum_{r=0}^\infty 
   \int \left( C[VC]^r\right)_{kk} \,dP}  \\
 && =\sum_{r=0}^\infty  {\ts {(-\lambda)^r\over \sqrt{L^d}^r}}    
  \sum_{p_2\cdots p_{r}} {\ts {1\over a_ka_{p_2}\cdots a_{p_{r}} 
  a_k}} \int v_{k-p_2} v_{p_2-p_3}\cdots v_{p_{r}-k} \,dP 
   \nonumber
\eeqn
For a given $r$, we may represent this by ($c_k:=1/a_k$)

\begin{figure}[thb]
 \centerline{\epsfbox{bild1.eps}}
 \caption{A string of particle lines with unpaired squiggles (dashed lines)}
 \label{Figure 1}
\end{figure}
\noindent
The integral over the $v$ gives a sum of $(r-1)!!$ terms where 
 each term is a product of $r/2$ Kroenecker-delta's, the terms for odd $r$ 
 vanish.  
If this is substituted in (2.24), the number of independent momentum
sums is cut down to $r/2$ and  each of the $(r-1)!!$ terms may be
represented by a diagram

\begin{figure}[thb]
 \centerline{\epsfbox{bild2.eps}}
 \caption{Lowest order diagrams contributing to (2.24) 
   for $r=2$ and $r=4$ }
 \label{Figure 2}
\end{figure}
\noindent
where, say, the value of the third diagram is given by 
 ${\lambda^4\over L^{2d}}\sum_{p_1,p_2} {1\over a_k
 a_{k+p_1} a_{k+p_1+p_2} a_{k+p_2}a_k}$. 
 For short: 
$$\la G\ra (k)={\rm sum\;of\; all\; two\; legged\; diagrams\,.}
  \eqno (2.25)$$
Since the value of a diagram depends on its subgraph structure, 
one distinguishes, in the easiest case, two types of diagrams. 
Diagrams with or without two-legged subdiagrams.  Those diagrams 
with two-legged subgraphs usually produce anomalously large 
 contributions. They are devided further into the one-particle  
 irreducible ones and the reducible ones. Thereby a diagram is 
 called one-particle reducible if it can be made disconnected 
 by cutting one solid or `particle' line (no squiggle or dashed line), see 
 also figure 3.  

\begin{figure}[thb]
 \centerline{\epsfbox{bild3.eps}}
 \caption{Reducible and irreducible diagrams}
 \label{Figure 3}
\end{figure} 
\noindent
The reason for introducing reducible and irreducible diagrams is 
 that the reducible ones can be easily resummed by writing down 
the Schwinger-Dyson equation which states that if 
the self energy $\Sigma_k$ is defined through 
$$\la G\ra (k)={1\over a_k-\Sigma_k(G_0)} \eqno (2.26)$$
then $\Sigma_k(G_0)$ is the sum of all 
 amputated (no $1/a_k$'s at the ends) one particle irreducible 
diagrams. Here we wrote $\Sigma_k(G_0)$ to indicate that the factors
(`propagators') assigned to the solid lines of the diagrams
contributing to $\Sigma_k$ are given by the free two point function 
 $G_0(p)={1\over a_p}$. However, the diagrams 
contributing to $\Sigma_k(G_0)$  still  
contain anomalously large contributions, namely irreducible 
graphs which contain two-legged subgraphs like diagram (c) in  
 figure 3. 

In the following we show, using the inversion formula (1.3) 
 including all higher loop terms, that all graphs with two-legged 
subgraphs can be eliminated or resummed by writing down 
the following integral equation for $\la G\ra$:
$$ \la G\ra (k)= {1\over a_k-\sigma_k(\la G\ra)} \eqno (2.27) $$
where 
\begin{quote}
 $\sigma_k(\la G\ra)$ 
  is the sum of all amputated two-legged diagrams 
 which do not contain any two-legged subdiagrams, but now 
 with propagators $\la G\ra(k)={1\over a_k-\sigma_k}$ 
instead of  $G_0={1\over a_k}$
\end{quote}
\noindent
which may be formalized as in (2.35) below. 
 The advantage of this formula is that the series for  
 $\sigma_k(\la G\ra)$ can be expected to be asymptotic, that is, its 
 lowest order contributions are a good approximation if the 
 coupling is small, but, usually, the series for $\Sigma_k(G_0)$ 
 is not asymptotic. Thus, 
in order to rigorously controll $\la G\ra (k)$,   one has to 
define a suitable space of propagators, to estimate the sum of all 
 amputated two-legged graphs without two-legged subgraphs on 
 that space and then finally to show that the equation (2.27) has a 
 solution on this space. We intend to address this problem in another
paper. 

\bigskip
We now show (2.27) for the Anderson model. For fixed $v$ one has 
$$G(k,v)={1\over a_k-\sigma_k(v)}  \eqno (2.28)$$
where 
$$\sigma_k(v)=\sum_{r=2}^{L^d}(-1)^{r}\!\!\!\!\!\!
    \sum_{p_2\cdots p_{r}
 \atop p_i\ne p_j,\> p_i\ne k}
  \!\! \ts {\lambda^r\over \sqrt{ L^d}^r} 
 \, G_k(p_2)\cdots G_{kp_2\cdots p_{r-1}}(p_r)\, v_{k-p_2}
  \cdots v_{p_r-k}  \eqno (2.29)$$
We cutoff the $r$-sum in (2.29) at some arbitrary but fixed order $\ell<L^d$ 
where $\ell$ is choosen to be independent of the volume. 
 Furthermore we substitute $G_{kp_2\cdots p_j}(p)$ by $G(p)$. 
Thus   
$$\la G\ra (k)=\biggl\la {1\over a_k-\sum_{r=2}^\ell 
   \sigma_k^r(v) } \biggr\ra  \eqno (2.30) $$
where 
$$\sigma_k^r(v)=(-1)^r{\ts {\lambda^r\over \sqrt{ L^d}^r}} \!\!\!\!
  \sum_{p_2\cdots p_{r} \atop p_i\ne p_j,\> p_i\ne k} \!\!\!\!
  G(p_2)\cdots G(p_r)\, v_{k-p_2} \cdots v_{p_r-k}\eqno (2.31)$$
Consider first two strings $s_k^{r_1}$, $s_k^{r_2}$ where 
$$ s_k^r(v)={\ts {\lambda^r\over \sqrt{ L^d}^r}} \!\!\!\!
  \sum_{p_2\cdots p_{r} \atop p_i\ne p_j,\> p_i\ne k} \!\!\!\!
   c^r_{kp_2\cdots p_r}\, v_{k-p_2} \cdots v_{p_r-k}\eqno (2.32)$$
and the $c^r_{kp_2\cdots p_r}$ are some numbers. Then in the
infinite volume limit 
$$\la s_k^{r_1} s_k^{r_2} \ra= \la s_k^{r_1}\ra\, 
  \la s_k^{r_2} \ra  \eqno (2.33)$$ 
because all pairings which connect the two strings have an extra 
 volume factor $1/L^d$. Namely, if the two strings are 
  disconnected, there
are $(r_1+r_2)/2$ loops and a volume factor of 
   $1/\sqrt{L^d}^{(r_1+r_2)}$ giving $(r_1+r_2)/2$ 
 Riemannian sums. If the two strings are connected, there are only 
 $(r_1+r_2-2)/2$ loops leaving an extra factor of $1/L^d$. By the 
 same argument one has in the infinite volume limit
$$\la (s_k^{r_1})^{n_1}\cdots (s_k^{r_m})^{n_m}\ra = 
  \la s_k^{r_1} \ra^{n_1}\cdots \la s_k^{r_m}\ra^{n_m}\eqno (2.34)$$
which results in 
$$\la G\ra(k)= {1\over \ds a_k -\sum_{r=2}^\ell 
 {\ts {(-\lambda)^r\over \sqrt{ L^d}^{\,r}}}\!\!\!\!
  \sum_{p_2\cdots p_{r} \atop p_i\ne p_j,\> p_i\ne k} \!\!\!\!
  \la G\ra(p_2)\cdots \la G\ra (p_r)\, \la v_{k-p_2} \cdots 
   v_{p_r-k} \ra} \eqno (2.35) $$
The condition $p_2,\cdots,p_r\ne k$ and $p_i\ne p_j$ means exactly 
 that two-legged subgraphs are forbidden. Namely, for a 
two-legged subdiagram  
as in (c) in figure 3, the incomming and outgoing momenta  
$p$, $p'$ (to which are
assigned propagators $\la G\ra(p)$, $\la G\ra(p')$)
must be equal which is forbidden by 
 the condidtion $p_i\ne p_j$ in (2.35).

However, we cannot take the limit $\ell\to\infty$ in (2.35) since the 
 series in the denominator of (2.35) is only an asymptotic one. 
 To see this a bit more clearly suppose for the moment that there 
 were no restrictions on the momentum sums. Then, if
$V=({\lambda\over\sqrt{L^d}}v_{k-p})_{k,p}$ and 
  $\la G\ra=(\la G\ra (k)\,\delta_{k,p})_{k,p}$, 
$${\ts {\lambda^r\over \sqrt{ L^d}^r}} 
   \sum_{p_2\cdots p_r}\la G\ra(p_2)\cdots \la G\ra(p_r)\,
  \la v_{k-p_2} \cdots v_{p_r-k} \ra 
 =\left\la (V[\la G\ra V]^{r-1})_{kk} \right\ra \eqno (2.36)$$ 
and for $\ell \to \infty$ we would get 
$$\la G\ra (k)={1\over  a_k-\la (V {\la G\ra V\over Id +\la G\ra V} 
    )_{kk} \ra }
 = {1\over  a_k-\la (V {1\over \la G\ra^{-1} + V} V
    )_{kk} \ra }   \eqno (2.37)$$
That is, the factorials produced by the number of diagrams in the
denominator of (2.35) are basically the same as those in the 
 expansion 
$$ \int_{\Bbb R}{\ts {x^2\over z+\lambda x}}\, e^{-{x^2\over 2}} 
  {\ts {dx\over\sqrt{2\pi}}}\> =\>\sum_{r=0}^\ell \ts 
  {\lambda^{2r}\over z^{2r+1}}\,(2r+1)!!\>
 +R_{\ell+1}(\lambda)\, \eqno (2.38)$$ 
where the remainder satisfies the bound  
$|R_{\ell+1}(\lambda)|\le \ell!\,const_{\! z}^\ell\, \lambda^\ell $. 

\bigskip
We close this section with two  further remarks. 
So far the computations were done in momentum space. One may wonder 
 what one gets if the inversion formula (1.3) is applied to 
 $[-\Delta+z+\lambda V]^{-1}$ in coordinate space. Whereas a 
 geometric series expansion of $[-\Delta+z+\lambda V]^{-1}$ gives 
 a representation in terms of the simple random walk, 
  application of (1.3) results in a representation in terms 
 of the self avoiding walk: 
$$[-\Delta+z+\lambda V]_{0,x}^{-1}=\sum_{\gamma:0\to x\atop 
  \gamma\;{\rm self\;avoiding}}
 {\det\left[ (-\Delta+z+\lambda V)_{y,y'\in\Gamma\setminus\gamma}
   \right] \over\det\left[ (-\Delta+z+\lambda V)_{y,y'\in\Gamma}
      \right]  }  \eqno (2.39)$$
where $\Gamma$ is the lattice in coordinate space.  Namely, if $|x|>1$, the 
inversion formula (1.4) for the off-diagonal elements gives 
\beq
 \lefteqn{[-\Delta+\lambda V]_{0,x}^{-1}=\sum_{r=3}^{L^d}(-1)^{r+1} 
 \!\!\!\!\!\!\!\!\! 
  \sum_{x_3\cdots x_r\in\Gamma\setminus\{0,x\}\atop x_i\ne x_j}
  \!\!\!\!\!\!\!\! G(0)G_0(x_3)\cdots G_{0x_3\cdots x_r}(x)
  \,(-\Delta)_{0 x_3}
  \cdots (-\Delta)_{x_r x}  }  \\
 &&=\sum_{r=3}^{L^d}  \sum_{x_2=0,x_3,\cdots ,x_r,x_{r+1}=x\in\Gamma\atop
    |x_i-x_{i+1}|=1\; \forall i=2\cdots r}
  \!\!\!\! {\det\left[ (-\Delta+\lambda V)_{y,y'\in\Gamma\setminus\{0\}}
   \right] \over\det\left[ (-\Delta+\lambda V)_{y,y'\in\Gamma}
      \right]  } \cdots 
   {\det\left[ (-\Delta+\lambda V)_{y,y'\in\Gamma\setminus\{0,x_3\cdots x_r,x\}}
   \right] \over\det\left[
   (-\Delta+\lambda V)_{y,y'\in\Gamma\setminus\{0,x_3\cdots x_r\}}
      \right]  }  
\eeq
which coincides with (2.39). 

Finally we remark that, while the argument following (2.32) leads to a 
factorization
property for on-diagonal elements in momentum space, 
 $\la G(k)\,G(p)\ra=\la G(k)\ra\,\la G(p)\ra$, 
 there is no such property for products of off-diagonal elements 
which appear in a quantity like 
$$\Lambda(q)= {\ts {1\over L^d}}\sum_{k,p}\ts \left\la 
 \left[ a_k\delta_{k,p}+{\lambda\over \sqrt L^{\,d}} v_{k-p}
  \right]^{-1}_{k,p}
  \left[ \bar a_k\delta_{k,p}+{\lambda\over \sqrt L^{\,d}} \bar v_{k-p}
  \right]^{-1}_{k-q,p-q} \right\ra  \eqno (2.40) $$ 
which is the Fourier transform of 
$\bigl\la \left| [-\Delta+z+\lambda V]^{-1}_{x,y}
   \right|^2\bigr\ra$. (Each off-diagonal inverse matrix element is proportional 
 to $1/\sqrt{L^d}$, therefore the prefactor of $1/L^d$ in (2.40) is correct.)
\bigskip
\bigskip
%
%
% S E C T I O N   I I I
%
%
\section{Proof of the Inversion Formula}
%\setcounter{equation}{5}
\label{sec3}

\bigskip
\noindent{\bf Lemma 3.1:} Let $B\in\Bbb C^{k\times n}$, 
 $C\in\Bbb C^{n\times k}$ and let $Id_k$ denote the identity 
 in $\Bbb C^{k\times k}$. 
Then:
\bitem
\item[{\bf (i)}] $Id_k-BC\;\;{\rm invertible}\;\;\;
  \Leftrightarrow\;\;\;Id_n-CB\;\;{\rm invertible}$. 
\item[{\bf (ii)}]  If the left or the right hand side of (i) fullfilled, 
 then
 $\ts C\,{1\over Id_k-BC} =\ts  {1\over Id_n-CB}\,C $.
\eitem

\bigskip
\noindent{\bf Proof:}  Let 
$$B=\left(\bmatr{ccc}
  - & \vec b_1 & - \\  
      & \vdots & \\ 
    - & \vec b_k & - 
\ematr\right),\;\;\;\;
  C=\left(\bmatr{ccc}
   |& &|\\
   \vec c_1&\cdots &\vec c_k\\
   |& &|
\ematr\right)  $$ 
where the $\vec b_j$ are $n$-component row vectors and the 
 $\vec c_j$ are $n$-component column vectors. Let $\vec x\in 
 {\rm Kern}(Id-CB)$.  Then $\vec x=CB \vec x=\sum_j \lambda_j \vec c_j$ if 
 we define $\lambda_j:= (\vec b_j,\vec x)$. Let 
 $\vec \lambda=(\lambda_j)_{1\le j\le k}$. Then $[(Id-BC)\vec\lambda]_i
  =\lambda_i-\sum_j(\vec b_i,\vec c_j)\lambda_j 
  =(\vec b_i,\vec x)-\sum_j(\vec b_i,\vec c_j)\lambda_j=0$ 
 since $\vec x=\sum_j \lambda_j \vec c_j$, thus 
$\vec \lambda \in {\rm Kern}(Id-BC)$. On the other hand, if 
 some $\vec\lambda\in {\rm Kern}(Id-BC)$, then $\vec x:=\sum_j\lambda_j 
 \vec c_j\in {\rm Kern}(Id-CB)$ which proves (i). Part (ii) then follows 
from 
$ C\ts={1\over Id_n-CB}\,(Id_n-CB)C 
  ={1\over Id_n-CB}\,C (Id_k-BC)$ $\blacksquare$

\bigskip
\bigskip
The inversion formula (1.3,4) is obtained by iterative application of 
 the next lemma, which states the Feshbach formula for finite dimensional 
matrices. For a more general version one may look in [BFS], Theorem 2.1. 

\bigskip
\bigskip
\noindent{\bf Lemma 3.2:} Let 
 $\ts h=\left({A\atop C}\;{B\atop D}\right)\in\Bbb C^{n\times n}$ where 
$A\in \Bbb C^{k\times k}$, 
 $D\in \Bbb C^{(n-k)\times (n-k)}$ are invertible and  
 $B\in \Bbb C^{k\times (n-k)}$, $C\in\Bbb C^{(n-k)\times k}$. Then 
$$ h\;\;{\rm invertible}\;\;\;\;
  \Leftrightarrow\;\;\;\;A-BD^{-1}C\;\;{\rm invertible}\;\;\;\;
  \Leftrightarrow\;\;\;\; D-CA^{-1}B\;\;{\rm invertible} \eqno (3.1)$$
and if one of the conditions in (3.1) is fullfilled, one has 
 $h^{-1} = \left({E\atop G}\;{F\atop H}\right)$ 
where 
\setcounter{equation}{1}
\beqn
\ts E={1\over A-BD^{-1}C}\>,&\hskip 0.7cm&\ts
      H={1\over D-CA^{-1}B}\>, \\
  F=-EBD^{-1}=-A^{-1}BH\>,&\hskip 0.7cm& G=-HCA^{-1}=-D^{-1}CE\>. 
\eeqn

\bigskip
\noindent{\bf Proof:} We have, using Lemma 3.1 in the second line, 
\beq
  A-BD^{-1}C\;\;{\rm inv.}&\Leftrightarrow &
    Id_k-A^{-1}BD^{-1}C\;\;{\rm inv.}  \\
 & \Leftrightarrow &
    Id_{n-k}-D^{-1}CA^{-1}B\;\;{\rm inv.}  \\
 & \Leftrightarrow & D-CA^{-1}B\;\;{\rm inv.}  
\eeq
Furthermore, again by Lemma 3.1, 
$$\ts D^{-1}C\,{1\over Id-A^{-1}BD^{-1}C}=
    {1\over Id-D^{-1}CA^{-1}B}\,D^{-1}C=
  {1\over D-CA^{-1}B}\,C=HC$$
and 
$$\ts A^{-1}B\,{1\over Id-D^{-1}CA^{-1}B}=
  {1\over Id-A^{-1}BD^{-1}C}\,A^{-1}B=
 {1\over A-BD^{-1}C}\,B=EB$$
which  proves the last equalities in (3.3), 
 $HCA^{-1}=D^{-1}CE$
and 
 $EBD^{-1}= A^{-1}BH $. 
Using these equations and the definition of $E,F,G$ and $H$ one computes 
$$\ts \left({A\atop C}\;{B\atop D}\right)
      \left({E\atop G}\;{F\atop H}\right)
     = \left({E\atop G}\;{F\atop H}\right)
      \left({A\atop C}\;{B\atop D}\right)
  =\left({Id\atop 0}\;{0\atop Id}\right)$$
It remains to show that the invertibility of $h$ implies the invertibility 
 of $A-BD^{-1}C$. To this end let $P=
  \left({Id\atop \phantom{C}}\;{\phantom{B}\atop 0}\right)$, 
 $\bar P=
  \left({0\atop \phantom{C}}\;{\phantom{B}\atop Id}\right)$ such that 
$A-BD^{-1}C=PhP-Ph\bar P(\bar P h\bar P)^{-1} \bar P h P$. Then 
\beq
  (A-BD^{-1}C)Ph^{-1}P&=&PhPh^{-1}P-
   Ph\bar P(\bar P h\bar P)^{-1} \bar P h Ph^{-1}P \\
 &=&Ph(1-\bar P)h^{-1}P-
   Ph\bar P(\bar P h\bar P)^{-1} \bar P h(1- \bar P)h^{-1}P \\
 &=&P-Ph\bar P h^{-1} P +Ph \bar P h^{-1} P=P  
\eeq
and similarly $Ph^{-1}P(A-BD^{-1}C)=P$ which proves the invertibility 
  of $A-BD^{-1}C$ $\blacksquare$

\bigskip
\bigskip
\noindent{\bf Theorem 3.3:} Let $B\in \Bbb C^{nN\times nN}$ be 
 given by  $B=(B_{kp})_{k,p\in\M}$, $\M$ some index set, $|\M|=N$, 
 and  $B_{kp}=
 (B_{k\sigma,p\tau})_{\sigma,\tau\in I}\in \Bbb C^{n\times n}$ 
  where $I$ is another index set, $|I|=n$. 
 Suppose that $B$ and, for any $\N\subset \M$, the 
  submatrix $(B_{kp})_{k,p\in\N}$ is invertible. 
For $k\in \M$ let 
$$G(k):=\left[B^{-1}\right]_{kk}\in \Bbb C^{n\times n}\eqno (3.4)$$ 
and, if $\N\subset\M$, $k\notin \N$, 
$$G_{\N}(k):= \left[\{(B_{st})_{s,t\in\M\setminus\N}\}^{-1}\right]_{kk}
    \in \Bbb C^{n\times n} \eqno (3.5)$$ 
Then one has 
\bitem
\item[\bf (i)] The on-diagonal block matrices of $B^{-1}$ are given by 
$$G(k)={1\over\ds B_{kk}-\sum_{r=2}^N (-1)^{r}
   \!\!\!\!\!\!\!\!\!\!\!
    \sum_{p_2\cdots p_{r}\in\M\setminus\{k\}\atop p_i\ne p_j}
  \!\!\!\!\!\!\! \!\! 
  B_{kp_2}G_k(p_2)B_{p_2 p_3}\cdots B_{p_{r-1} p_r} 
  G_{kp_2\cdots p_{r-1}}(p_r)B_{p_r k}  }  \eqno (3.6)$$
where $1/\;\cdot\;$ is inversion of $n\times n$ matrices. 
\item[\bf (ii)] Let $k,p\in\M$, $k\ne p$. Then the 
  off-diagonal block matrices of $B^{-1}$ can be expressed in terms 
 of the $G_{\N}(s)$ and the $B_{st}$, 
$$[B^{-1}]_{kp}=-G(k) B_{kp} G_k(p)-
  \sum_{r=3}^N(-1)^{r} \!\!\!\!\!\!\!  
 \!\!\! \sum_{t_3\cdots t_r\in\M\setminus\{k,p\}\atop t_i\ne t_j}
  \!\!\!\!\!\! G(k) B_{kt_3} G_k(t_3) B_{t_3t_4} 
  \cdots B_{t_rp}G_{kt_3\cdots t_r}(p) \eqno (3.7) $$
\eitem

\bigskip
\noindent{\bf Proof:} Let $k$ be fixed and let $p,p'\in\M\setminus\{k\}$ 
 below label columns and rows.  By Lemma 3.2 we have 
$$ \left(\bmatr{cccc}
  G(k) & && \\ & && \\ &&*& \\ &&& 
\ematr\right)\;=\;\left( 
 \bmatr{cccc} B_{kk} & \;-&B_{kp}&-\; \\
  | & && \cr B_{p'k}&&B_{p'p}& \\ |&&& 
 \ematr\right)^{-1}
 =\;\;
\left(\bmatr{cccc}
  E &\;- &F&-\; \\ |& && \\ G&&H& \\ |&&& 
\ematr\right)  $$
where 
\setcounter{equation}{7}
\beqn
  G(k)=E&
   =&{1\over\ds B_{kk}-\sum_{p,p'\ne k}B_{kp}\left[\{(B_{p' p})_{ 
  p',p\in\M\setminus\{k\}}\}^{-1}\right]_{pp'} B_{p'k} }  \\
 &=&{1\over \ds B_{kk}-\sum_{p\ne k}B_{kp} G_k(p) B_{pk}-
  \sum_{p,p'\ne k\atop p\ne p'} B_{kp}\left[\{ (B_{p' p})_{ 
  p',p\in\M\setminus\{k\}}\}^{-1}\right]_{pp'} B_{p'k} }\;\; \nonumber
\eeqn 
and 
\beqn F_{kp}=\left[ B^{-1}\right]_{kp}&=& 
  -G(k)\sum_{t\ne k} B_{kt} 
 \left[\{ (B_{p' p})_{p',p\in\M\setminus\{k\}}\}^{-1}\right]_{tp} \\
 &=&-G(k)B_{kp}G_k(p)-G(k)\sum_{t\ne k,p} B_{kt} 
 \left[\{ (B_{p' p})_{p',p\in\M\setminus\{k\}}\}^{-1}\right]_{tp}\nonumber
\eeqn
Apply Lemma 3.2 now to the matrix 
 $\{ (B_{p' p})_{p',p\in\M\setminus\{k\}}\}^{-1}$ and proceed by 
 induction to obtain after $\ell$ steps 
\beqn
  G(k)&=&{1\over\ds B_{kk}-\sum_{r=2}^\ell (-1)^{r}
   \!\!\!\!\!\!\!\!\!\!\!
    \sum_{p_2\cdots p_{r}\in\M\setminus\{k\}\atop p_i\ne p_j}
  \!\!\!\!\!\!\! \!\! 
  B_{kp_2}G_k(p_2)B_{p_2 p_3}\cdots B_{p_{r-1} p_r} 
  G_{kp_2\cdots p_{r-1}}(p_r)B_{p_r k} -R_{\ell+1}  } \\
 F_{kp}&=&-G(k) B_{kp} G_k(p)-
  \sum_{r=3}^\ell (-1)^{r} \!\!\!\!\!\!\!  
 \!\!\! \sum_{t_3\cdots t_r\in\M\setminus\{k,p\}\atop t_i\ne t_j}
  \!\!\!\!\!\! G(k) B_{kt_3} G_k(t_3) B_{t_3t_4} 
  \cdots B_{t_rp}G_{kt_3\cdots t_r}(p) -\tilde R_{\ell+1} \nonumber \\
  &&
\eeqn
where 
\beqn
  R_{\ell+1}&=&(-1)^\ell
   \!\!\!\!\!\!\!\!\!\!\!
    \sum_{p_2\cdots p_{\ell+1}\in\M\setminus\{k\}\atop p_i\ne p_j}
  \!\!\!\!\!\!\! \!\! 
  B_{kp_2}G_k(p_2)\cdots B_{p_{\ell-1} p_\ell} 
   \left[\{ (B_{p' p})_{ 
  p',p\in\M\setminus\{kp_2\cdots p_\ell\}}\}^{-1}\right]_{p_\ell
    p_{\ell+1}}
    B_{p_{\ell+1} k}\nonumber\\
 &&\\
 &&\nonumber\\
 \tilde R_{\ell+1}&=& (-1)^\ell \!\!\!\!\!\!\!  
 \!\!\! \sum_{t_3\cdots t_{\ell+1}\in\M\setminus\{k,p\}\atop t_i\ne t_j}
  \!\!\!\!\!\! G(k) B_{kt_3} 
  \cdots G_{kt_3\cdots t_{\ell-1}}(t_\ell)B_{t_\ell t_{\ell+1}} 
  \left[\{ (B_{p' p})_{p',p\in\M\setminus\{k t_3\cdots t_\ell\}}\}^{-1}
   \right]_{t_{\ell+1} p} \nonumber\\
 &&
\eeqn
Since $R_{N+1}=\tilde R_{N+1}=0$ the theorem follows
 $\blacksquare$ 
\goodbreak
%
%
% S E C T I O N   I V
%
%

\bigskip
\bigskip
\bigskip
\section{Application to the Many-Electron System and to 
  the $\varphi^4$-Model}

\bigskip
\bigskip
\subsection{\bf The Many-Electron System}

\bigskip
\setcounter{equation}{4}
\label{sec4}
We consider the  many-electron system in the grand canonical 
 ensemble in finite volume 
 $[0,L]^d$ and at some small but positive temperature $T=1/\beta>0$ 
 with attractive delta-interaction given by the Hamiltonian 
$$H=H_0-\lambda H_{\rm int}={\ts
{1\over L^d}}\sum_{\bk\sigma}({\ts {\bk^2\over 2m}}-\mu)a_{\bk\sigma}^+ 
 a_{\bk\sigma}-{\ts {\lambda\over L^{3d}}} \sum_{\bk\p\q} a_{\bk\up}^+ 
 a_{\q-\bk\down}^+ a_{\q-\p\down} a_{\p\up} \eqno (4.1)$$
Our normalization conventions concerning the volume factors are such 
 that the canonical anticommutation relations read $\{a_{\bk\sigma},
  a_{\p\tau}^+\}=L^d\,\delta_{\bk,\p}\delta_{\sigma,\tau}$. The  
 momentum sums range over some subset of $\left({2\pi\over L}\Bbb Z\right)^d$, 
 say ${\cal M}=\bigl\{\bk\in \left({2\pi\over L}\Bbb Z\right)^d\,\bigr|\> 
  |e_\bk|\le 1\bigr\}$, $e_\bk=\bk^2/2m-\mu$, 
  and $\q\in\{\bk-\p\,|\bk,\p\in\M\}$. 

We are interested in the momentum distribution 
$$\la a_{\bk\sigma}^+a_{\bk\sigma}\ra= Tr[e^{-\beta H}a_{\bk\sigma}^+
  a_{\bk\sigma}]\bigr/ Tr\,e^{-\beta H}  \eqno (4.2)$$
and in the expectation value of the energy 
$$\la H_{\rm int}\ra=\sum_{\q} \Lambda(\q)  \eqno (4.3)$$
where 
$$\Lambda(\q)={\ts {\lambda\over L^{3d}}}\sum_{\bk,\p} Tr[e^{-\beta H}
    a_{\bk\up}^+ a_{\q-\bk\down}^+ a_{\q-\p\down} a_{\p\up}]
  \bigr/ Tr\,e^{-\beta H} \eqno (4.4)$$

By writing down the perturbation series for the partition function, rewriting 
 it as a Grassmann integral
\beqn
 {\ts {Tr\,e^{-\beta( H_0-\lambda H_{\rm int})}
   \over Tr\, e^{-\beta H_0}}}&=&\int e^{ {\lambda\over 
  (\beta L^d)^3}\sum_{kpq}\bar\psi_{k\up}\bar\psi_{q-k\down}
    \psi_{q-p\down}\psi_{p\up} } d\mu_C(\psi,\bar\psi)\\
 d\mu_C&=&\pro_{k\sigma}{\ts {\beta L^d\over ik_0-e_\bk}}\> 
   e^{-{1\over \beta L^d}\sum_{k\sigma}(ik_0-e_\bk)\bar\psi_{k\sigma}\psi_{k\sigma}} 
   \pro_{k\sigma} d\psi_{k\sigma} d\bar\psi_{k\sigma} \>, \nonumber
\eeqn
 performing a Hubbard-Stratonovich 
 transformation ($\phi_q=u_q+iv_q$, 
   $d\phi_q d\bar\phi_q:=du_q dv_q$)
$$ e^{-\sum_q a_q b_q}= \int  e^{i\sum_q(a_q\phi_q+b_q\bar\phi_q)}  
    e^{-\sum_q|\phi_q|^2}\pro_q\ts {d\phi_q d\bar\phi_q\over \pi}
  \eqno (4.6)$$
with 
$$ a_q={\ts {\lambda^{1\over 2}\over (\beta L^d)^{3\over2}}}\sum_k
        \bar\psi_{k\up} \bar\psi_{q-k\down},\;\;\;
  b_q={\ts {\lambda^{1\over 2}\over (\beta L^d)^{3\over2}}}
   \sum_p \psi_{p\up} \psi_{q-p\down} \eqno (4.7)$$
and then integrating out the $\psi,\bar\psi$ variables, 
  one arrives at the following representation which is the 
starting point for our analysis (for more details, see [FKT] or [L1]):
$$ {\ts {1\over L^d}}\la a_{\bk\sigma}^+a_{\bk\sigma}\ra=
 {\ts {1\over \beta L^d}\>{1\over \beta}}\!\!\!\!\sum_{k_0\in {\pi\over\beta}
  (2\Bbb Z+1)} \!\!\la \psi_{\bk k_0\sigma}^+\psi_{\bk k_0\sigma}\ra 
   \eqno (4.8)$$
where, abbreviating $k=(\bk,k_0)$, $\kappa=\beta L^d$, $a_k=ik_0-e_\bk$, 
  $g=\lambda^{1\over2}$,  
$${\ts {1\over\kappa}}\,\la \bar\psi_{t\sigma}\psi_{t\sigma}\ra=\int \left[ 
 \bmatr{cc} a_{k}\delta_{k,p} & {ig\over \sqrt{\kappa}}\,
  \bar\phi_{p-k} \\ {ig\over \sqrt{\kappa}}\, \phi_{k-p} & 
  a_{-k}\delta_{k,p} \ematr \right]_{t\sigma,t\sigma}^{-1} 
 \!\!\! dP(\phi) \eqno(4.9)$$
and $dP(\phi)$ is the probability measure 
$$dP(\phi)={\ts {1\over Z}} \>\det\left[ 
  \bmatr{cc} a_{k}\delta_{k,p} & {ig\over \sqrt{\kappa}}\,
  \bar\phi_{p-k} \\ {ig\over \sqrt{\kappa}}\, \phi_{k-p} & 
  a_{-k}\delta_{k,p} \ematr
  \right]  e^{-\sum_q|\phi_q|^2} \pro_q {\ts {d\phi_q 
 d\bar\phi_q}}  \eqno (4.10)$$
 Furthermore 
$$\Lambda(\q)={\ts {1\over\beta}}\sum_{q_0\in {2\pi\over\beta}\Bbb Z} 
  \Lambda(\q,q_0) \eqno (4.11)$$
where
\setcounter{equation}{11}
\beqn
 \Lambda(q)&=&{\ts {\lambda\over (\beta L^d)^3}}\sum_{k,p} 
  \la \bar\psi_{k\up}\bar\psi_{q-k\down}\psi_{q-p\down}\psi_{p\up}\ra
  \nonumber \\
 &=& \la |\phi_q|^2\ra -1 
\eeqn
and the expectation in the last line is integration with respect 
 to $dP(\phi)$. 
  The expectation on the 
 $\psi$ variables 
 $\la \bar\psi_{k\sigma}\psi_{k\sigma}\ra={\ts {1\over{\cal Z}}} 
 \int \bar\psi_{k\sigma}\psi_{k\sigma}\,
   e^{ {\lambda\over \kappa^3}\sum_{k,p,q}\bar\psi_{k\up}
    \bar\psi_{q-k\down}\psi_{q-p\down}\psi_{p\up} } d\mu_C$
 is Grassmann integration,  but these 
representations are not used in the following. 
 The matrix and the integral in (4.9) become finite dimensional if we
choose some cutoff on the $k_0$ variables which is removed in the
end. The set $\M$ for the spatial momenta is already finite since we 
have chosen a fixed UV-cuttoff $|e_\bk|=|\bk^2/2m-\mu|\le 1$ which
  will not be removed 
 in the end since we are interested in the infrared properties at 
 $\bk^2/2m=\mu$. 

Our goal is to apply the inversion formula to the inverse matrix element in
(4.9). Instead of writing the matrix in terms of four $N\times N$ blocks 
 $(a_k\delta_{k,p})_{k,p}$, $(\bar\phi_{p-k})_{k,p}$, 
 $(\phi_{k-p})_{k,p}$ and $(a_{-k}\delta_{k,p})_{k,p}$
 where $N$ is the number of the $d+1$-dimensional momenta $k,p$, 
 we interchange rows and columns to rewrite it in terms of $N$ blocks 
 of size $2\times 2$ (the matrix $U$ in the next line interchanges the rows and 
 columns): 
$$U\left[ 
  \bmatr{cc} a_{k}\delta_{k,p} & {ig\over \sqrt{\kappa}}\,
  \bar\phi_{p-k} \\ {ig\over \sqrt{\kappa}}\, \phi_{k-p} & 
  a_{-k}\delta_{k,p} \ematr \right]U^{-1}=B=(B_{kp})_{k,p}$$
 where the  $2\times 2$ blocks $B_{kp}$ are given by  
$$ B_{kk}=\left(
  \bmatr{cc} a_{k} & {ig\over \sqrt{\kappa}}\,
  \bar\phi_{0} \\ {ig\over \sqrt{\kappa}}\, \phi_{0} & 
  a_{-k} \ematr \right),\;\;\;B_{kp}={\ts {ig\over \sqrt{\kappa}}} 
 \left( \bmatr{cc} 0  &\bar\phi_{p-k} \\  \phi_{k-p} & 0 
    \ematr \right)\;\;{\rm if}\;k\ne p\,. \eqno (4.13)   $$
We want to compute the $2\times 2$ matrix 
$$ \la G\ra(k)=\int G(k)\,dP(\phi) \eqno (4.14)$$
where 
$$G(k)=[B^{-1}]_{kk} \eqno (4.15)$$

\bigskip
We start again with the two loop approximation which retains only the
$r=2$ term in the  denominator of (1.3). The result will be equation  
 (4.20) below where the quantities $\la\sigma_k\ra$ and $\la|\phi_0|^2\ra$
 appearing in (4.20) have to satisfy the equations (4.21) and (4.24) which have to be 
solved in conjunction with (4.26). The solution to these equations is discussed 
 below (4.27) and leads to (4.32).
\par
We first derive (4.20). In the two loop approximation,
\setcounter{equation}{15} 
\beqn
  G(k)&\approx& \biggl[ B_{kk}-\sum_{p\ne k} B_{kp}\,
    G_k(p)\, B_{pk} \biggr]^{-1}  \nonumber \\
 &=&\biggl[\left(\bmatr{cc} a_{k} & {ig\over \sqrt{\kappa}}\,
  \bar\phi_{0} \\ {ig\over \sqrt{\kappa}}\, \phi_{0} & 
  a_{-k} \ematr\right) +{\ts {\lambda\over\kappa}} 
  \sum_{p\ne k} \left(\bmatr{cc}   & 
  \bar\phi_{p-k} \\ \phi_{k-p} & 
    \ematr\right) \,G_k(p)\, 
  \left(\bmatr{cc}   & 
  \bar\phi_{k-p} \\ \phi_{p-k} & 
    \ematr\right) \biggr]^{-1}\nonumber \\
 &=:& \biggl[\left(\bmatr{cc} a_{k} & {ig\over \sqrt{\kappa}}\,
  \bar\phi_{0} \\ {ig\over \sqrt{\kappa}}\, \phi_{0} & 
  \bar a_{k} \ematr\right) 
  + \Sigma(k) \biggr]^{-1}  
\eeqn
where, substituting again $G_k(p)$ by $G(p)$  in the infinite volume limit, 
$$\Sigma(k)={\ts {\lambda\over\kappa}} 
  \sum_{p\ne k} \left(\bmatr{cc}   & 
  \bar\phi_{p-k} \\ \phi_{k-p} & 
    \ematr\right)\, \biggl[ \left(\bmatr{cc}
   a_{p} & {ig\over \sqrt{\kappa}}\,
  \bar\phi_{0} \\ {ig\over \sqrt{\kappa}}\, \phi_{0} & 
  \bar a_{p} \ematr\right)+ \Sigma(p) \biggr]^{-1}\, 
  \left(\bmatr{cc}   & 
  \bar\phi_{k-p} \\ \phi_{p-k} & 
     \ematr\right)   \eqno (4.17)$$
Anticipating the fact that the off-diagonal elements of $\la\Sigma\ra(k)$ 
 will be zero (for `zero external field'), we make the Ansatz 
$$\Sigma(k)=\left(\bmatr{cc} \sigma_k& \\ & \bar\sigma_k\ematr\right)
    \eqno (4.18) $$
and obtain 
$$ \left(\bmatr{cc} \sigma_k& \\ & \bar\sigma_k\ematr\right)=
   {\ts {\lambda\over\kappa}} 
  \sum_{p\ne k} \ts {1\over (a_p+\sigma_p)(\bar a_p+\bar\sigma_p) 
   +{\lambda\over\kappa}|\phi_0|^2} 
\left(\bmatr{cc} (a_k+\sigma_k)|\phi_{p-k}|^2 
 & -{ig\over \sqrt{\kappa}}\, \phi_{0}\bar\phi_{k-p}\bar\phi_{p-k}\\
   -{ig\over \sqrt{\kappa}}\, \bar\phi_{0} \phi_{k-p}\phi_{p-k} 
  &  (\bar a_k+\bar\sigma_k)|\phi_{k-p}|^2\ematr\right)
  \eqno(4.19)$$
As for the Anderson model, we perform the functional integral by substituting 
 the quantities $|\phi_q|^2$ by their expectation values $\la|\phi_q|^2\ra$. 
 Apparently this is less obvious in this case since  $dP(\phi)$ is 
 no longer Gaussian and the $|\phi_q|^2$ are no longer identically, independently 
 distributed.  We will comment on this  
  after (4.37) below and at the end of  
 the next section by reinterpreting this procedure as a 
 resummation of diagrams. 
  For now, we simply continue in this way. Then 
$$\la G\ra(k)=\ts {1\over |a_k+\la\sigma_k\ra|^2
   +{\lambda\over\kappa}\la|\phi_0|^2\ra}\left(\bmatr{cc} \bar a_k
   +\la\bar\sigma_k\ra & \ts -{ig\over \sqrt\kappa}\,\la\bar\phi_0\ra \\ 
  -{ig\over \sqrt\kappa} \,\la\phi_0\ra  & a_k+\la\sigma_k\ra 
   \ematr\right) \eqno (4.20)$$
where the quantity $\la\sigma_k\ra$ has to satisfy the equation 
$$\la\sigma_k\ra= {\ts {\lambda\over\kappa}} 
  \sum_{p\ne k} \ts {\bar a_p+\la\bar\sigma_p\ra 
  \over |a_p+\la\sigma_p\ra|^2
   +{\lambda\over\kappa}\la |\phi_0|^2\ra }\, 
   \la |\phi_{p-k}|^2\ra \eqno (4.21) $$   

\medskip
Since $dP(\phi)$ is not Gaussian, we do not know the expectations 
 $\la|\phi_q|^2\ra$. However, by partial integration, we obtain 
$$ \la |\phi_q|^2\ra =1+ {\ts {ig\over \sqrt\kappa}} 
   \sum_p   \int \phi_q\, [B^{-1}(\phi)]_{p\up,p+q\down}\, dP(\phi) 
  \eqno(4.22) $$
Namely, 
\beq
  \la |\phi_q|^2\ra&=&{\ts {1\over Z}}\int \phi_q\bar\phi_q\, 
   \det\left[ \{B_{kp}(\phi)\}_{k,p}\right]\, e^{-\sum_q |\phi_q|^2}
      d\phi_qd\bar\phi_q \\
 &=&1+{\ts {1\over Z}}\int \phi_q\,\ts \Bigl( {\pt\over \pt\phi_q}
   \det\left[ \{B_{kp}(\phi)\}_{k,p}\right]\Bigr)\, e^{-\sum_q |\phi_q|^2}
      d\phi_qd\bar\phi_q \\
 &=&1+{\ts {1\over Z}}\int \phi_q\,\sum_{p,\tau} \ts 
   \det\left[\bmatr{ccc} |&|&|\\ 
   B_{k\sigma,p'\tau'}& \ts  {\pt B_{k\sigma,p\tau}\over \pt\phi_q}& 
   B_{k\sigma,p''\tau''} \\  |&|&|\ematr
    \right] e^{-\sum_q |\phi_q|^2}
      d\phi_qd\bar\phi_q 
\eeq
Since 
$$\ts  {\pt \over \pt\phi_q}B_{kp}
  ={ig\over \sqrt\kappa}\left(\bmatr{cc} 0&0 \cr1&0 \ematr\right)
   \,\delta_{k-p,q}$$
we have 
$$ {\det\left[\bmatr{ccc} |&|&|\\ 
   B_{k\sigma,p'\tau'}& \ts  {\pt B_{k\sigma,p\tau}\over \pt\phi_q}& 
   B_{k\sigma,p''\tau''} \\  |&|&|\ematr 
    \right]\Bigr/ \det\left[ \{B_{kp}\}_{k,p}\right]} 
  =\left\{\begin{array}{ll} 0& {\rm if}\; \tau=\down\\ & \\
  \ts {ig\over \sqrt\kappa}\, [B^{-1}]_{p\up,p+q\down}& 
 {\rm if}\; \tau=\up\end{array}\right. $$
which results in (4.22). 

The inverse matrix element in (4.22) we compute again with (1.3,4) in the 
 two loop approximation.  
Consider first the case $q=0$. Then one gets 
\setcounter{equation}{22}
\beqn
 \la |\phi_0|^2\ra&=&1+ {\ts {ig\over \sqrt\kappa}} 
   \sum_p   \int \phi_0\, G(p)_{\up\down}\, dP(\phi)  \nonumber\\
 &=&1+ {\ts {ig\over \sqrt\kappa}} 
   \sum_p   \int \phi_0\,  \ts {1\over |a_p+\sigma_p|^2
   +{\lambda\over\kappa}|\phi_0|^2}\left(\bmatr{cc}\bar a_p
   +\bar\sigma_p & \ts -{ig\over \sqrt\kappa}\,\bar\phi_0 \\ 
  -{ig\over \sqrt\kappa} \,\phi_0  & a_p+\sigma_p\ematr\right)_{\up\down}  
   \, dP(\phi)  \nonumber\\
  &=&1+ {\ts {\lambda\over \kappa}} 
   \sum_p   \int \phi_0\,  \ts {\bar\phi_0\over |a_p+\sigma_p|^2
   +{\lambda\over\kappa}|\phi_0|^2} \, dP(\phi) 
\eeqn
Performing the functional integral by substitution of expectation values
gives 
$$\la |\phi_0|^2\ra=1+ {\ts {\lambda\over \kappa}} 
   \sum_p  \la |\phi_0|^2\ra\,  \ts {1\over |a_p+\la\sigma_p\ra|^2
   +{\lambda\over\kappa}\la|\phi_0|^2\ra }  $$
or 
$$\la |\phi_0|^2\ra={1\over 1-{\ts {\lambda\over \kappa}} 
  {\ds \sum_p^{\phantom{,}}} \ts {1\over |a_p+\la\sigma_p\ra|^2
   +{\lambda\over\kappa}\la|\phi_0|^2\ra }  } \eqno (4.24) $$
Before we discuss (4.24), we write down the equation 
 for $q\ne 0$. In that case 
 we use (1.4) to compute $[B^{-1}(\phi)]_{p\up,p+q\down}$ in the two loop 
 approximation. We get 
\setcounter{equation}{24}
\beqn
\lefteqn{ [B^{-1}(\phi)]_{p\up,p+q\down}
    \approx -\left[G(p)B_{p,p+q}G(p+q)\right]_{\up\down} }\nonumber \\
 & &\nonumber \\
 \lefteqn{ =- \ts {1\over |a_p+\sigma_p|^2
   +{\lambda\over\kappa}|\phi_0|^2}
  \, \ts {1\over |a_{p+q}+\sigma_{p+q}|^2
   +{\lambda\over\kappa}|\phi_0|^2}\,{ig\over\sqrt\kappa}\times }\nonumber \\
 & & \nonumber \\    
 & & \left(\bmatr{cc} -{ig\over \sqrt\kappa}[
  (\bar a+\bar \sigma)_{p+q}\bar\phi_0\phi_{-q}+(\bar a+\bar \sigma)_{p} 
  \phi_0\bar\phi_q]& 
   (\bar a+\bar \sigma)_{p}(a+\sigma)_{p+q}\bar\phi_q 
  -{\lambda\over\kappa} \bar\phi_0^2\phi_{-q}\\
  ( a+ \sigma)_{p}(\bar a+\bar\sigma)_{p+q}\phi_{-q}
  -{\lambda\over\kappa} \phi_0^2\bar\phi_{q} & 
  -{ig\over \sqrt\kappa}[
  ( a+ \sigma)_{p+q}\phi_0\bar\phi_{q}+( a+ \sigma)_{p} 
  \bar\phi_0\phi_{-q}]  \ematr\right)_{\up\down} \nonumber \\
 & & \nonumber \\
  \lefteqn{ =- \ts{ig\over\sqrt\kappa}
   {(\bar a+\bar \sigma)_{p}(a+\sigma)_{p+q}\bar\phi_q 
  -{\lambda\over\kappa} \bar\phi_0^2\phi_{-q}
       \over \left(|a_p+\sigma_p|^2
   +{\lambda\over\kappa}|\phi_0|^2\right)\left( |a_{p+q}+\sigma_{p+q}|^2
   +{\lambda\over\kappa}|\phi_0|^2\right)}   }
\eeqn
which gives 
\beq
 \la |\phi_q|^2\ra&=&1+ {\ts {\lambda\over \kappa}} 
   \sum_p   \int \phi_q\, \ts
  {(\bar a+\bar \sigma)_{p}(a+\sigma)_{p+q}\bar\phi_q 
  -{\lambda\over\kappa} \bar\phi_0^2\phi_{-q}
       \over \left(|a_p+\sigma_p|^2
   +{\lambda\over\kappa}|\phi_0|^2\right)\left( |a_{p+q}+\sigma_{p+q}|^2
   +{\lambda\over\kappa}|\phi_0|^2\right)} \, dP(\phi)  \\
 &=&1+ \la |\phi_q|^2\ra  {\ts {\lambda\over \kappa}} 
   \sum_p   \ts
  {(\bar a_p+ \la\bar\sigma_p\ra)(a_{p+q}+\la\sigma_{p+q}\ra)
       \over \left(|a_p+\la\sigma_p\ra|^2
   +{\lambda\over\kappa}\la|\phi_0|^2\ra\right)
  \left( |a_{p+q}+\la\sigma_{p+q}\ra|^2
   +{\lambda\over\kappa}\la|\phi_0|^2\ra \right)}   
\eeq
or 
$$\la |\phi_q|^2\ra={1\over 1- 
       {\ts {\lambda\over \kappa}} 
  {\ds \sum_p^{\phantom{,}} }  \ts
  {(\bar a_p+ \la\bar\sigma_p\ra)(a_{p+q}+\la\sigma_{p+q}\ra)
       \over \left(|a_p+\la\sigma_p\ra|^2
   +{\lambda\over\kappa}\la|\phi_0|^2\ra\right)
  \left( |a_{p+q}+\la\sigma_{p+q}\ra|^2
   +{\lambda\over\kappa}\la|\phi_0|^2\ra \right)}   }\>,\;\;\;q\ne 0
    \eqno (4.26)$$

\bigskip
We now discuss the solutions to (4.24) and (4.26). We assume that the solution 
 $\la \sigma_k\ra$ of (4.21) is sufficiently small such that the 
 BCS equation 
$$  {\ts {\lambda\over \kappa}} 
   \sum_p^{\phantom{,}} { \ts {1\over |a_p+\la\sigma_p\ra|^2
   + |\Delta|^2  }} = 1 \eqno (4.27)$$
has a nonzero solution $\Delta\ne 0$ (in particular this excludes Luttinger 
 liquid behaviour, for $d=1$ one should make a seperate analysis),  
and make the Ansatz 
$$\lambda\la|\phi_0|^2\ra= \beta L^d\, |\Delta|^2 +\eta \eqno (4.28) $$
where $\eta$ is independent of the volume. Then 
\setcounter{equation}{28}
\beqn
  {\ts {\lambda\over \kappa}} 
 \sum_p^{\phantom{,}} \ts {1\over |a_p+\la\sigma_p\ra|^2
   +{\lambda\over\kappa}\la|\phi_0|^2\ra } &= &
  {\ts {\lambda\over \kappa}} 
 \sum_p^{\phantom{,}} \ts {1\over |a_p+\la\sigma_p\ra|^2
   + |\Delta|^2+{\eta\over\kappa} }  \nonumber\\
 &=&{\ts {\lambda\over \kappa}} 
 \sum_p^{\phantom{,}} {\ts {1\over |a_p+\la\sigma_p\ra|^2
   + |\Delta|^2 }} -
  {\ts {\lambda\over \kappa}} 
 \sum_p^{\phantom{,}} \ts {\eta/\kappa\over ( |a_p+\la\sigma_p\ra|^2
   + |\Delta|^2)^2 } +O\left(({\eta\over\kappa})^2\right) \nonumber\\
 &=&\ts 1 - c_\Delta\,{\eta\over\kappa}+O\left(({\eta\over\kappa})^2\right) 
\eeqn
where we put $c_\Delta=
  {\ts {\lambda\over \kappa}} 
 \sum_p \ts {1\over ( |a_p+\la\sigma_p\ra|^2
   + |\Delta|^2)^2 }$ and used the BCS equation (4.27) in the last line. 
Equation (4.24) becomes 
\beqn
  \kappa\, |\Delta|^2 +\eta &=&{\lambda\over 
    c_\Delta \,{\eta\over\kappa} 
      +O\left(({\eta\over\kappa})^2\right) } 
  =\kappa\,{\lambda\over c_\Delta\,\eta} +O(1) 
\eeqn
 and has a solution $\eta=\lambda/(c_\Delta|\Delta|^2)$.  Now consider 
 $\la|\phi_q|^2\ra$ for small but nonzero $q$. Substituting (4.28) and 
Taylor expanding around $q=0$ we get 
\beqn
  \la |\phi_q|^2\ra&=&{1\over 1- 
       {\ts {\lambda\over \kappa}} 
  {\ds \sum_p^{\phantom{,}} }  \ts
  {(\bar a_p+ \la\bar\sigma_p\ra)(a_{p}+\la\sigma_{p}\ra)
       \over \left(|a_p+\la\sigma_p\ra|^2
   +|\Delta|^2 \right)
  \left( |a_{p}+\la\sigma_{p}\ra|^2
   +|\Delta|^2 \right)} +O(q)   } \nonumber \\
 &=&{1\over 
   {\ts {\lambda\over \kappa}} 
  {\ds \sum_p^{\phantom{,}} }  \ts
  { |\Delta|^2
       \over \left(|a_p+\la\sigma_p\ra|^2
   +|\Delta|^2 \right)^2} +O(q)   } 
\eeqn
which is bounded for $q\to 0$.  Thus, in the infinite volume limit 
 and at zero temperature we find 
$$ \lambda\la|\phi_q|^2\ra=|\Delta|^2\,\delta(q)+{\rm regular}
   \eqno (4.32)$$
\par 
The function $\Lambda(q)={\ts {\lambda\over (\beta L^d)^3}}\sum_{k,p} 
  \la \bar\psi_{k\up}\bar\psi_{q-k\down}\psi_{q-p\down}\psi_{p\up}\ra 
 = \la |\phi_q|^2\ra -1$ or some closely related four-point functions 
have been discussed  by some authors [FMRT,CFS,B]. The conclusion is 
$\Lambda(q)\sim 1/q^2$ for small $q$. Basically this is infered from 
the second order Taylor expansion of the effective potential
$$V_{\rm eff}(\{\phi_q\})=\sum_q|\phi_q|^2-\log
  \det\left[ 
  \bmatr{cc} \delta_{k,p} & {ig\over \sqrt{\kappa}}\,
  {\bar\phi_{p-k}\over a_k} \\
  {ig\over \sqrt{\kappa}}\, {\phi_{k-p}\over a_{-k}} & 
  \delta_{k,p} \ematr
  \right]   \eqno (4.33) $$
around its global minimum [L2]
$$\phi_q^{\rm min}=\sqrt{\beta L^d}\,\ts {|\Delta|\over \sqrt\lambda}\,\delta_{q,0}\, 
  e^{i\theta_0}  \eqno (4.34) $$
where the phase $\theta_0$ of $\phi_0$ is arbitrary. If one expands $V_{\rm eff}$ up to 
 second order in 
$$ \xi_{q}=\phi_{q}-\delta_{q,0}\sqrt{\beta L^d}\,
   \ts {|\Delta|\over \sqrt\lambda}\,
    e^{i\theta_0}  
  =\left\{ \begin{array}{ll}
    \left(\rho_0-\sqrt{\beta L^d}\,\ts {|\Delta|\over \sqrt\lambda}\right)
   \,e^{i\theta_0} &\mbox{for $q=0$ }\\ 
   \>\rho_{q}\, e^{i\theta_{q}} &\mbox{for $q\ne 0$} 
 \end{array}\right. \eqno(4.35) $$
one obtains [L2] 
\setcounter{equation}{35}
\beqn
  V_{\rm eff}(\{\phi_q\})&=&V_{\rm min}+2\beta_0 \,
  (\rho_0-\sqrt{\beta L^d}\,{\ts {|\Delta|\over \sqrt\lambda}})^2 +
  \sum_{q\ne 0} (\alpha_q+i\gamma_q)\, \rho_q^2 \nonumber\\
 &\phantom{m}& + {\ts{1\over2}}\sum_{q\ne 0} 
   \beta_q \,|e^{-i\theta_0} \phi_q
   +e^{i\theta_0} \bar\phi_{-q}|^2 
  +O(\xi^3)  
\eeqn
where for small $q$ one has $\alpha_q,\gamma_q\sim q^2$. 
Hence, if $V_{\rm eff}$ is substituted by the right hand side of (4.36) 
 one obtains 
 $\la |\phi_q|^2\ra\sim 1/q^2$. Thus, in view of the above computations, it is 
 questionable to what extent one can draw any conclusions 
from the quadratic 
 approximation. We believe that the probability of the configurations 
 $\{\phi_q\}$ for which (4.36) is a good approximation , the small field 
 region, is too small to give a significant contribution to 
 the functional integral. 

\bigskip
Finally we argue why it is reasonable to substitute $|\phi_0|^2$ by its 
expectation
value while performing the functional integral. 
 We may write the effective potential (4.33) 
as 
$$V_{\rm eff}(\{\phi_q\})=V_1(\phi_0)+ V_2(\{\phi_q\}) \eqno (4.37)$$
where 
\setcounter{equation}{37}
\beqn
  V_1(\phi_0)&=&|\phi_0|^2- \sum_{k}\ts
 \log\left[ 1+ {\lambda\over\kappa}
  { |\phi_0|^2 \over k_0^2+e_\bk^2}\right] \nonumber \\
  &=& \kappa \Bigl({\ts {|\phi_0|^2\over\kappa}- {1\over\kappa}}\sum_{k}\ts
 \log\left[ 1+ {\lambda {|\phi_0|^2\over\kappa}\over k_0^2+e_\bk^2}\right]
   \Bigr)\equiv \kappa\, 
 V_{\rm BCS}\ts\left({|\phi_0|\over\sqrt\kappa}\right)
\eeqn
and
$$ V_2(\{\phi_q\})=  \sum_{q\ne 0}|\phi_q|^2-\log
  \det\left[ 
  \left(\bmatr{cc} \delta_{k,p} & {ig\over \sqrt{\kappa}}\,
  {\bar\phi_{0}\over a_k}\delta_{k,p} \\
   {ig\over \sqrt{\kappa}}\, {\phi_{0}\over a_{-k}}\delta_{k,p} & 
  \delta_{k,p} \ematr\right)^{-1}
  \left(\bmatr{cc} \delta_{k,p} & {ig\over \sqrt{\kappa}}\,
  {\bar\phi_{p-k}\over a_k} \\
  {ig\over \sqrt{\kappa}}\, {\phi_{k-p}\over a_{-k}} & 
  \delta_{k,p} \ematr\right)
  \right]  \eqno(4.39)$$
If we ignore the $\phi_0$-dependence of $V_2$, then the $\phi_0$-integral
\setcounter{equation}{39} 
\beqn
  {\int F\left({1\over\kappa}|\phi_0|^2\right)\,
   e^{-V_{1}(\phi_0)} d\phi_0
  d\bar\phi_0 \over  
  \int e^{-V_{1}(\phi_0)} d\phi_0
  d\bar\phi_0} 
  &=& {\int F\left( \rho^2\right)\, e^{-\kappa V_{\rm BCS}(\rho)} 
   \rho\,d\rho \over  
  \int e^{-\kappa V_{\rm BCS}(\rho)}  \rho\,d\rho} 
   \buildrel \kappa\to\infty\over \to F(\rho_{\rm min}^2)
  =\ts F\left({1\over\kappa}\la|\phi_0|^2\ra\right)\nonumber \\
 & & 
\eeqn
simply puts $|\phi_0|^2$ at the global minimum 
  of the (BCS) effective potential. 
\bigskip
\bigskip \goodbreak
%
%
% P H I   4
%
%
\subsection{The $\vp^4$-Model}
\bigskip
In this section we choose the  $\varphi^4$-model as a typical
bosonic model to demonstrate our method. As in section 2, we start in 
 finite volume $[0,L]^d$ on a lattice with lattice spacing $1/M$. 
The two point function is given by 
$$S(x,y)=\la \vph_x\vph_y\ra:={ \int_{\Bbb R^{N^d}} 
     \vph_x\vph_y\> e^{-{g^2\over 2}{1\over M^d} 
  \sum_x \vph_x^4}\> e^{-{1\over M^{2d}}\sum_{x,y} 
  (-\Delta+m^2)_{x,y}\vph_x\vph_y} \pro_x d\vph_x \over 
   \int_{\Bbb R^{N^d}} e^{-{g^2\over 2}{1\over M^d} 
  \sum_x \vph_x^4}\> e^{-{1\over M^{2d}}\sum_{x,y} 
  (-\Delta+m^2)_{x,y}\vph_x\vph_y} \pro_x d\vph_x} \eqno (4.41)$$
where 
$$(-\Delta+m^2)_{x,y}=M^d\Bigl[
  -M^2\sum_{i=1}^d(\delta_{x,y-e_i/M} +\delta_{x,y+e_i/M} 
  -2\delta_{x,y} ) +m^2\delta_{x,y}\Bigr]\,  \eqno (4.42) $$
First we have to bring this into the form $\int[P+Q]^{-1}_{x,y}d\mu$, 
 $P$ diagonal in momentum space, $Q$ diagonal in coordinate space.  
This is done again by making a Hubbard Stratonovich  transformation 
 which in this case reads 
$$  e^{-{1\over 2} \sum_x a_x^2}=\int e^{i\sum_x a_x u_x} 
  e^{-{1\over 2} \sum_x u_x^2}\pro_x  
    {\ts {d u_x\over\sqrt{2 \pi}}} \eqno (4.43)$$
with 
$$a_x={\ts   {g\over \sqrt{M^d}}}\vph_x^2  \eqno (4.44)  $$
The result is Gaussian in the $\vp_x$-variables and the integral over 
 these variables gives 
$$S(x,y)= \int_{\Bbb R^{N^d}}\ts \left[ {1\over M^{2d}} 
  (-\Delta+m^2)_{x,y}-{ig\over \sqrt{M^d}} u_x\delta_{x,y} 
   \right]_{x,y}^{-1} dP(u)  \eqno (4.45)$$
where 
$$ dP(u)={\ts{1\over  Z}} \,
  {\ts\det\left[ {1\over M^{2d}} 
  (-\Delta+m^2)_{x,y}-{ig\over \sqrt{M^d}} u_x\delta_{x,y} 
   \right]^{-{1\over2}} } e^{-{1\over 2}\sum_x u_x^2}\pro_x
    du_x \eqno  (4.46)   $$
Since we have  bosons, the determinant comes with a power of 
 $-1/2$ which is the only difference compared to a fermionic system. 
 In momentum space this reads (compare equations (2.7-11)) 
$$S(x-y)={\ts {1\over L^d}}\sum_k e^{ik(x-y)} \la G\ra (k) \eqno (4.47)$$
where ($\gamma_q=v_q+iw_q$, $\gamma_{-q}=\bar\gamma_q$, 
   $d\gamma_qd\bar\gamma_q:=dv_qdw_q$)
$$\la G\ra (k)= \int_{\Bbb R^{N^d}}\ts \left[
   a_k\delta_{k,p} -{ig\over \sqrt{L^d}}
   {\gamma_{k-p}} 
    \right]^{-1}_{kk}  dP(\gamma) \eqno (4.48)$$ 
and 
$$dP(\gamma)={\ts {1\over Z}}\, {\ts \det\left[
   a_k\delta_{k,p} -{ig\over \sqrt{L^d}}
   {\gamma_{k-p}} 
    \right]^{-{1\over 2}}} e^{-{1\over 2}v_0^2}dv_0 \pro_{q\in \M^+} 
   e^{-|\gamma_q|^2} {\ts {d\gamma_q 
  d\bar\gamma_q}} \eqno (4.49)$$
and $\M^+$ again is a set such that either $q\in \M^+$ or 
  $-q\in \M^+$. Furthermore 
$$a_k= 4M^2\sum_{i=1}^d\ts  \sin^2\left[ {k_i\over 2M}\right] \;+\; 
  m^2  \eqno (4.50)$$

\medskip
Equation (4.48) is our starting point. We apply (1.3) to the inverse matrix 
 element in (4.48). In the two loop 
 approximation one obtains $(\gamma_0=v_0\in\Bbb R)$
$${\ts \left[a_k\delta_{k,p} -{ig\over \sqrt{L^d}}
   {\gamma_{k-p}} 
    \right]^{-1}_{kk} }\approx {1\over a_k
  -{igv_0\over \sqrt{L^d}} +{g^2\over L^d}\sum_{p\ne k}G_k(p)|\gamma_{k-p}|^2 }
  =: {1\over a_k+\sigma_k}  \eqno (4.51)$$
where 
$$\sigma_k =-{\ts {ig\over \sqrt{L^d}}}v_0+
  {\ts {g^2\over L^d}}\sum_{p\ne k}{ |\gamma_{k-p}|^2
  \over a_p-{igv_0\over \sqrt{L^d}} +\sigma_p} \eqno (4.52) $$
which results in 
$$\la G\ra(k)= {1\over a_k +\la\sigma_k\ra } \eqno (4.53)$$
where $\la\sigma_k\ra$ has to satisfy the equation 
\setcounter{equation}{53}
\beqn
 \la\sigma_k\ra & =&
   -{\ts {ig\over \sqrt{L^d}}}\la v_0\ra +{\ts {g^2\over L^d}}
   \sum_{p\ne k}{ \la |\gamma_{k-p}|^2\ra
  \over a_p +\la\sigma_p\ra }\nonumber  \\
 &=& {\ts {g^2\over 2 L^d}}\sum_{p} \la G\ra(p) +{\ts {g^2\over L^d}}
   \sum_{p\ne k}{ \la |\gamma_{k-p}|^2\ra
  \over a_p +\la\sigma_p\ra } 
 ={\ts {g^2\over L^d}}
   \sum_{p\ne k}{ \la |\gamma_{k-p}|^2\ra +{1\over 2}
  \over a_p +\la\sigma_p\ra } 
\eeqn
where the last line is due to 
\beqn
  \la v_0\ra&=&{\ts {1\over Z}}\int v_0\> 
  {\ts \det\left[
   a_k\delta_{k,p} -{ig\over \sqrt{L^d}}
   {\gamma_{k-p}} 
    \right]^{-{1\over 2}}} e^{-{1\over 2}v_0^2}dv_0 \pro_{q\in \M^+} 
   e^{-|\gamma_q|^2} {\ts {d\gamma_q 
  d\bar\gamma_q}}   \nonumber \\ 
 &=&{\ts {1\over Z}}\int \Bigl\{ {\ts {\pt\over \pt v_0}}
   {\ts \det\left[
   a_k\delta_{k,p} -{ig\over \sqrt{L^d}}
   {\gamma_{k-p}} 
    \right]^{-{1\over 2}}}\Bigr\} e^{-{1\over 2}v_0^2}dv_0 \pro_{q\in \M^+} 
   e^{-|\gamma_q|^2} {\ts {d\gamma_q 
  d\bar\gamma_q}}   \nonumber\\ 
 &=&{\ts -{1\over 2}}\sum_p {\ts (-{ig\over \sqrt{L^d}}) }\int \ts
   \left[
   a_k\delta_{k,p} -{ig\over \sqrt{L^d}}
   {\gamma_{k-p}} 
    \right]^{-1}_{pp} dP(\gamma) 
\eeqn

\smallskip
As for the Many-Electron system, we can derive an equation 
 for $\la|\gamma_q|^2\ra$ by partial integration:
\beqn
  \la|\gamma_q|^2\ra&=&{\ts {1\over Z}} \int 
  \gamma_q \bar\gamma_q\>  {\ts \det\left[
   a_k\delta_{k,p} -{ig\over \sqrt{L^d}}
   {\gamma_{k-p} } 
    \right]^{-{1\over 2}}} e^{-{v_0^2\over 2}}dv_0 \pro_{q} 
   e^{-|\gamma_q|^2} {\ts {d\gamma_q 
  d\bar\gamma_q}}   \nonumber \\ 
 &=&\;1\;+\; {\ts {1\over Z}} \int 
  \gamma_q {\ts{\partial\over \partial \gamma_q}}\biggl\{
    {\ts \det\left[
  a_k \delta_{k,p} -{ig\over \sqrt{L^d}}
   {\gamma_{k-p}} 
    \right]^{-{1\over 2}}}  \biggr\}
   e^{-{v_0^2\over 2}}dv_0 \pro_{q} 
   e^{-{1\over 2}|\gamma_q|^2} {\ts {d\gamma_q 
  d\bar\gamma_q}}  \nonumber\\  
 &=&\;1\;-\;{\ts{1\over2}}\>  \int 
  \gamma_q\,{ {\partial\over \partial \gamma_q}
    {\ts \det\left[
   a_k \delta_{k,p} -{ig\over \sqrt{L^d}}
   {\gamma_{k-p}}  \right]} \over 
 {\ts \det\left[
   a_k\delta_{k,p} -{ig\over \sqrt{L^d}}
   {\gamma_{k-p}} \right] }  }  \,dP(\gamma) \nonumber\\  
 &=&\;1\;-\;{\ts{1\over2}}\>\sum_p {\ts {-ig\over \sqrt{L^d}}}
   \int \ts \gamma_q\, \left[ a_k\delta_{k,p} -{ig\over \sqrt{L^d}}
   {\gamma_{k-p}}  \right]_{p,p+q}^{-1}
    \,dP(\gamma) 
\eeqn
Computing the inverse matrix element in (4.56) again in the two loop 
approximation, one arrives at 
$$\la |\gamma_q|^2\ra=1-\la |\gamma_q|^2\ra 
  {\ts {g^2\over 2 L^d}} \sum_p \ts {1\over 
   (a_p+\la\sigma_p\ra)(a_{p+q}+\la\sigma_{p+q}\ra)}$$
or  
$$\la |\gamma_q|^2\ra={1\over 1+
   {\ts {g^2\over 2}} \int_{[0,2\pi M]^d} {d^dp\over (2\pi)^d}\, {1\over 
   (a_p+\la\sigma_p\ra)(a_{p+q}+\la\sigma_{p+q}\ra)}  }  \eqno (4.57)$$
which has to be solved in conjunction with 
$$\la\sigma_k\ra  =\ts {g^2}
   \int_{[0,2\pi M]^d}\ts   {d^dp\over (2\pi)^d}\, 
   { \la |\gamma_{k-p}|^2\ra+{1\over2} \over a_p +\la\sigma_p\ra }
 \eqno (4.58) $$
Introducing the rescaled quantities 
$$\la \sigma_k\ra= M^2 s_{p\over M}\,,\;\;\;\;
  \la |\gamma_q|^2\ra=\lambda_{q\over M}\,,\;\;\;\;
   a_k=M^2 \vep_{k\over M}\,,\;\;\vep_k=\sum_{i=1}^d \sin^2\ts{k_i\over2}
  +{m^2\over M^2}\eqno (4.60)$$
 (4.57,58) read 
\setcounter{equation}{60}
\beqn
   s_k & =&\ts M^{d-4} {g^2}
   \int_{[0,2\pi ]^d}\ts   {d^dp\over (2\pi)^d}\, 
   { \lambda_{k-p}+{1\over2} \over \vep_p + s_p}  \\
 \lambda_q&=&{1\over 1+
   {\ts M^{d-4}{g^2\over 2}} \int_{[0,2\pi ]^d} {d^dp\over (2\pi)^d}\,
   {1\over 
   (\vep_p+ s_p)(\vep_{p+q}+s_{p+q})}  } 
\eeqn
Unfortunately we cannot check this result with the rigorously proven 
 triviality theorem since 
$\la\sigma_k\ra$ and $\la|\gamma_q|^2\ra$ only 
 give information on the 2-point function $S(x,y)$, (4.41), and 
 on ${g^2\over M^d}\sum_x \la \vp(x)^4\ra=\sum_q\Lambda(q)$ where 
 $\Lambda(q)=\la|\gamma_q|^2\ra-1$. However, the triviality theorem [F,FFS] 
 makes a statement on the connected 4-point function 
 $S_{4,c}(x_1,x_2,x_3,x_4)$ at noncoinciding 
 arguments, namely that this function vanishes in the continuum limit 
 in dimension $d>4$. 

\bigskip
We now include the higher loop terms of (1.3,4) and give an interpretation in terms 
 of diagrams. 
The exact equations for $\la G\ra(k)$ and $\la |\gamma_q|^2\ra$ are 
\setcounter{equation}{62}
\beqn
  \la G\ra (k)&=& \int \ts \left[
   a_k\delta_{k,p} -{ig\over \sqrt{L^d}}
   {\gamma_{k-p} }  \right]^{-1}_{kk}  dP(\gamma)
  \;=\;\left\la {1\over a_k+\sigma_k}
    \right\ra  \\
 \sigma_k&=& -{\ts {ig\over \sqrt{L^d}}}\, v_0 +\sum_{r=2}^{N^d}  
  {\ts \left({ig\over \sqrt{L^d}}\right)^r}\!\!\!\!
  \sum_{p_2\cdots p_r\ne k\atop p_i\ne p_j} 
  G_k(p_2)\cdots G_{kp_2\cdots p_{r-1}}(p_r)\,\gamma_{k-p_2}\gamma_{p_2-p_3}\cdots 
  \gamma_{p_r-k} \nonumber 
\eeqn
and 
\beqn
  \la |\gamma_q|^2\ra&=&1+{\ts {ig\over 2\sqrt{L^d}}}\sum_p 
   \int \ts \gamma_q \left[
   a_k\delta_{k,p} -{ig\over \sqrt{L^d}}
   {\gamma_{k-p} }  \right]^{-1}_{p,p+q} dP(\gamma) \\
 &\buildrel p\to p_2\over=& 1+{\ts {1\over 2}}
   \sum_{r=2}^{N^d} \left( {\ts {ig\over\sqrt{L^d}}}\right)^r 
   \!\!\!\!\!\sum_{p_2\cdots p_r\ne p_2+q\atop p_i\ne p_j}  \Bigl\la 
  G(p_2)G_{p_2}(p_3) 
   \cdots G_{p_2\cdots p_{r-1}}(p_r) G_{p_2\cdots p_r}(p_2+q)\times 
  \nonumber \\ 
 & &\phantom{mmmmmmmmmmmmmmmm } 
 \gamma_{p_2-p_3}\cdots \gamma_{p_{r-1}-p_r}\gamma_{p_r-p_2-q}\gamma_{p_2+q-p_2}
  \Bigr\ra  \nonumber
\eeqn
For $r>2$, we obtain terms $\la \gamma_{k_1}\cdots\gamma_{k_r}\ra$ 
whose connected contributions 

\begin{figure}[thb]
 \centerline{\epsfbox{bild4.eps}}
 \caption{three and higher loop contributions}
 \label{Figure 4}
\end{figure} 
\noindent 
are, in terms of the electron or $\vp^4$-lines, are at least six-legged. 
Since for the many-electron system and for the $\vp^4$-model (for $d=4$) 
 the relevant  diagrams are two- and four-legged [FT,R], one may start with an
approximation which ignores the connected $r$-loop contributions for $r>2$. 
This is obtained by  writing 
$$\bigl\la\gamma_{k_1}\cdots \gamma_{k_{n}} \bigr\ra \approx 
  \bigl\la\gamma_{k_1}\cdots \gamma_{k_{n}} \bigr\ra_2 \eqno (4.65)$$
where (the index `2' for `retaining only two-loop contributions')
$$\bigl\la\gamma_{k_1}\cdots \gamma_{k_{2n}} \bigr\ra_2
  :=\sum_{{\rm pairings}\;\sigma} \la\gamma_{k_{\sigma 1}}
   \gamma_{k_{\sigma 2}} \ra\cdots \la\gamma_{k_{\sigma (2n-1)}}
   \gamma_{k_{\sigma 2n}} \ra  =\int \gamma_{k_1}\cdots \gamma_{k_{2n}} \,
  dP_2(\gamma) \eqno (4.66) $$
if we define 
$$dP_2(\gamma):= e^{-\sum_q{|\gamma_q|^2\over \la |\gamma_q|^2\ra} } 
  \pro_q \ts {d\gamma_q d\bar\gamma_q\over \pi\, \la |\gamma_q|^2\ra } 
  \eqno (4.67)$$
Substituting $dP$ by $dP_2$ in (4.63,64), we obtain a model which differs 
 from the original model only by irrelevant contributions and for which we
are able to write down a closed set of equations for the two-legged 
 particle correlation function $\la G\ra (k)$ and the two-legged squiggle
 correlation function $\la |\gamma_q|^2\ra$ by resumming all two-legged 
 (particle and squiggle) subdiagrams. The exact equations of this model are
\setcounter{equation}{67} 
\beqn
  \la G\ra (k)&=& \int \ts \left[
   a_k\delta_{k,p} -{ig\over \sqrt{L^d}}
   {\gamma_{k-p} }  \right]^{-1}_{kk}  dP_2(\gamma) \\
 \la |\gamma_q|^2\ra&=&1+{\ts {ig\over 2\sqrt{L^d}}}\sum_p 
   \int \ts \gamma_q \left[
   a_k\delta_{k,p} -{ig\over \sqrt{L^d}}
   {\gamma_{k-p} }  \right]^{-1}_{p,p+q} dP_2(\gamma)
\eeqn
and  the resummation 
 of the two-legged particle and squiggle subdiagrams is 
 obtained by applying the inversion formula (1.3,4) to the 
 inverse matrix elements in (4.68,69). A discussion similar to those of  
 section 2 gives the following closed set of equations for the 
 quantities $\la G\ra(k)$ and $\la |\gamma_q|^2\ra$: 
$$\la G\ra(k)= {1\over a_k+\la \sigma_k\ra}\,,\;\;\;\;\;\;
  \la |\gamma_q|^2\ra={1\over 1+\la\pi_q\ra}  \eqno (4.70)$$
where 
\setcounter{equation}{70} 
\beqn
 \la \sigma_k\ra&=&{\ts {g^2\over 2L^d}}\sum_p\la G\ra(p)
 +\sum_{r=2}^{\ell}  
  {\ts \left({ig\over \sqrt{L^d}}\right)^r}\!\!\!\!
  \sum_{p_2\cdots p_r\ne k\atop p_i\ne p_j} 
  \la G\ra(p_2)\cdots \la G\ra (p_r)\,
  \la\gamma_{k-p_2}\gamma_{p_2-p_3}\cdots 
  \gamma_{p_r-k}\ra_2  \nonumber\\
 & & \\
 \la\pi_q\ra&=&-{\ts {1\over 2}}\sum_{r=2}^\ell 
  {\ts \left({ig\over \sqrt{L^d}}\right)^r} \sum_{s=3}^{r-1} 
  \sum_{p_2\cdots p_r\ne p_2+q\atop p_i\ne p_j} \!\!\!
 \Bigl( \delta_{q,p_{s+1}-p_s}\, \la G\ra(p_2)\cdots \la G\ra(p_r)\,
   \la G\ra(p_2+q)\times  \nonumber \\
 & &\phantom{mmmmmmmmmmmmmmmmm  }
  \la \gamma_{p_2-p_3}\cdots 
   \widehat{\gamma}_{p_s-p_{s+1}}  \cdots \gamma_{p_{r-1}-p_r} 
   \gamma_{p_r-p_2-q}\ra_2 \Bigr)  \nonumber\\ 
 & &  
\eeqn
In the last line we used that $\gamma_q$ in (4.64) cannot 
 contract to $\gamma_{p_2-p_3}$ or to $\gamma_{p_r-p_2-q}$. 
 If the expectations of the $\gamma$-fields on the right hand side of 
 (4.71,72) are computed according to (4.66), one obtains the expansion 
 into diagrams. The graphs contributing to $\la \sigma_k\ra$ 
have exactly one string of 
 particle lines, each line having $\la G\ra$ as propagator, 
  and no particle loops (up to the tadpole diagram). 
 Each squiggle corresponds to a factor $\la |\gamma|^2\ra$. 
 The diagrams contributing to $\la \pi\ra$ have exactly one 
 particle loop, the propagators being again the interacting 
 two point functions, $\la G\ra$ for the particle lines and 
 $\la|\gamma|^2\ra$ for the squiggles. 
 In both cases there are no two-legged subdiagrams. 
 However, although the equation 
 $\la |\gamma_q|^2\ra={1\over 1+\la\pi_q\ra}$ resums ladder 
 or bubble  diagrams 
 (which is apparent from (4.57) or (4.26)) and 
 more general four-legged particle subdiagrams if the terms for $r\ge 4$ 
  in (4.72) are taken into account, the right hand side 
 of (4.71,72) still contains diagrams with four-legged particle 
 subdiagrams. Thus, the resummation of four-legged particle subdiagrams 
 is only partially through the complete resummation of two-legged 
 squiggle diagrams.     

Finally  observe that, in going from (4.68,69) to (4.70-72), 
  we cut off the $r$-sum at some 
 fixed order $\ell$ independent of the volume since we can only expect 
 that the expansions are asymptotic ones, 
   compare the discussion in section 2. 

\bigskip
\bigskip
\bigskip
\noindent{\Mittel References} 
\bigskip
\bitem
\item[{[AG]}]  M. Aizenman, G.M. Graf, {\it Localization 
  Bounds for an Electron Gas}, Journ. Phys. A, Bd. 31, S. 6783, 1998.
\item[{[B]}] N. N. Bogoliubov, {\it Lectures on Quantum Statistics}, Vol.2, 
    Gordon and Breach, 1970
\item[{[BFS]}] V. Bach, J. Fr\"ohlich, and I. M. Sigal, 
  {\it Renormalization group analysis of spectral problems 
      in quantum field theory}, 
  Adv. in Math. 137, 205-298, 1998.
\item[{[CFS]}]  T. Chen,  J. Fr\"ohlich,  M. Seifert, {\it Renormalization Group 
  Methods: Landau-Fermi Liquid and BCS Superconductor}, Proceedings of the 
 Les Houches session {\it Fluctuating Geometries in Statistical Mechanics 
 and Field Theory}, eds. F. David, P. Ginsparg, J. Zinn-Justin, 1994. 
\item[{[F]}] J. Fr\"ohlich, {\it On the Triviality of $\lambda\vp_d^4$ Theories 
  and the Approach to the Critical Point}, Nucl. Physics B200, 281-296, 1982. 
\item[{[FFS]}] R. Fernandez, J. Fr\"ohlich, A. Sokal, {\it Random Walks, 
    Critical Phenomena and Triviality in Quantum Field Theory}, Texts and Mongraphs 
    in Physics, Springer 1992.
\item[{[FKT]}]  J. Feldman, H. Kn\"orrer, E. Trubowitz, {\it Mathematical Methods of 
  Many Body Quantum Field Theory}, Lecture Notes, ETH Z\"urich.
\item[{[FMRT]}] J. Feldman, J. Magnen, V. Rivasseau, E. Trubowitz, {\it Ward Identities 
 and a Perturbative Analysis of a U(1) Goldstone Boson in a Many Fermion 
 System}, Helvetia Physica Acta 66, 1993, 498-550.
\item[{[FT]}] J. Feldman, E. Trubowitz,  {\it Perturbation Theory for Many Fermion 
   Systems}, Helvetia Physica Acta 63,  p.156-260, 1990; 
     {\it The Flow of an Electron-Phonon 
    System to the Superconducting State}, Helv. Phys. Acta 64, p.214-357, 1991.
\item[{[K]}] A. Klein, {\it The Supersymmetric Replica Trick and Smoothness 
  of the Density of States for Random Schr\"odinger Operators}, Proceedings of 
 Symposia in Pure Mathematics, vol. 51, part 1, 1990.
\item[{[L1]}]  D. Lehmann, {\it The Many-Electron System in the Forward, 
 Exchange and BCS Approximation}, Comm. Math.  
Phys. 198, 427-468, 1998.
\item[{[L2]}] D. Lehmann, {\it The Global Minimum of the Effective Potential 
  of the Many-Electron System with
   Delta-Interaction}, to appear in Rev. Math. Phys. Vol. 12, No 9, Sept. 2000.
\item[{[MPR]}] J. Magnen, G. Poirot, V. Rivasseau, 
  {\it Ward Type Identities for the 2D Anderson Model at Weak Disorder}, cond-mat/9801217; 
  {\it The Anderson Model as a Matrix Model}, Nucl. Phys. B (Proc. Suppl.) 58, S.149, 1997. 
\item[{[P]}] G. Poirot, {\it Mean Green's Function of the 
  Anderson Model at Weak Disorder with an Infrared Cutoff}, cond-mat/9702111
\item[{[R]}] V. Rivasseau, {\it From Perturbative to Constructive
   Renormalization}, Princeton Univ. Press 1991.
\item[{[W]}]  Wei-Min Wang, {\it Supersymmetry, Witten Complex and Asymptotics for 
  Directional Lyapunov Exponents in $\mathbb Z^d$}, mp-arc/99-355; {\it Localization 
  and Universality of Poisson Statistics Near Anderson Transition}, mp-arc/99-473.


\eitem





\end{document}


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 \caption{Figure 4}
 \label{Figure 4}
\end{figure} 
\noindent
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gs  [90] 0 sd
n 6300.0 4515.2 450.2 178.1 -1.9 arc
gs col0 s gr
 gr
 [] 0 sd
% Arc
gs  [90] 0 sd
n 6982.7 4522.3 667.7 179.3 -1.9 arcn
gs col0 s gr
 gr
 [] 0 sd
% Arc
gs  [90] 0 sd
n 10012.5 4507.5 337.6 178.7 -1.3 arc
gs col0 s gr
 gr
 [] 0 sd
% Arc
gs  [90] 0 sd
n 11144.8 4500.2 330.2 177.4 -0.0 arc
gs col0 s gr
 gr
 [] 0 sd
% Polyline
n 900 4515 m 4275 4515 l gs col0 s gr 
% Polyline
n 5175 4500 m 8115 4515 l gs col0 s gr 
% Polyline
n 9015 4515 m 12150 4500 l gs col0 s gr 
% Polyline
 [90] 0 sd
n 9240 4515 m 9241 4514 l 9243 4512 l 9247 4509 l 9253 4503 l 9263 4495 l
 9275 4484 l 9290 4470 l 9309 4452 l 9332 4432 l 9358 4409 l
 9388 4382 l 9422 4352 l 9459 4320 l 9498 4286 l 9541 4249 l
 9585 4210 l 9632 4171 l 9680 4130 l 9729 4089 l 9779 4048 l
 9829 4007 l 9879 3967 l 9928 3928 l 9977 3890 l 10025 3854 l
 10073 3820 l 10119 3787 l 10163 3758 l 10206 3730 l 10248 3705 l
 10289 3683 l 10328 3663 l 10366 3646 l 10403 3632 l 10438 3620 l
 10473 3611 l 10508 3605 l 10541 3601 l 10575 3600 l 10609 3601 l
 10642 3604 l 10677 3610 l 10712 3619 l 10748 3630 l 10785 3644 l
 10823 3660 l 10863 3680 l 10903 3702 l 10946 3726 l 10989 3753 l
 11035 3782 l 11081 3814 l 11129 3848 l 11177 3883 l 11227 3920 l
 11277 3959 l 11328 3998 l 11379 4039 l 11429 4079 l 11479 4120 l
 11528 4160 l 11575 4199 l 11620 4237 l 11663 4273 l 11703 4308 l
 11740 4340 l 11774 4369 l 11805 4395 l 11832 4418 l 11855 4438 l
 11874 4455 l 11890 4469 l 11902 4480 l 11911 4488 l 11918 4494 l
 11922 4497 l 11924 4499 l 11925 4500 l gs col0 s gr  [] 0 sd
/Times-Roman-iso ff 240.00 scf sf
1725 5610 m
gs 1 -1 sc (\(a\) reducible) col0 sh gr
/Times-Roman-iso ff 240.00 scf sf
5175 5625 m
gs 1 -1 sc (\(b\) irrdeducible, no two-legged ) col0 sh gr
/Times-Roman-iso ff 240.00 scf sf
9045 5610 m
gs 1 -1 sc (\(c\) irreducible, with two-legged ) col0 sh gr
/Times-Roman-iso ff 240.00 scf sf
10530 5865 m
gs 1 -1 sc (subgraphs) col0 sh gr
/Times-Roman-iso ff 240.00 scf sf
6765 5850 m
gs 1 -1 sc (subgraphs) col0 sh gr
$F2psEnd
rs

---------------0003061656364
Content-Type: application/postscript; name="bild4.eps"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="bild4.eps"

%!PS-Adobe-2.0 EPSF-2.0
%%Title: bild4.eps
%%Creator: fig2dev Version 3.2 Patchlevel 1
%%CreationDate: Mon Feb 28 23:44:31 2000
%%For: root@seven (root)
%%Orientation: Portrait
%%BoundingBox: 0 0 263 34
%%Pages: 0
%%BeginSetup
%%EndSetup
%%Magnification: 0.5000
%%EndComments
/MyAppDict 100 dict dup begin def
/$F2psDict 200 dict def
$F2psDict begin
$F2psDict /mtrx matrix put
/col-1 {0 setgray} bind def
/col0 {0.000 0.000 0.000 srgb} bind def
/col1 {0.000 0.000 1.000 srgb} bind def
/col2 {0.000 1.000 0.000 srgb} bind def
/col3 {0.000 1.000 1.000 srgb} bind def
/col4 {1.000 0.000 0.000 srgb} bind def
/col5 {1.000 0.000 1.000 srgb} bind def
/col6 {1.000 1.000 0.000 srgb} bind def
/col7 {1.000 1.000 1.000 srgb} bind def
/col8 {0.000 0.000 0.560 srgb} bind def
/col9 {0.000 0.000 0.690 srgb} bind def
/col10 {0.000 0.000 0.820 srgb} bind def
/col11 {0.530 0.810 1.000 srgb} bind def
/col12 {0.000 0.560 0.000 srgb} bind def
/col13 {0.000 0.690 0.000 srgb} bind def
/col14 {0.000 0.820 0.000 srgb} bind def
/col15 {0.000 0.560 0.560 srgb} bind def
/col16 {0.000 0.690 0.690 srgb} bind def
/col17 {0.000 0.820 0.820 srgb} bind def
/col18 {0.560 0.000 0.000 srgb} bind def
/col19 {0.690 0.000 0.000 srgb} bind def
/col20 {0.820 0.000 0.000 srgb} bind def
/col21 {0.560 0.000 0.560 srgb} bind def
/col22 {0.690 0.000 0.690 srgb} bind def
/col23 {0.820 0.000 0.820 srgb} bind def
/col24 {0.500 0.190 0.000 srgb} bind def
/col25 {0.630 0.250 0.000 srgb} bind def
/col26 {0.750 0.380 0.000 srgb} bind def
/col27 {1.000 0.500 0.500 srgb} bind def
/col28 {1.000 0.630 0.630 srgb} bind def
/col29 {1.000 0.750 0.750 srgb} bind def
/col30 {1.000 0.880 0.880 srgb} bind def
/col31 {1.000 0.840 0.000 srgb} bind def

end
save
-88.0 159.0 translate
1 -1 scale
.9 .9 scale % to make patterns same scale as in xfig

% This junk string is used by the show operators
/PATsstr 1 string def
/PATawidthshow { 	% cx cy cchar rx ry string
  % Loop over each character in the string
  {  % cx cy cchar rx ry char
    % Show the character
    dup				% cx cy cchar rx ry char char
    PATsstr dup 0 4 -1 roll put	% cx cy cchar rx ry char (char)
    false charpath		% cx cy cchar rx ry char
    /clip load PATdraw
    % Move past the character (charpath modified the
    % current point)
    currentpoint			% cx cy cchar rx ry char x y
    newpath
    moveto			% cx cy cchar rx ry char
    % Reposition by cx,cy if the character in the string is cchar
    3 index eq {			% cx cy cchar rx ry
      4 index 4 index rmoveto
    } if
    % Reposition all characters by rx ry
    2 copy rmoveto		% cx cy cchar rx ry
  } forall
  pop pop pop pop pop		% -
  currentpoint
  newpath
  moveto
} bind def
/PATcg {
  7 dict dup begin
    /lw currentlinewidth def
    /lc currentlinecap def
    /lj currentlinejoin def
    /ml currentmiterlimit def
    /ds [ currentdash ] def
    /cc [ currentrgbcolor ] def
    /cm matrix currentmatrix def
  end
} bind def
% PATdraw - calculates the boundaries of the object and
% fills it with the current pattern
/PATdraw {			% proc
  save exch
    PATpcalc			% proc nw nh px py
    5 -1 roll exec		% nw nh px py
    newpath
    PATfill			% -
  restore
} bind def
% PATfill - performs the tiling for the shape
/PATfill { % nw nh px py PATfill -
  PATDict /CurrentPattern get dup begin
    setfont
    % Set the coordinate system to Pattern Space
    PatternGState PATsg
    % Set the color for uncolored pattezns
    PaintType 2 eq { PATDict /PColor get PATsc } if
    % Create the string for showing
    3 index string		% nw nh px py str
    % Loop for each of the pattern sources
    0 1 Multi 1 sub {		% nw nh px py str source
	% Move to the starting location
	3 index 3 index		% nw nh px py str source px py
	moveto			% nw nh px py str source
	% For multiple sources, set the appropriate color
	Multi 1 ne { dup PC exch get PATsc } if
	% Set the appropriate string for the source
	0 1 7 index 1 sub { 2 index exch 2 index put } for pop
	% Loop over the number of vertical cells
	3 index 		% nw nh px py str nh
	{			% nw nh px py str
	  currentpoint		% nw nh px py str cx cy
	  2 index show		% nw nh px py str cx cy
	  YStep add moveto	% nw nh px py str
	} repeat		% nw nh px py str
    } for
    5 { pop } repeat
  end
} bind def

% PATkshow - kshow with the current pattezn
/PATkshow {			% proc string
  exch bind			% string proc
  1 index 0 get			% string proc char
  % Loop over all but the last character in the string
  0 1 4 index length 2 sub {
				% string proc char idx
    % Find the n+1th character in the string
    3 index exch 1 add get	% string proe char char+1
    exch 2 copy			% strinq proc char+1 char char+1 char
    % Now show the nth character
    PATsstr dup 0 4 -1 roll put	% string proc chr+1 chr chr+1 (chr)
    false charpath		% string proc char+1 char char+1
    /clip load PATdraw
    % Move past the character (charpath modified the current point)
    currentpoint newpath moveto
    % Execute the user proc (should consume char and char+1)
    mark 3 1 roll		% string proc char+1 mark char char+1
    4 index exec		% string proc char+1 mark...
    cleartomark			% string proc char+1
  } for
  % Now display the last character
  PATsstr dup 0 4 -1 roll put	% string proc (char+1)
  false charpath		% string proc
  /clip load PATdraw
  neewath
  pop pop			% -
} bind def
% PATmp - the makepattern equivalent
/PATmp {			% patdict patmtx PATmp patinstance
  exch dup length 7 add		% We will add 6 new entries plus 1 FID
  dict copy			% Create a new dictionary
  begin
    % Matrix to install when painting the pattern
    TilingType PATtcalc
    /PatternGState PATcg def
    PatternGState /cm 3 -1 roll put
    % Check for multi pattern sources (Level 1 fast color patterns)
    currentdict /Multi known not { /Multi 1 def } if
    % Font dictionary definitions
    /FontType 3 def
    % Create a dummy encoding vector
    /Encoding 256 array def
    3 string 0 1 255 {
      Encoding exch dup 3 index cvs cvn put } for pop
    /FontMatrix matrix def
    /FontBBox BBox def
    /BuildChar {
	mark 3 1 roll		% mark dict char
	exch begin
	Multi 1 ne {PaintData exch get}{pop} ifelse  % mark [paintdata]
	  PaintType 2 eq Multi 1 ne or
	  { XStep 0 FontBBox aload pop setcachedevice }
	  { XStep 0 setcharwidth } ifelse
	  currentdict		% mark [paintdata] dict
	  /PaintProc load	% mark [paintdata] dict paintproc
	end
	gsave
	  false PATredef exec true PATredef
	grestore
	cleartomark		% -
    } bind def
    currentdict
  end				% newdict
  /foo exch			% /foo newlict
  definefont			% newfont
} bind def
% PATpcalc - calculates the starting point and width/height
% of the tile fill for the shape
/PATpcalc {	% - PATpcalc nw nh px py
  PATDict /CurrentPattern get begin
    gsave
	% Set up the coordinate system to Pattern Space
	% and lock down pattern
	PatternGState /cm get setmatrix
	BBox aload pop pop pop translate
	% Determine the bounding box of the shape
	pathbbox			% llx lly urx ury
    grestore
    % Determine (nw, nh) the # of cells to paint width and height
    PatHeight div ceiling		% llx lly urx qh
    4 1 roll				% qh llx lly urx
    PatWidth div ceiling		% qh llx lly qw
    4 1 roll				% qw qh llx lly
    PatHeight div floor			% qw qh llx ph
    4 1 roll				% ph qw qh llx
    PatWidth div floor			% ph qw qh pw
    4 1 roll				% pw ph qw qh
    2 index sub cvi abs			% pw ph qs qh-ph
    exch 3 index sub cvi abs exch	% pw ph nw=qw-pw nh=qh-ph
    % Determine the starting point of the pattern fill
    %(px, py)
    4 2 roll				% nw nh pw ph
    PatHeight mul			% nw nh pw py
    exch				% nw nh py pw
    PatWidth mul exch			% nw nh px py
  end
} bind def

% Save the original routines so that we can use them later on
/oldfill	/fill load def
/oldeofill	/eofill load def
/oldstroke	/stroke load def
/oldshow	/show load def
/oldashow	/ashow load def
/oldwidthshow	/widthshow load def
/oldawidthshow	/awidthshow load def
/oldkshow	/kshow load def

% These defs are necessary so that subsequent procs don't bind in
% the originals
/fill	   { oldfill } bind def
/eofill	   { oldeofill } bind def
/stroke	   { oldstroke } bind def
/show	   { oldshow } bind def
/ashow	   { oldashow } bind def
/widthshow { oldwidthshow } bind def
/awidthshow { oldawidthshow } bind def
/kshow 	   { oldkshow } bind def
/PATredef {
  MyAppDict begin
    {
    /fill { /clip load PATdraw newpath } bind def
    /eofill { /eoclip load PATdraw newpath } bind def
    /stroke { PATstroke } bind def
    /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def
    /ashow { 0 0 null 6 3 roll PATawidthshow }
    bind def
    /widthshow { 0 0 3 -1 roll PATawidthshow }
    bind def
    /awidthshow { PATawidthshow } bind def
    /kshow { PATkshow } bind def
  } {
    /fill   { oldfill } bind def
    /eofill { oldeofill } bind def
    /stroke { oldstroke } bind def
    /show   { oldshow } bind def
    /ashow  { oldashow } bind def
    /widthshow { oldwidthshow } bind def
    /awidthshow { oldawidthshow } bind def
    /kshow  { oldkshow } bind def
    } ifelse
  end
} bind def
false PATredef
% Conditionally define setcmykcolor if not available
/setcmykcolor where { pop } {
  /setcmykcolor {
    1 sub 4 1 roll
    3 {
	3 index add neg dup 0 lt { pop 0 } if 3 1 roll
    } repeat
    setrgbcolor - pop
  } bind def
} ifelse
/PATsc {		% colorarray
  aload length		% c1 ... cn length
    dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor
  } ifelse } ifelse
} bind def
/PATsg {		% dict
  begin
    lw setlinewidth
    lc setlinecap
    lj setlinejoin
    ml setmiterlimit
    ds aload pop setdash
    cc aload pop setrgbcolor
    cm setmatrix
  end
} bind def

/PATDict 3 dict def
/PATsp {
  true PATredef
  PATDict begin
    /CurrentPattern exch def
    % If it's an uncolored pattern, save the color
    CurrentPattern /PaintType get 2 eq {
      /PColor exch def
    } if
    /CColor [ currentrgbcolor ] def
  end
} bind def
% PATstroke - stroke with the current pattern
/PATstroke {
  countdictstack
  save
  mark
  {
    currentpoint strokepath moveto
    PATpcalc				% proc nw nh px py
    clip newpath PATfill
    } stopped {
	(*** PATstroke Warning: Path is too complex, stroking
	  with gray) =
    cleartomark
    restore
    countdictstack exch sub dup 0 gt
	{ { end } repeat } { pop } ifelse
    gsave 0.5 setgray oldstroke grestore
  } { pop restore pop } ifelse
  newpath
} bind def
/PATtcalc {		% modmtx tilingtype PATtcalc tilematrix
  % Note: tiling types 2 and 3 are not supported
  gsave
    exch concat					% tilingtype
    matrix currentmatrix exch			% cmtx tilingtype
    % Tiling type 1 and 3: constant spacing
    2 ne {
	% Distort the pattern so that it occupies
	% an integral number of device pixels
	dup 4 get exch dup 5 get exch		% tx ty cmtx
	XStep 0 dtransform
	round exch round exch			% tx ty cmtx dx.x dx.y
	XStep div exch XStep div exch		% tx ty cmtx a b
	0 YStep dtransform
	round exch round exch			% tx ty cmtx a b dy.x dy.y
	YStep div exch YStep div exch		% tx ty cmtx a b c d
	7 -3 roll astore			% { a b c d tx ty }
    } if
  grestore
} bind def
/PATusp {
  false PATredef
  PATDict begin
    CColor PATsc
  end
} bind def

% right30
11 dict begin
/PaintType 1 def
/PatternType 1 def
/TilingType 1 def
/BBox [0 0 1 1] def
/XStep 1 def
/YStep 1 def
/PatWidth 1 def
/PatHeight 1 def
/Multi 2 def
/PaintData [
  { clippath } bind
  { 32 16 true [ 32 0 0 -16 0 16 ]
	{<00030003000c000c0030003000c000c0030003000c000c00
	30003000c000c00000030003000c000c0030003000c000c0
	030003000c000c0030003000c000c000>}
     imagemask } bind
] def
/PaintProc {
	pop
	exec fill
} def
currentdict
end
/P2 exch def
1.1111 1.1111 scale %restore scale

/cp {closepath} bind def
/ef {eofill} bind def
/gr {grestore} bind def
/gs {gsave} bind def
/sa {save} bind def
/rs {restore} bind def
/l {lineto} bind def
/m {moveto} bind def
/rm {rmoveto} bind def
/n {newpath} bind def
/s {stroke} bind def
/sh {show} bind def
/slc {setlinecap} bind def
/slj {setlinejoin} bind def
/slw {setlinewidth} bind def
/srgb {setrgbcolor} bind def
/rot {rotate} bind def
/sc {scale} bind def
/sd {setdash} bind def
/ff {findfont} bind def
/sf {setfont} bind def
/scf {scalefont} bind def
/sw {stringwidth} bind def
/tr {translate} bind def
/tnt {dup dup currentrgbcolor
  4 -2 roll dup 1 exch sub 3 -1 roll mul add
  4 -2 roll dup 1 exch sub 3 -1 roll mul add
  4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb}
  bind def
/shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul
  4 -2 roll mul srgb} bind def
 /DrawEllipse {
	/endangle exch def
	/startangle exch def
	/yrad exch def
	/xrad exch def
	/y exch def
	/x exch def
	/savematrix mtrx currentmatrix def
	x y tr xrad yrad sc 0 0 1 startangle endangle arc
	closepath
	savematrix setmatrix
	} def

/$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def
/$F2psEnd {$F2psEnteredState restore end} def
%%EndProlog

$F2psBegin
10 setmiterlimit
n -1000 6047 m -1000 -1000 l 12130 -1000 l 12130 6047 l cp clip
 0.03150 0.03150 sc
15.000 slw
% Ellipse
n 4065 4515 450 450 0 360 DrawEllipse gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def
15.00 15.00 sc P2 [16 0 0 -8 241.00 271.00] PATmp PATsp ef gr PATusp gs col0 s gr

% Ellipse
n 8100 4515 456 456 0 360 DrawEllipse gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def
15.00 15.00 sc P2 [16 0 0 -8 509.60 270.60] PATmp PATsp ef gr PATusp gs col0 s gr

% Ellipse
n 10350 4515 30 30 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 10710 4515 30 30 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 11085 4515 30 30 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
 [90] 0 sd
n 2925 4515 m 3615 4515 l gs col0 s gr  [] 0 sd
% Polyline
 [90] 0 sd
n 4350 4905 m 4995 4905 l gs col0 s gr  [] 0 sd
% Polyline
 [90] 0 sd
n 7785 4185 m 7230 4170 l gs col0 s gr  [] 0 sd
% Polyline
 [90] 0 sd
n 7845 4920 m 7200 4920 l gs col0 s gr  [] 0 sd
% Polyline
 [90] 0 sd
n 8415 4185 m 9045 4185 l gs col0 s gr  [] 0 sd
% Polyline
 [90] 0 sd
n 8385 4920 m 9030 4920 l gs col0 s gr  [] 0 sd
% Polyline
n 2925 4530 m 2820 4665 l gs col0 s gr 
% Polyline
n 2940 4530 m 2820 4365 l gs col0 s gr 
% Polyline
 [90] 0 sd
n 4305 4125 m 4995 4125 l gs col0 s gr  [] 0 sd
% Polyline
n 4935 4140 m 5055 4005 l gs col0 s gr 
% Polyline
n 4965 4155 m 5040 4245 l gs col0 s gr 
% Polyline
n 4980 4920 m 5070 4800 l gs col0 s gr 
% Polyline
n 5063 4800 m 5078 4800 l gs col0 s gr
% Polyline
n 4995 4935 m 5085 5025 l gs col0 s gr 
% Polyline
n 5078 5025 m 5093 5025 l gs col0 s gr
% Polyline
n 7245 4185 m 7140 4290 l gs col0 s gr 
% Polyline
n 7245 4185 m 7140 4050 l gs col0 s gr 
% Polyline
n 7215 4935 m 7095 5025 l gs col0 s gr 
% Polyline
n 7200 4905 m 7095 4830 l gs col0 s gr 
% Polyline
n 9045 4200 m 9165 4065 l gs col0 s gr 
% Polyline
n 9075 4215 m 9165 4305 l gs col0 s gr 
% Polyline
n 9030 4935 m 9135 4815 l gs col0 s gr 
% Polyline
n 9045 4905 m 9135 4995 l gs col0 s gr 
$F2psEnd
rs
end

---------------0003061656364--
