\magnification=\magstep1
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\font\mittel=cmbx10 scaled \magstep0
\font\Gross=cmr12 scaled \magstep2
\font\Mittel=cmr12 scaled \magstep0
\overfullrule=0pt
%\input amssym
\def\Bbb#1{{\bf #1}}
\def\blacksquare{\bullet}
\baselineskip=12pt
\def\up{\uparrow}
\def\down{\downarrow}
\def\k{{\bf k}}
\def\n{{\bf n}}
\def\m{{\bf m}}
\def\p{{\bf p}}
\def\q{{\bf q}}
\def\x{{\bf x}}
\def\y{{\bf y}}
\def\I{{\rm I}}
\def\II{{\rm II}}
\def\III{{\rm III}}
\def\IV{{\rm IV}}
\def\ts{\textstyle}
\def\ds{\displaystyle}
\def\tr{\Delta}
\def\la{\langle}
\def\ra{\rangle}
\def\pro{\mathop\Pi}
\def\1cm{\hskip 1cm}
\def\1k{{\textstyle{1\over\kappa}}}
\def\db#1{{\ts{d^d\k\over (2\pi)^d}}}
\def\vp{\varphi}
\def\sl{g}
\def\slb{\sqrt{|\lambda|}\>}
\def\U{{\cal U}}
\def\V{{\cal V}}
\def\su{\mathop{\Sigma}}
\def\u{\underline}
\def\Eta#1{\u\zeta_{\phantom{.}\!#1}}
\def\Pf{{\rm Pf}}
\def\ep{\epsilon}
\def\O{O}

{
\nopagenumbers
\baselineskip=13pt
$$ $$
\vskip 2.4cm
\centerline{\Gross An Explicit Solution of the BCS Model and}
\centerline{\Gross Demonstration of Symmetry Breaking}
\bigskip
\bigskip
\centerline{by}
\bigskip
\bigskip
\centerline{\Mittel Detlef Lehmann}
\centerline{\Mittel University of British Columbia}
\centerline{\Mittel Department of Mathematics} 
\centerline{\Mittel Vancouver, B.C.}
\centerline{\Mittel V6T 1Z2, Canada}
\vskip 2cm 
\noindent{\bf Abstract:} The BCS model, 
whose partition function 
is given by $Tr e^{-\beta H(\lambda)}$ where 
$H(\lambda)={1\over L^d} \sum_{k,\sigma} e(k) a_{k\sigma}^+ a_{k\sigma}
 +{\lambda\over L^{3d}}\sum_{k,p} a_{k\uparrow}^+
 a_{-k\downarrow}^+ a_{-p\downarrow}a_{p\uparrow}$ is solved explicitely as 
 a quartic model without using mean field theory. 
The Greens functions and their generating functional 
can be computed explicitely in arbitrary dimension $d$ and for arbitrary 
energy momentum relation $e(k)$. Renormalization effects as well as 
 symmetry breaking can be seen explicitely.  
 The more general interaction ${1\over L^{3d}}
  \sum_{\sigma\tau} \sum_{k,p,q} 
  \sum_{l=0}^N\lambda_l \,\bar y_l(k') y_l(p')\,[\delta_{k,p}+\delta_{k,-p}+
  \delta_{q,0}]\,
  a_{k\sigma}^+ a_{q-k\tau}^+ a_{q-p\tau}a_{p\sigma}$ which 
contains a forward, an 
 exchange and a BCS term 
 can also be treated. For pure BCS, one obtains a  $6(N+1)$ dimensional integral 
 representation for the two point functions and for the generating functional. 
 The existence and symmetry of a gap is determined by the global minimum 
of an effective potential of $6(N+1)$ variables. We show, in 3 space 
 dimensions, that the usual mean field approach may be misleading if the 
electron electron interaction contains higher angular momentum terms. 
In particular, if the interaction is given by a single attractive even $l$
 term ${\lambda_l\over L^{3d}}\sum_{m=-l}^l Y_{l m}(k')\bar 
   Y_{l m}(p')\,a_{k\up}^+a_{-k\down}^+ a_{-p\down}a_{p\up}$, then, 
 if $e(k)$ has $SO(3)$ symmetry, the expectations $\langle a_{k\sigma}^+
   a_{k\sigma}\rangle$  also have to have $SO(3)$ symmetry. 
The property which makes the model explicitely solvable is 
the fact that the interaction is given by a finite sum of products of 
quadratic factors $\sum_{\sigma\tau}\sum_{l=0}^N 
   \bigl(\sum_k \bar y_l(k')\>a_{k\sigma}^+
 a_{-k\tau}^+\bigr)\bigl(\sum_p y_l(p') \>a_{-p\tau}a_{p\sigma}\bigr)$.
\vfill
\eject  }  
%
%  
%  CHAPTER I
%
%
\pageno=1
\noindent {\gross I. Introduction} 
\bigskip
In this paper, we present an explicit solution of the BCS model. The quartic
Hamiltonian is not approximated by a mean field Hamiltonian. The partition 
function of the BCS model in $d$ dimensions at temperature $T={1\over\beta} 
  >0$ is given by 
$$Z=Z(\beta,L,\lambda)=Tr\, e^{-\beta H(\lambda)} \eqno (\I.1)$$
where  
$$H(\lambda)=H_0+I={\ts {1\over L^d}} \sum_{\k,\sigma} e_\k \,a_{\k\sigma}^+ 
    a_{\k\sigma}+{\ts {\lambda\over L^d} {1\over L^{2d}} } 
   \sum_{\k,\p} a_{\k\up}^+ a_{-\k\down}^+ a_{\p\up} a_{-\p\down} \eqno (\I.2) $$
The spatial momenta range over some subset of 
 $\left( {2\pi\over L} \Bbb Z\right)^d$, say, $\k\in M=\{\k\in ({2\pi\over L} \Bbb Z)^d
  \>|\> |e_\k|\le \omega_D\}$ where $L^d$ is the spatial volume of the system 
and $e_\k$ denotes the energy momentum relation. We assume $e_\k= 
  e_{-\k}$. A positive $\lambda$ corresponds to an attractive interaction. 
\par
We also consider the more general interaction 
$$I\left(\{\lambda_\ell\}\right)={\ts {1\over L^{2d}} }
   \sum_{\sigma,\tau\in \{\up,\down\}} \sum_{\k,\p} \ts 
    {1\over L^d}U(\k'-\p')\, a_{\k\sigma}^+ a_{-\k\tau}^+ a_{\p\sigma} a_{-\p\tau} 
    \eqno (\I.3)$$
where, for $e_\k={\k^2\over 2m}-\mu$,  $\k'=\sqrt{2m\mu}{\k\over|\k|}=:
    k_F{\k\over|\k|}$ and 
$$U(\k'-\p')=\sum_{\ell=0}^n\lambda_\ell P_\ell\left({\ts{\k'\p'\over k_F^2}}
    \right)=\sum_{\ell=0}^n \sum_{m=-\ell}^\ell \lambda_\ell
   Y_{\ell m}\left({\ts{\k'\over k_F}}\right)
     \bar Y_{\ell m}\left({\ts{\p'\over k_F}}\right)\>,\;\;\; d=3\eqno (\I.4)$$
$$U(\k'-\p')=\sum_{\ell=0}^n \lambda_\ell 
    \cos\left[\ell(\varphi_\k-\varphi_\p)\right]=
  {1\over2} \sum_{\ell=-n}^n \lambda_{|\ell|}
     e^{i\ell\varphi_\k}e^{-i\ell\varphi_\p}
    +{\lambda_0\over2}\>,\;\;\; d=2 \eqno (\I.5)$$
which reduces to (I.2) for $n=0$. To simplify notation, 
 suppose that 
$$U(\k'-\p')=\sum_{l=0}^N\lambda_l\> y_l(\k')\>\bar y_l(\p') \eqno (\I.6)$$
where $l=(\ell,m)$ or $l=\ell$ depending on the dimension of space. 
\medskip
In section IV.2, by specializing Theorem III.1, in which a more general model 
 with forward, exchange and BCS interaction is solved, 
  we prove the following $6(N+1)$ dimensional 
 integral representation 
for the $\la a^+ a\ra$ expectation values: 
$$\eqalignno{   \la a_{p_0,\p,\up}^+ 
    a_{p_0',\p',\up} \ra_{\beta,L}&= { Tr\, 
    e^{-\beta[ H_0+I(\{\lambda_l\})] }a_{p_0,\p,\up}^+ 
    a_{p_0',\p',\up} \over Tr\, e^{-\beta [ H_0+I(\{\lambda_l\})]} }  \cr
 &=  \beta  L^d\delta_{p,p'} { \int^{\phantom{I}} F_{\p}^{\up\up}(\phi)\>
   e^{-\beta L^d V(\phi)}\ds \pro_{\sigma\tau} 
  \pro_{l=0}^N du_{\sigma\tau}^l dv_{\sigma\tau}^l \over 
   \int e^{-\beta L^d V(\phi)} \ds \pro_{\sigma\tau} 
  \pro_{l=0}^N du_{\sigma\tau}^l dv_{\sigma\tau}^l }&(\I.7)  \cr}$$
Here $\phi^l_{\sigma\tau}=u^l_{\sigma\tau}+i v^l_{\sigma\tau}$, $\sigma\tau\in 
  \{\up\up,\down\down,\up\down\}$,  
$$F_{\p}^{\up\up}(\phi)=\ts { (ip_0+e_\p) [ p_0^2+e_\p^2+ \Phi_{-\p\up\down} 
     \bar\Phi_{-\p\up\down}+\Phi_{\p\down\down}\bar\Phi_{\p\down\down}] 
  \over (p_0^2+e_\p^2+\Omega_\p^+)(p_0^2+e_\p^2+\Omega_\p^-) } $$
$$\Phi_{\k\up\down}=\sum_{l=0}^N\lambda_l^{1\over2}\phi^l_{\up\down} 
    \, y_l(\k')\>,\;\;\;\;
  \bar\Phi_{\k\up\down} 
   =\sum_{l=0}^N \lambda_l^{1\over2} \bar\phi^l_{\up\down} 
    \, \bar y_l(\k')\>,$$
$$\Phi_{\k\sigma\sigma}=\sum_{l=0}^N \lambda_l^{1\over2}
    \phi_{\sigma\sigma}^l\,{\ts [y_l(\k')-y_l(-\k')] } \>,\;\;\;
  \bar\Phi_{\k\sigma\sigma}=\sum_{l=0}^N \lambda_l^{1\over2} 
     \bar\phi_{\sigma\sigma}^l\,  {\ts [\bar y_l(\k')-\bar y_l(-\k')]} $$
and $\Omega^{\pm}_\k$ are the solutions of the quadratic equation 
$$\eqalignno{  \Omega^2-&\Bigl( \bar\Phi_{\k\up\down} 
  \Phi_{\k\up\down}+\bar\Phi_{-\k\up\down} \Phi_{-\k\up\down}
  +\Phi_{\k\up\up}\bar\Phi_{\k\up\up} 
  +\Phi_{\k\down\down}\bar\Phi_{\k\down\down}\Bigr)\Omega   
  +\Phi_{\k\up\up}\Phi_{\k\down\down}
    \bar\Phi_{\k\up\down}\bar\Phi_{-\k\up\down}  \cr  
 &+\bar\Phi_{\k\up\up}\bar\Phi_{\k\down\down}
    \Phi_{\k\up\down}\Phi_{-\k\up\down} 
   +\Phi_{\k\up\up}\Phi_{\k\down\down}
    \bar\Phi_{\k\up\up}\bar\Phi_{\k\down\down} 
  +\bar\Phi_{\k\up\down}  \Phi_{\k\up\down}
   \bar\Phi_{-\k\up\down}  \Phi_{-\k\up\down} =0 &(\I.8)  \cr}$$
Observe that $\bar\Phi_{\k\sigma\tau}$ is not necessarily the complex conjugate 
 of $\Phi_{\k\sigma\tau}$, depending on the signs of the coupling constants 
 $\lambda_l$.       
The effective potential $V_\beta$ is given by 
$$V_\beta(\phi )= 
   \sum_{l=0}^N\left(  |\phi_{\up\down}^l|^2+ |\phi_{\up\up}^l|^2+
    |\phi_{\down\down}^l|^2\right)   -\sum_{\epsilon\in\{+,-\}} 
    \int_M \ts 
   {d^d\k\over (2\pi)^d}\>{1\over\beta} 
    \log\left[ {\cosh({\beta\over2}\sqrt{ e_\k^2+\Omega^\epsilon_\k }) 
    \over \cosh {\beta\over 2} e_\k} \right]\; \eqno (\I.9) $$
\smallskip
For a pure even interaction, that is, if $\lambda_\ell=0$ for all odd angular 
 momentum $\ell$, one has $\Phi_{\k\up\up}=\Phi_{\k\down\down}=0$ and 
 the expectation value simplifies to 
$$\eqalignno{   \la a_{p_0,\p,\sigma}^+ 
    a_{p_0',\p',\sigma} \ra_{\beta,L}
 &=  \beta  L^d\delta_{p,p'} \> 
  { \int {ip_0+e_\p \over p_0^2+e_\p^2+\Phi_{\p\up\down}\bar\Phi_{\p\up\down}} 
    e^{ -\beta L^d V_\beta(\phi_{\up\down}) }
      \ds\pro_{l=0}^N du^l_{\up\down} dv^l_{\up\down}
   \over \int e^{ -\beta L^d V_\beta(\phi_{\up\down} ) } 
    \ds\pro_{l=0}^N du^l_{\up\down} dv^l_{\up\down} }\> &(\I.10)  \cr}$$
with an effective potential 
$$V_\beta(\phi_{\up\down})= \sum_{l=0}^N |\phi_{\up\down}^l|^2-
   \int_M \ts 
   {d^d\k\over (2\pi)^d}\>{1\over\beta} 
    \log\left[ {\cosh({\beta\over2}
     \sqrt{ e_\k^2+\Phi_{\k\up\down}\bar\Phi_{\k\up\down} }) 
    \over \cosh {\beta\over 2} e_\k} \right]^2\; \eqno (\I.11) $$
In particular, for a delta function interaction one obtains the two dimensional 
 integral representation 
$$\eqalignno{   \la a_{p_0,\p,\sigma}^+ 
    a_{p_0',\p',\sigma} \ra_{\beta,L}&=\beta L^d \delta_{p,p'}\> 
  { \int { ip_0+e_\p
    \over p_0^2+e_\p^2+\lambda(u^2+v^2)} \> e^{-\beta L^d 
   V_{\beta}(u,v)} dudv \over \int e^{-\beta L^d V_{\beta}(u,v)} dudv }
    & (\I.12) \cr}$$
where
$$V_{\beta}(u,v)=u^2+v^2-
     \int_M   \ts{d^d\k\over (2\pi)^d}\>{1\over\beta}  \log\left[ 
   {\cosh({\beta\over2}\sqrt{ e_\k^2+\lambda(u^2+v^2)})\over 
    \cosh {\beta\over2} e_\k}\right]^2 \eqno (\I.13)$$
A positive $\lambda$ corresponds to an attractive interaction. 
\medskip\goodbreak
Since the only place where the volume $L^d$ shows up in the 
 integral  representations (I.7,10) is the prefactor in the exponential 
 $e^{-\beta L^d V(\phi)}$, the computation of the infinite volume 
 limit comes down to the determination of the global minimum of the real part 
 of the effective potential $V$ as a function of the $\phi^l_{\sigma\tau}$'s. 
\bigskip
The expectation values $\la a^+a^+\ra$ and $\la a a\ra$ are also computed. 
 To make them nonzero,  
  we introduce a small external field $r=|r| e^{i\alpha}$. 
That is, we substitute $H\left(\{\lambda_l\}\right)$ by 
  (we do not consider here $\la a_\up a_\up \ra$ expectations)
$$H_r=H\left(\{\lambda_l\}\right)+{\ts {1\over L^d}} \sum_\k [ 
   r \,a_{\k\up} a_{-\k\down}-\bar r\, a_{\k\up}^+ a_{-\k\down}^+] \eqno (\I.15)$$
One obtains again a $6(N+1)$ dimensional integral representation. 
For a delta function interaction, that is for $N=0$ in (I.6),  it 
 reduces to a two dimensional integral: 
$$\eqalignno{   \la a_{p_0,\p,\up}^+ 
    a_{-p_0',-\p',\down}^+ \ra_{\beta,L}&=\beta L^d \delta_{p,p'}\> 
  { \int {-i\sqrt\lambda \> e^{i\alpha} 
   (u+iv) \over p_0^2+e_\p^2+\lambda(u^2+v^2)} \> e^{-\beta L^d 
   V_{\beta,r}(u,v)} dudv \over \int e^{-\beta L^d V_{\beta,r}(u,v)} dudv }
    & (\I.16) \cr}$$
where 
$$V_{\beta,r}(u,v)=u^2+{\ts \left( v+{|r|\over \sqrt \lambda}\right)^2 
   - } \int_M \ts {d^d\k\over (2\pi)^d}\>{1\over\beta} \log\left[ 
   {\cosh({\beta\over2}\sqrt{ e_\k^2+\lambda(u^2+v^2)})\over 
    \cosh {\beta\over2} e_\k}\right]^2 \eqno (\I.17)$$
\par
For attractive $\lambda>0$ and sufficiently small $T={1\over\beta}$, 
  the potential $V_{\beta,r}$ has the 
 form of a mexican hat. 
For $r=0$, the global minimum of $V_{\beta,r}$ is degenerated and lies on a 
 circle in the $u,v$ plane. In particular, $V_{\beta,0}$  is an even function 
 of $u$ and $v$ and 
 $\la a_{p\up}^+a_{-p\down}^+\ra$ vanishes by symmetry. 
\par
For $r\ne 0$,   $V_{\beta,r}$  has a unique 
global minimum at $(u,v)=(0,v_0)$ where $v_0$ is given by the negative solution
of 
$$ v_0\left\{ \lambda \int_M \ts {d^d\k\over (2\pi)^d} 
   {\tanh({\beta\over2}\sqrt{e_\k^2+\lambda v_0^2})\over 2\sqrt{ e_\k^2+\lambda 
   v_0^2}}-1 \right\}=2|r|  \eqno (\I.18)$$
which, in the limit $r\to 0$,  becomes the BCS equation 
 for $|\tr|^2=\lambda v_0^2$. Thus 
$$\lim_{|r|\to 0}\lim_{L\to\infty} {\ts {e^{-\beta L^d V_{\beta,r}(u,v)} \over 
   \int e^{-\beta L^d V_{\beta,r}(u,v)} dudv }  }=
   \lim_{|r|\to 0}\lim_{L\to\infty} \ts {e^{-\beta L^d
      [V_{\beta,r}(u,v)-V_{\beta,r}(0,v_0)] } \over 
   \int e^{-\beta L^d [V_{\beta,r}(u,v)-V_{\beta,r}(0,v_0)]} dudv }
  =\delta(u)\delta\left(v+{|\tr|\over \sqrt\lambda}\right)  \eqno (\I.19) $$
and $\la a_{p\up}^+a_{-p\down}^+\ra$ becomes nonzero. 
\par
For repulsive $\lambda<0$, $V_{\beta,r}$ is complex and the real part of 
 $V_{\beta,r}$ has a unique global minimum at $(u,v)=(0,0)$ which results in 
$$\lim_{|r|\to 0}\lim_{L\to\infty}\ts {e^{-\beta L^d V_{\beta,r}(u,v)}\over \int 
   e^{-\beta L^d V_{\beta,r}(u,v)}dudv }=\delta(u)\delta(v)$$
and $\lim_{|r|\to 0}\lim_{L\to\infty}\la a_{p\up}^+a_{-p\down}^+\ra=0$. 
\bigskip
The starting point for our analysis is the perturbation series for the partition 
 function which in terms of fermionic functional integrals is given by 
 (for $N=0$) 
$$\eqalignno{ Z(\beta,L,\lambda)&={  Tr\,e^{
  -\beta\bigl[ {\ts {1\over L^d}}\sum_{\k,\sigma} e_\k\,a_{\k,\sigma}^+ 
  a_{\k,\sigma} +{\ts {\lambda\over L^d}{1\over L^{2d}}} \sum_{\k,\p} 
   a_{\k\up}^+a_{-\k\down}^+ a_{\p\up}a_{-\p\down} \bigr] }   \over   
  Tr\,e^{
  -\beta\bigl[ {\ts {1\over L^d}}\sum_{\k,\sigma} e_\k\,a_{\k,\sigma}^+ 
  a_{\k,\sigma}\bigr] }   }   \cr
 & \cr
  &=\int e^{-{\lambda\over (\beta L^d)^3}  \sum_{k,p,q_0} 
    \bar\psi_{k\up} \bar\psi_{q_0-k\down} \psi_{p\up} \psi_{q_0-p\down} } 
   d\mu_C(\psi,\bar\psi) &(\I.20) \cr}$$
where 
$$d\mu_C(\psi,\bar\psi)=\pro_{k,\sigma}{\ts  {\beta L^d\over ik_0-e_\k}}\; 
   e^{-{1\over \beta L^d} \sum_{k,\sigma}(ik_0-e_\k) \bar\psi_{k,\sigma} 
   \psi_{k,\sigma} } \pro_{k,\sigma} d\psi_{k,\sigma} d\bar\psi_{k,\sigma} \eqno 
  (\I.21)$$
and $k=(k_0,\k)\in {\pi\over\beta}(2\Bbb Z+1)\times M$. 
 The variables $k_0$, $p_0$ and $q_0$  which do 
 not appear in the first line 
 of (I.20) are the Fourier transform variables of a variable $\tau$ which enters 
the perturbation series because of 
${d\over d\lambda}_{|_{\lambda=0}} 
     e^{-\beta(H_0+\lambda I)}=\int_0^\beta e^{-(\beta-\tau)H_0} 
   I e^{-\tau H_0} d\tau$. 
\par
The perturbation series for a $\delta$-function interaction without BCS 
approximation reads 
$$\int e^{-\lambda {1\over (\beta L^d)^3}\sum_{k,p,q}
   \bar\psi_{k\up}\bar\psi_{q-k\down}\psi_{p\up}\psi_{q-p\down} }
  d\mu_C(\psi,\bar\psi)  \eqno (\I.22)$$
Thus the BCS approximation consists in putting the spatial 
 transfer momentum $\q$ equal to zero. For $e_\k={\k^2\over 2m}-\mu$, 
Feldman and Trubowitz [FT] proved that the most singular contributions 
 to the   Feynman diagramms come from $q_0=0$. Therefore we put also 
 $q_0$ equal to zero which (obviously) makes the model explicitly solvable. 
 Thus as our starting point we choose an approximation where 
  $\q=0$ and $q_0=0$. 
\par
That is, the model we solve in this paper is not exactly the model defined by 
 (I.1) and (I.2) or (I.3), but is the model which is defined by the 
 perturbation series (for $N$=0)
$$ Z(\beta,L,\lambda)=\int e^{-{\lambda\over \beta L^d} {1\over (\beta L^d)^2}
   \sum_{k,p} \bar\psi_{k\up}\bar\psi_{-k\down}\psi_{p\up}\psi_{-p\down} } 
   d\mu_C(\psi,\bar\psi) \eqno (\I.23) $$
In particular, it does not come from a Hamiltonian. 
\bigskip
We start in section II.1 with a solution of the model with $\delta$-function 
 interaction. This is simply done using a standard method. 
 Namely, introduce a bosonic
 field $\phi$, integrate out the fermions and obtain a quotient of determinants 
 which gives the exponential of an effective potential. One just has to observe 
 that for $\q=0,\;q_0=0$ the bosonic field $\phi$ has to be only two dimensional 
 and that the determinant involving $\phi$ can be computed explicitely. 
 That is, use the identity ($\phi=u+iv$, $\bar\phi=u-iv\in \Bbb C$)
$$e^{2ab}=\ts {1\over 2\pi}\int_{\Bbb R^2} e^{a\phi+b\bar\phi}  e^{-{1\over2} 
    |\phi|^2} dudv  \eqno (\I.24)$$
to obtain 
$$\eqalignno{ Z&=
   \int e^{-\left({\lambda\over \beta L^d}\right)^{1\over2} {1\over \beta L^d}
   \sum_{k} \bar\psi_{k\up}\bar\psi_{-k\down}
     \left({\lambda\over \beta L^d}\right)^{1\over2}{1\over \beta L^d}
   \sum_p \psi_{p\up}\psi_{-p\down} } 
   d\mu_C(\psi,\bar\psi)   \cr
 &=\int_{\Bbb R^2}
    \int e^{i\left({\lambda\over\beta L^d}\right)^{1\over2} 
    \phi\, {1\over\beta L^d}\sum_k\psi_{k\up}\psi_{-k\down}\,+\,
   i\left({\lambda\over\beta L^d}\right)^{1\over2} 
   \bar\phi \,{1\over\beta L^d}\sum_k\bar\psi_{k\up}\bar\psi_{-k\down}  } 
   d\mu_C(\psi,\bar\psi) \ts {1\over 2\pi} e^{-{1\over2} 
    |\phi|^2} dudv  \cr
 &=\int_{\Bbb R^2} \pro_k{\ts{1\over k_0^2+e_\k^2}} \ts  \det\left(
 { ik_0-e_\k\;\;\;\; i
    \lambda^{1\over2}\bar\phi \atop 
   i\lambda^{1\over2} \phi \;\;\;\; -ik_0-e_\k} \right)
   \ts {\beta L^d\over \pi} e^{-\beta L^d |\phi|^2 } dudv  \cr
  &={\ts {\beta L^d\over \pi}}\int_{\Bbb R^2} e^{-\beta L^d V_\beta(u,v)} 
      dudv    &(\I.25) \cr}$$
\par
In the same way one obtains  integral representations for the two point 
 functions and even for the generating functional of the connected amputated 
 Greens functions. 
\medskip
In section II.2 we show that the partition function may  also be computed 
 by a direct summation of the perturbation series. Instead of using (I.24) and 
 integrating out the fermions, the perturbation series (I.23) 
$$Z(\beta,L,\lambda)=\sum_{n=0}^\infty { \left( {\lambda\over\kappa^3}\right)^n 
  \over n!} \sum_{k_1,\cdots,k_n}\sum_{p_1,\cdots,p_n} 
  \det\left[ \kappa\delta_{k_i,p_j}C(k_i)\right]_{1\le i,j\le n}  
 \det\left[ \kappa\delta_{k_i,p_j}C(-k_i)\right]_{1\le i,j\le n} $$ 
can be summed up explicitly by expanding the determinants 
   and by  an appropriate  
reordering of the resulting diagramms. Here $\kappa=\beta L^d$. 
\medskip
In section III we consider the model for arbitrary $N$ (see I.6). Furthermore, we 
 include a forward and an exchange scattering term in the interaction. That is, 
we consider the partition function $Z=\int e^{-{\cal U}(\psi,\bar\psi)} d\mu_C$ 
 where 
$$ {\cal U}(\psi,\bar\psi)={\ts {1\over \kappa^3}}
   \sum_{\sigma,\tau\in\{\up,\down\}}
    \sum_{k,p,q} U(\k'-\p')\bigl[\delta_{k,p}+\delta_{k,-p}
  +\delta_{q,0} \bigr] \>\bar\psi_{k,\sigma}
   \bar\psi_{{q}-k,\tau}\psi_{p,\sigma} 
  \psi_{{q}-p,\tau} \eqno (\I.26)  $$
This model can also be solved. The only difference is that more bosonic fields 
are needed. But still, the dimension of the resulting integral 
 representations does not depend on the volume or on other cuttoffs. 
\medskip
In the fourth section we specialize to the following cases. In IV.1 we consider 
a $\delta$-function interaction with a forward, an exchange and a BCS term.  
In section IV.2 we discuss the case of a pure BCS interaction for arbitrary $N$. 
That is, the interaction includes higher angular momentum terms. 
Using the integral representation (I.10), we prove in Theorem IV.2 that, in 3 
 dimensions, if the interaction is given by a single attractive even $\ell$ term 
 ${\lambda_\ell\over L^{3d}} \sum_{m=-\ell}^\ell \bar Y_{\ell m}(\k') Y_{\ell m} 
  (\p')\, a_{\k\up}^+ a_{-\k\down}^+ a_{-\p\down}a_{\p\up}$, then, if $e_\k$ 
 has $SO(3)$ symmetry, the expectations $\la a_{\k\sigma}^+ a_{\k\sigma}\ra$ 
 also have  to have $SO(3)$ symmetry. In conjunction with a result of Feldman, 
 Kn\"orrer and Trubowitz [FKT], who proved that the BCS $2\times 2$ matrix 
 gap equation [AB,BW] does not have unitary isotropic solutions for $\ell\ge 2$, 
this indicates that in 3 dimensions for $\ell\ge 2$ the standard mean field 
 approach may be misleading. 
\bigskip
Finally one may hope that the method presented here turns out to be useful 
 also for other models, including bosonic ones. The strategy would be to 
 write down the perturbation expansion in fermionic or bosonic functional 
 integral form and to approximate all quartic terms by a sum of type (I.26). Then 
 one can use (I.24) to transform each quartic term into a quadratic one 
 and the fermionic or bosonic functional integral can be performed. One ends 
 up with a finite dimensional integral, the dimension $D$ being two times 
the number of quartic terms in the approximation  (I.26), where the 
 integrand, an infinite product over momenta, gives the exponential of an 
 effective potential. Then if, as it is the case for the BCS model, the cuttoffs 
 (here volume) only show up as prefactors of the effective potential, then 
 removing the cuttoffs is equivalent to finding the global minimum of the real 
 part of $V$ which is a function of $D/2$ complex variables. 
\bigskip
\bigskip
\noindent{\bf Acknowledgements} 
\bigskip
I am grateful to Horst Kn\"orrer and Eugene Trubowitz and to the 
 Forschungsinstitut f\"ur Mathematik at ETH Z\"urich for the hospitality and 
 the support during the summer 1996. Furthermore I would like to thank 
 Joel Feldman who made it possible for me to visit the University of British 
 Columbia in Vancouver in the academic year 1996/97. 
\vfill
\eject   
%
%
% Chapter 2
%
%
\noindent{\gross II. The Model with Delta Function Interaction}
\bigskip
\bigskip
\bigskip
\noindent{\mittel II.1 Computation of the Two Point Functions 
   and the Generating}
\par
\noindent$\phantom{II.1.. }${\mittel  Functional }
\bigskip
\bigskip
In this section, we compute the partition function 
$$\eqalignno{
    Z(\beta,L,\{s_k\},\{r_k\})&=\int \exp\biggl\{-{\ts  {\lambda\over\kappa}
  {1\over\kappa^2}}  \sum_{k,p} \psi_{k,\up}\psi_{-k,\down}
    \bar\psi_{p,\up}\bar\psi_{-p,\down} 
     + {\ts {1\over\kappa}}\sum_k[s_{k,\up}\bar\psi_{k,\up}\psi_{k,\up}  \cr
  &+
    s_{k,\down}\bar\psi_{k,\down}\psi_{k,\down}   
   +r_k\psi_{k,\up}\psi_{-k,\down}
   -\bar r_k\bar\psi_{k,\up}\bar\psi_{-k,\down} ]  \biggr\} 
   d\mu_C(\psi,\bar\psi) &(\II.1)  \cr}$$ 
for arbitrary numbers  $s_{k,\up},s_{k,\down},r_k,\bar r_k$ and the two point
functions 
$$\eqalignno{\1k \la\bar\psi_{p,\sigma}\psi_{p,\sigma}\ra_{\beta,L,r}
   &={1\over Z(\beta,L,r)} \int\1k \bar\psi_{p,\sigma}\psi_{p,\sigma}\>
   \exp\biggl\{ -{\ts {\lambda\over\kappa^3}}
    \sum_{k,p} \psi_{k,\up}\psi_{-k,\down}
    \bar\psi_{p,\up}\bar\psi_{-p,\down}  \cr
 &\phantom{mmmmmmm}    + {\ts{1\over\kappa}}
      \sum_k [r\psi_{k,\up}\psi_{-k,\down}
   -\bar r\bar\psi_{k,\up}\bar\psi_{-k,\down}]   \biggr\} 
   d\mu_C(\psi,\bar\psi)  \cr   
 &={\ts {\partial \over \partial s_{k,\sigma}}} \log Z(\beta,L,\{s_k\},\{r_k\})\bigr|_{
   s_k=0,r_k=r} &(\II.2) \cr}$$ 
$$\eqalignno{\1k \la \psi_{p,\up}\psi_{-p,\down}\ra_{\beta,L,r}
   &={1\over Z(\beta,L,r)} \int\1k \psi_{p,\up}\psi_{-p,\down}\>
   \exp\biggl\{- {\ts {\lambda\over\kappa^3}}
    \sum_{k,p} \psi_{k,\up}\psi_{-k,\down}
    \bar\psi_{p,\up}\bar\psi_{-p,\down}  \cr
 &\phantom{mmmmmmm}    +{\ts {1\over\kappa}}
        \sum_k [r\psi_{k,\up}\psi_{-k,\down}
   -\bar r\bar\psi_{k,\up}\bar\psi_{-k,\down}]   \biggr\} 
   d\mu_C(\psi,\bar\psi)  \cr
 &={\ts {\partial \over \partial r_{k}}} \log Z(\beta,L,\{s_k\},\{r_k\})\bigr|_{
   s_k=0,r_k=r} &(\II.3) \cr  }$$
Here $k=(k_0,\k)\in {\pi\over\beta}(2\Bbb Z+1)\times M$ where $M=
  \left\{\k\in \left({2\pi\over L}\Bbb Z\right)^d\>\bigl|\>|e(\k)|\le \omega_D\right\}$ 
and $\kappa=\beta L^d$. 
\bigskip
\bigskip
\noindent{\bf Proposition II.1:}  Let  $\phi=u+iv,\;\bar\phi=u-iv$ 
where $u$ and $v$ are  one dimensional
real variables. Let $g=\sqrt\lambda\in \Bbb C$ 
  and let $\kappa=\beta L^d$.  Then, if 
 $a_k=ik_0-e_\k$, one has the two dimensional integral representations 
$$Z(\beta,L,\{s_k\},\{r_k\})=\int 
   \pro_k{\ts {(a_k-s_{k,\up})(a_{-k}-s_{-k,\down})
  +\left(  g \phi -ir_k\right) 
   \left(  g \bar\phi+i\bar r_k\right) 
    \over a_k a_{-k} } }\>{\ts {\kappa\over\pi}}
       e^{-\kappa|\phi|^2} dudv  \leqno {\bf a)}$$
\par In particular, for $r_k=r$ and $s_k=0$ for all $k$ 
$$Z(\beta,L,r)= \int 
   \pro_k \ts \Bigl\{  {\ts 1+{\left(  g \phi  -ir\right) 
   \left(  g \bar\phi+i\bar r\right)
    \over a_k a_{-k} } } \Bigr\} {\ts {\kappa\over\pi}} e^{-\kappa|\phi|^2}dudv   $$
\item{\bf b)} The $\bar\psi\psi$ expectation values  are given by 
$$\eqalignno{\1k \la\bar\psi_{p,\sigma}\psi_{p,\sigma}\ra_{\beta,L,r}   
 &={1\over Z}
   \int {\ts {-a_{-p}\over a_pa_{-p}+\left( g \phi-ir\right)
   \left( g \bar\phi+i\bar r\right)}}\>   
   \pro_k\Bigl\{  {\ts 1+{\left(  g \phi -ir\right) 
   \left(  g \bar\phi+i\bar r\right)
    \over a_k a_{-k} } } \Bigr\}  \ts {\kappa\over\pi} e^{-\kappa|\phi|^2}dudv 
    \cr}$$
\item{\bf c)} The $\psi\psi$ and $\bar\psi\bar\psi$ expectation 
    values  are given by 
$$\eqalignno{\1k \la \psi_{p,\up}\psi_{-p,\down}\ra_{\beta,L,r} 
 &={1\over Z}\int {\ts {- i\left(  g \bar\phi+i\bar r\right)
             \over a_pa_{-p}+\left( g \phi-ir\right)
   \left( g \bar\phi+i\bar r\right)}}\>   
   \pro_k\Bigl\{  {\ts 1+{\left(  g \phi -ir\right) 
   \left(  g \bar\phi+i\bar r\right)
    \over a_k a_{-k} } } \Bigr\} \ts{\kappa\over\pi} 
    e^{-\kappa|\phi|^2}dudv     \cr}$$ 
$$\eqalignno{\1k \la \bar\psi_{p,\up}\bar\psi_{-p,\down}\ra_{\beta,L,r}   
 &={1\over Z}\int {\ts { -i\left( g \phi-i  r\right)
             \over a_pa_{-p}+\left( g \phi-ir\right)
   \left(g \bar\phi +i\bar r\right)}}\>   
   \pro_k\Bigl\{  {\ts 1+{\left( g \phi -ir\right) 
   \left(  g \bar\phi+i\bar r\right)
    \over a_k a_{-k} } } \Bigr\}\ts {\kappa\over\pi}
      e^{-\kappa|\phi|^2}dudv     \cr}$$ 
\bigskip
\noindent{\bf Proof:} {\bf a)} Define the measure 
$$d\nu(\phi,\bar\phi)={\ts{1\over 2\pi}}e^{-{1\over2}(u^2+v^2)}du\>dv$$ 
Then, for some commuting elements $a,b$, 
$$\eqalignno{ \int e^{a\phi+b\bar\phi} d\nu(\phi,\bar\phi)&=
  {\ts {1\over \sqrt{2\pi}}}\int e^{(a+b)u}e^{-{1\over2}u^2}du
   {\ts{1\over \sqrt{2\pi}}}\int e^{i(a-b)v}e^{-{1\over2}v^2}dv  \cr
 &=e^{{1\over2}(a+b)^2}e^{-{1\over2}(a-b)^2} =e^{2ab}  \cr}$$
and we may write 
$$\eqalignno{ e^{ -{\ts{\lambda\over\kappa^3}}
    \sum_{k,p} \psi_{k,\up}\psi_{-k,\down}
    \bar\psi_{p,\up}\bar\psi_{-p,\down} }&= 
   e^{-2 
   {\ts\left({\lambda\over 2\kappa}\right)^{1\over2}}\1k \sum_{k} 
      \psi_{k,\up}\psi_{-k,\down}{\ts\left({\lambda\over
    2\kappa}\right)^{1\over2}} \1k 
    \sum_{p}  \bar\psi_{p,\up}\bar\psi_{-p,\down}  }   \cr  
 &=\int e^{ 
    {i\phi\>{\ts\left({\lambda\over 2\kappa}\right)^{1\over2}}
     \1k \sum_{k} \psi_{k,\up}\psi_{-k,\down} +i\bar\phi \>
   {\ts\left({\lambda\over  2\kappa}\right)^{1\over2}} \1k
    \sum_{k}  \bar\psi_{k,\up}\bar\psi_{-k,\down}} 
      }  d\nu(\phi,\bar\phi) \cr
 &=\int  e^{ 
    {i\phi\>g
     \1k \sum_{k} \psi_{k,\up}\psi_{-k,\down} +i\bar\phi \>
   g \1k
    \sum_{k}  \bar\psi_{k,\up}\bar\psi_{-k,\down}} 
       }  {\ts{\kappa\over\pi}} \>e^{-\kappa|\phi|^2}dudv \cr}$$
Then the partition function becomes 
$$\eqalignno{ Z&(\beta,L)={\ts{\kappa\over\pi}}\int \int e^{ 
    {ig \phi
     \1k \sum_{k} \psi_{k,\up}\psi_{-k,\down} +ig \bar\phi \1k
    \sum_{k}  \bar\psi_{k,\up}\bar\psi_{-k,\down}} 
      }   \times  \cr
 &\phantom{mmmm}  e^{ { \1k
     \sum_k[s_{k,\up}\bar\psi_{k,\up}\psi_{k,\up}+
    s_{k,\down}\bar\psi_{k,\down}\psi_{k,\down}+r_k\psi_{k,\up}\psi_{-k,\down}
   -\bar r_k\bar\psi_{k,\up}\bar\psi_{-k,\down}}   } 
   d\mu_C(\psi,\bar\psi) \>e^{-\kappa|\phi|^2}dudv  \cr  
 &\phantom{mmm}={\ts{\kappa\over\pi}}\int \int e^{ 
    { \1k \sum_{k}i \gamma_k \psi_{k,\up}\psi_{-k,\down} 
  + \1k
    \sum_{k} i\bar\gamma_k \bar\psi_{k,\up}\bar\psi_{-k,\down}} 
      }   \times  \cr   
 &\phantom{mmm={\ts{\kappa\over\pi}}\int \int }  e^{ \1k
       \sum_k[s_{k,\up}\bar\psi_{k,\up}\psi_{k,\up}+
    s_{k,\down}\bar\psi_{k,\down}\psi_{k,\down}     } 
   d\mu_C(\psi,\bar\psi) \> e^{-\kappa|\phi|^2}dudv    \cr  }$$
where we defined 
$$\gamma_k=g \phi -ir_k,\;\;\bar\gamma_k=g \bar\phi 
     +i\bar r_k $$
Observe that $\bar\gamma_k$ is not necessarily the complex
conjugate of $\gamma_k$. 
Now the Fermionic functional integral can be done. Let
$$a_k=ik_0-e_\k$$
One obtains 
$$\eqalignno{ \int& e^{ 
     \1k \sum_{k}  [i\gamma_k \psi_{k,\up}\psi_{-k,\down} 
  + i \bar\gamma_k \bar\psi_{k,\up}\bar\psi_{-k,\down}  
   +s_{k,\up}\bar\psi_{k,\up}\psi_{k,\up}+
    s_{k,\down}\bar\psi_{k,\down}\psi_{k,\down}  ]
       }  d\mu_C(\psi,\bar\psi) \cr
 &=\pro_{k}{\ts {\kappa^2\over a_k a_{-k} }}
   \int e^{ \1k\sum_{k}
    [i\gamma_k \psi_{k,\up}\psi_{-k,\down} 
  + i \bar\gamma_k \bar\psi_{k,\up}\bar\psi_{-k,\down}  
   -(a_k-s_{k,\up})\bar\psi_{k,\up}\psi_{k,\up}  -
    (a_k-s_{k,\down})\bar\psi_{k,\down}\psi_{k,\down}  ]   }
    \pro_{k,\sigma}d\psi_{k,\sigma}d\bar\psi_{k,\sigma}  \cr
  & =\pro_{k}{\ts {\kappa^2\over a_k a_{-k} }}
   \int e^{ -\1k\sum_{k} (\bar\psi_{k,\up},\psi_{-k,\down}) 
   \left( { a_k-s_{k\up}\;\; \;\; -i\bar\gamma_k \atop  i\gamma_k \;\; \;\;
    -a_{-k}+s_{-k\down} }\right)  
   \left(  {\psi_{k,\up} \atop \bar\psi_{-k,\down}  } \right) 
      } \pro_{k}(-d\psi_{k\up}d\bar\psi_{k\up} d\bar\psi_{-k\down} 
    d \psi_{-k\down} )     \cr      
 &= \pro_{k}{\ts {\kappa^2\over a_k a_{-k} }}\pro_k
   \left( {\ts {-1\over \kappa^2}}
   \det \left[ {\ts { a_k-s_{k\up} \;\;\;\; -i\bar\gamma_k \atop  i\gamma_k \;\; \;\;
    -a_{-k}+s_{-k\down}) }}\right] \right)     \cr   
 &=\pro_k{\ts  {(a_k-s_{k,\up})(a_{-k}-s_{-k,\down})+\gamma_k \bar\gamma_k
    \over a_k a_{-k} } }  \cr}$$   
This proves the formula under a). Part b) and c) follow by differentiation with 
respect to $s_{k,\sigma}$ or $r_k,\>\bar r_k \;\blacksquare$
\bigskip
\bigskip
\noindent{\bf Lemma II.2:} Let $a_k=ik_0-e_\k$.  Using
   the approximation $\sum_{\k\in M}\approx 
 L^d\int_M {d^d\k\over (2\pi)^d}$ one has 
$$\prod_{\k\in M}\prod_{k_0\in {\pi\over\beta}(2\Bbb Z+1)} \ts 
   \left\{ 1+{\xi\over a_k a_{-k}} \right\}=e^{\kappa W_\beta(\xi)}
  \eqno (\II.4)$$
where $\kappa=\beta L^d$ and 
$$W_\beta(\xi)=\int_M \db k\ts \>{\ts {1\over \beta}}\log\left[ 
   {\cosh\left( {\beta\over2}\sqrt{e_\k^2+\xi}\right)\over \cosh
    \left({\beta\over2}e_\k\right)}\right]^2 \eqno (\II.5) $$
\bigskip
\noindent{\bf Proof:} The product over $k_0$ may be computed exactly by using
the formula [H]
$$\prod_{n=0}^\infty\ts \left( 1+{\xi\over(2n+1)^2+a^2}\right)=
   {\cosh\left({\pi\over2}\sqrt{a^2+\xi}\right)\over \cosh{\pi\over2}a}\>.$$
One obtains 
$$\prod_{k_0\in{\pi\over\beta}(2\Bbb Z+1)}\ts\left(1+{\xi\over k_0^2+e_\k^2}\right)
   =\biggl\{ {\cosh\left( {\beta\over2}\sqrt{e_\k^2+\xi}\right)\over \cosh
    \left({\beta\over2}e_\k\right)} \biggr\}^2 \eqno (\II.6)$$
which results in the formula stated above $\blacksquare$
\bigskip
\noindent Substitution of (II.4) in Proposition II.1 and a transformation of variables
gives 
\bigskip
\noindent{\bf Theorem II.3:} Let $u,v$ be one dimensional real variables and 
 let $r=|r| e^{i\alpha}$. 
 Define the effective potential 
$$\ts V_{\beta,r}(u,v)= u^2+(v+{|r|\over\sqrt\lambda})^2
    -W_\beta\bigl(\lambda(u^2+v^2)\bigr)
           \eqno (\II.7)$$
\item{\bf a)} There are the two dimensional integral representations 
$$Z(\beta,L,r)={\ts{\kappa\over\pi}} 
    \ts \int e^{-\kappa V_{\beta,r}(u,v)} dudv  \eqno$$
$$\1k \la\bar\psi_{p,\sigma}\psi_{p,\sigma}\ra_{\beta,L,r}=
   {\int  {-a_{-p}\over a_pa_{-p}+\lambda(u^2+v^2)}
     \>e^{-\kappa V_{\beta,r}(u,v)} dudv 
  \over  \int e^{-\kappa V_{\beta,r}(u,v)} dudv}  \eqno$$
$$\1k \la \psi_{p,\up}\psi_{-p,\down}\ra_{\beta,L,r}={
   \int {\ts { -i \sqrt\lambda\,e^{-i\alpha}(u-iv)
             \over a_pa_{-p}+ \lambda(u^2+v^2)}}\>
      e^{-\kappa V_{\beta,r}(u,v)} dudv
  \over \int e^{-\kappa V_{\beta,r}(u,v)} dudv}    $$
$$\1k \la \bar\psi_{p,\up}\bar\psi_{-p,\down}\ra_{\beta,L,r}={
   \int {\ts {- i \sqrt\lambda\, e^{i\alpha}(u+iv)
             \over a_pa_{-p}+\lambda(u^2+v^2)}}\>
    e^{-\kappa V_{\beta,r}(u,v)} dudv
  \over \int e^{-\kappa V_{\beta,r}(u,v)} dudv}\eqno   $$
\item{\bf b)} Let $V_\beta(\rho):=\rho^2-W_\beta(\lambda\rho^2)$ 
 and $|\tr|^2=\lambda \rho_0^2$ be given by 
$$V_\beta(\rho_0)=\min_{\rho\ge 0} V_\beta(\rho) \eqno (\II.8)$$ 
Then, if $\tr=\tr(\lambda,\beta)=|\tr|\>e^{-i\alpha}$,  one has 
$$\lim_{|r|\to 0}\lim_{L\to\infty} {\ts{1\over\beta L^d}}\log Z(\beta,L,r)=
      -V_\beta(|\tr|)  \eqno $$
$$\lim_{|r|\to 0}\lim_{L\to\infty}
      \1k \la\bar\psi_{p,\sigma}\psi_{p,\sigma}\ra_{\beta,L,r}= 
   \ts {-a_{-p}\over a_pa_{-p}+|\tr|^2}  \eqno$$
$$\lim_{|r|\to 0}\lim_{L\to\infty}
    \1k \la \psi_{p,\up}\psi_{-p,\down}\ra_{\beta,L,r}=\ts 
    {\tr \over a_pa_{-p}+|\tr|^2}  $$
$$\lim_{|r|\to 0}\lim_{L\to\infty}
    \1k \la \bar\psi_{p,\up}\bar\psi_{-p,\down}\ra_{\beta,L,r}=\ts 
    {-\bar\tr \over a_pa_{-p}+|\tr|^2}  \eqno  $$
but 
$$\lim_{L\to\infty}\lim_{|r|\to 0}
   \1k \la \psi_{p,\up}\psi_{-p,\down}\ra_{\beta,L,r}=
    \lim_{L\to\infty}\lim_{|r|\to 0}
    \1k \la\bar\psi_{p,\up}\psi_{-p,\down}\ra_{\beta,L,r}=
   0   \eqno  $$
\bigskip
\noindent{\bf Proof:} a) Substituting the product over the momenta 
 by the exponential 
 (II.4) one obtains, if $g=\sqrt\lambda$ 
$$\eqalignno{  \1k \la\bar\psi_{p,\sigma}\psi_{p,\sigma}\ra_{\beta,L,r}&=
  {\int  {-a_{-p}\over a_pa_{-p}+\left(g \phi-ir\right)
   \left(g \bar\phi+i\bar r\right)}
     \>e^{-\kappa\left[ |\phi|^2-W_\beta\left( (g\phi-ir)
    (g\bar\phi+i\bar r)\right) \right] } dudv 
  \over  \int e^{-\kappa\left[ |\phi|^2-W_\beta\left( (g\phi-ir)
    (g\bar\phi+i\bar r)\right) \right] } dudv} \cr}$$ 
and analog expressions for $Z$ and the  $\la\psi\psi\ra$ and $\la\bar\psi 
 \bar\psi\ra$ expectations. 
Thus one has to consider integrals of the form 
$$ \int F\left( \sl \phi-i r,\sl \bar\phi+i\bar r\right) \> e^{-\kappa |\phi|^2} 
    dudv $$ 
By a substitution of variables one finds for both signs of $\lambda$ 
$$ \int F\left( \sl \phi-i r,\sl \bar\phi+i\bar r\right)
     \> e^{-\kappa |\phi|^2} dudv =   \int F\left(e^{i\alpha}\sl \phi, 
    e^{-i\alpha}\sl\bar\phi \right) 
   e^{-{\kappa} \left(u^2+(v+{|r|\over\sl})^2\right)}
    dudv \eqno (\II.9)$$
which proves part a). To obtain part b), 
 one has to compute the limit of 
$$\delta_{\kappa,r} (u,v)= { e^{-\kappa V_{\beta,r}(u,v)}\over \int
    e^{-\kappa V_{\beta,r}(u,v)} dudv}  \eqno (\II.10)$$ 
where 
$$\ts V_{\beta,r}(u,v)= u^2+(v+{|r|\over\sl})^2
   -W_\beta\bigl(\lambda(u^2+v^2)\bigr)
           \eqno (\II.7)$$
Recall that 
$$W_\beta(\xi)={\ts {2\over \beta}}\int_M \db k\ts \>\log\left[ 
   {\cosh\left( {\beta\over2}\sqrt{e_\k^2+\xi}\right)\over \cosh
    \left({\beta\over2}e_\k\right)}\right]$$
In particular, 
$${\rm sign}W_\beta\bigl(\lambda(u^2+v^2)\bigr)={\rm sign}\lambda$$
For positive $\lambda$ and nonzero $r$, 
   $V_{\beta,r}(u,v)$ is real and has a unique global minimum 
 determined by 
$$\eqalignno{ 2u-2u\lambda W_\beta'\bigl(\lambda(u^2+v^2)\bigr)&=0  \cr
  2(v+|r|)-2v\lambda W_\beta'\bigl(\lambda(u^2+v^2)\bigr)&=0  \cr}$$
Since $\lambda W_\beta'=1$ does not solve the second equation, the only 
solution of the first equation is $u=0$ and one is left with 
$$ v\left[ \lambda W_\beta'(\lambda v^2)-1\right] =|r| \eqno $$
which  has a solution $v=O(|r|)$ which 
 is a local maximum and two  
 nontrivial solutions $\lambda v^2=|\tr|^2+O(|r|)$ where  the positive one is only a 
 local minimum and the negative one, $v_0$,  is the global minimum. 
Therefore 
$$\lim_{\kappa\to\infty} \delta_{\kappa,r}(u,v)=\lim_{\kappa\to\infty} 
   { e^{-\kappa\left[ V_{\beta,r}(u,v)-V_{\beta,r}(0,v_0)\right]}\over \int
    e^{-\kappa\left[ V_{\beta,r}(u,v)-V_{\beta,r}(0,v_0)\right]} dudv}
   =\delta(u)\,\delta(v-v_0)  \eqno $$
and 
$$\lim_{|r|\to 0}\lim_{\kappa\to\infty} \delta_{\kappa,r}(u,v)=\ts 
    \delta(u)\,\delta\left(v+{|\tr|\over g}\right) \eqno (\II.11)$$
This proves the formulae under (b)  for attractive $\lambda$. Since  
$$\lim_{|r|\to 0} V_{r,\beta}(u,v)=u^2+v^2-W_\beta\left(\lambda(u^2+v^2)\right)$$
is an even function in $u$ and $v$, $\lim_{r\to 0}
  \la \psi_{p,\up}\psi_{-p,\down}\ra_{\beta,L,r}$=$\lim_{r\to 0}
   \la \bar\psi_{p,\up}\bar\psi_{-p,\down}\ra_{\beta,L,r}=0$. 
The limit of the logarithm of the partition function becomes 
$$\eqalignno{ {\ts {1\over \kappa}}\log Z&={\ts {1\over \kappa}}\log 
  {\ts{\kappa\over\pi}} \int e^{-\kappa V_{\beta,r}(u,v) }dudv   \cr
 &= {\ts {1\over \kappa}}\log 
  {\ts{\kappa\over\pi}} \int e^{-\kappa \left[V_{\beta,r}(u,v)
     -V_{\beta,r}(0,v_0)\right] }dudv +{\ts {1\over \kappa}}\log e^{-\kappa
   V_{\beta,r}(0,v_0)}  \cr}$$
The first term on the right hand side may be approximated by ($V_{uv}=
  {\partial^2V_{\beta,r}\over \partial u\partial v}(0,v_0)=0$)
$$ {\ts {1\over \kappa}}\log 
  {\ts{\kappa\over\pi}} \int
     e^{-{\kappa\over2}  (V_{uu}u^2+V_{vv}(v-v_0)^2)}dudv=
  {\ts {1\over \kappa}}\log 
  {\ts{1\over\pi }}
    \int e^{- {1\over2}(V_{uu}u^2+V_{vv}v^2)}dudv\buildrel
   \kappa\to\infty\over \to 0$$
which results in 
$$\lim_{\kappa\to\infty} {\ts{1\over \kappa}}\log Z=-V_{\beta,r}(0,v_0)$$
\medskip
Now let $\lambda$ be negative. In that case the effective potential (II.7) 
 is complex:  
$$\eqalignno{ V_{\beta,r}(u,v)&=u^2+v^2-\ts {|r|^2\over |\lambda|}
   -2iv{|r|\over \sqrt{|\lambda|} }-W_\beta\left(\lambda(u^2+v^2)\right) \cr
 &\ts =\rho^2  +|W_\beta\left(\lambda\rho^2\right)|
  -{|r|^2\over |\lambda|}-2i\rho\cos\vp{|r|\over \sqrt{|\lambda|} }  \cr}$$
Since the real part $U_{\beta,r}={\rm Re}V_{\beta,r}$ has 
 a global minimum at $u=v=0$ one has, since $U''={\partial^2U_{\beta,r}
   \over \partial u^2 }(0,0)={\partial^2U_{\beta,r}\over \partial v^2 }(0,0)>0$, 
$$\eqalignno{ \delta_{\kappa,r}(u,v)&={ e^{-\kappa\bigl[U_{\beta,r}(u,v)
     -2iv{|r|\over \sqrt{|\lambda|} } \bigr]} \over \int 
    e^{-\kappa\bigl[U_{\beta,r}(u,v)
     -2iv{|r|\over \sqrt{|\lambda|} } \bigr]} dudv } 
  \approx { e^{-\kappa\bigl[ {U''\over2}(u^2+v^2)
     -2iv{|r|\over \sqrt{|\lambda|} } \bigr]} \over \int 
    e^{-\kappa\bigl[{U''\over 2}(u^2+v^2)
     -2iv{|r|\over \sqrt{|\lambda|} } \bigr]} dudv }  \cr
 &= { e^{-\kappa {U''\over2}u^2  } \over \int 
    e^{-\kappa {U''\over 2}u^2} du } \;
     { e^{-\kappa\bigl[ {U''\over2}v^2
     -2iv{|r|\over \sqrt{|\lambda|} } \bigr]} \over \int 
    e^{-\kappa\bigl[{U''\over 2}v^2
     -2iv{|r|\over \sqrt{|\lambda|} } \bigr]} dv } \cr
 & = { e^{-\kappa {U''\over2}u^2  } \over \int 
    e^{-\kappa {U''\over 2}u^2} du } \;
     { e^{-\kappa  {U''\over2}\bigl[v-i{2|r|\over \sqrt{|\lambda|}U'' }
       \bigr]^2} \over \int 
    e^{-\kappa {U''\over 2}\bigl[v-i{2|r|\over \sqrt{|\lambda|}U'' }
       \bigr]^2} dv } 
   \buildrel \kappa\to\infty \over \to \ts \delta(u)\,\delta\Bigl(
     v-i{2|r|\over \sqrt{|\lambda|}U'' } \Bigr)   \cr  }$$
which results in 
 $$\lim_{|r|\to 0}\lim_{\kappa\to\infty}\delta_{\kappa,r}(u,v)=\delta(u)\,\delta(v)
    \eqno (\II.12)$$
and 
$$\lim_{|r|\to 0} \lim_{\kappa\to\infty}\ts {1\over \kappa}
   \la \bar\psi_{p,\up}\psi_{-p,\down}\ra_{\beta,L,r}=\int\ts   {-a_{-p}\over 
   a_pa_{-p}+\lambda(u^2+v^2)}\>\delta(u)\delta(v)\>dudv 
    ={-a_{-p}\over a_pa_{-p}}= 
   -{1\over a_p} $$ 
$$\lim_{|r|\to 0} \lim_{\kappa\to\infty}{\ts {1\over \kappa}}
   \la \psi_{p,\up}\psi_{-p,\down}\ra_{\beta,L,r}=\lim_{|r|\to 0}
    \lim_{\kappa\to\infty}\ts {1\over \kappa}
   \la \bar\psi_{p,\up}\bar \psi_{-p,\down}\ra_{\beta,L,r}=0\eqno \blacksquare$$
\bigskip
In the same way one obtains a two dimensional integral representation for 
 the generating functional for the connected amputated Greens functions. 
 It  is defined 
by 
$$\eqalignno{ G&(\eta)= G_r(\beta,L,\eta)= \log 
  \int e^{ \U(\psi+\eta,\bar\psi+\bar\eta)} 
   d\mu_C(\psi,\bar\psi) &(\II.13) \cr} $$
where
$$\U(\psi,\bar\psi)={\ts -  {\lambda\over\kappa}
  {1\over\kappa^2}}  \sum_{k,p}\psi_{k\up}\psi_{-k\down}\bar\psi_{p\up} 
  \bar\psi_{-p\down}+{\ts {1\over\kappa}}\sum_k[r\psi_{k\up}\psi_{-k\down} 
   -\bar r\bar\psi_{k\up}\bar\psi_{-k\down}]$$
and the $\eta_{k\sigma}$ are some Grassmann variables. 
\bigskip
\noindent{\bf Theorem II.4:} {\bf a)} Let $u,v$ be one dimensional real variables,  
 $r=|r|e^{i\alpha}$ and 
 $\gamma=ge^{i\alpha}(u+iv)$,
  $\bar\gamma=g e^{-i\alpha}(u-iv)$. Let  
 $V_{\beta,r}(u,v)$  be the effective potential (II.7). 
Then the generating functional for the connected amputated
Greens functions is given by 
$$\eqalignno{ G&(\eta)-G(0)= 
     -\1k\sum_k\Bigl[ a_k\bar\eta_{k\up}\eta_{k\up}
    +a_{-k}\bar\eta_{-k\down}\eta_{-k\down}\Bigr] \;+\;   
    \cr
 & + \log{ \int dudv\> 
    e^{-\kappa V_{\beta,r}(u,v) }
      \exp\biggl\{ \1k\sum_k   (a_k\bar\eta_{k\up},
    -a_{-k}\eta_{-k\down} ) {1\over a_ka_{-k}+\gamma\bar\gamma} 
   \left( {a_{-k}\atop i\gamma}\; {-i\bar\gamma \atop  -a_{k}} \right)
    \left( a_k\eta_{k\up} \atop 
    -a_{-k}\bar\eta_{-k\down}\right) \biggr\} 
   \over \int dudv\> 
    e^{-\kappa V_{\beta,r}(u,v) }  }  \cr}$$
{\bf b)} For attractive $\lambda>0$ the infinite volume limit of the generating
functional is given by 
$$\eqalignno{ \lim_{|r|\to 0}\lim_{L\to\infty}
    \{G(\beta,&L,r,\eta)-G(\beta,L,r,0)\}=  \cr
  &-\1k\sum_k\Bigl[\ts  a_k\,{|\tr|^2\over a_ka_{-k}+|\tr|^2} \>
      \bar\eta_{k\up}\eta_{k\up}
    +a_{-k}\,{|\tr|^2\over a_ka_{-k}+|\tr|^2} \>
    \bar\eta_{-k\down}\eta_{-k\down}  \cr
 &\phantom{ -\1k\sum_k\Bigl[} 
      \ts  +\tr\, {a_ka_{-k} \over a_ka_{-k}+|\tr|^2}\>\eta_{-k\down}\eta_{k\up} 
    +\bar\tr\,   {a_ka_{-k} \over a_ka_{-k}+|\tr|^2} \> \bar\eta_{k\up}
    \bar\eta_{-k\down} \Bigr] \cr}   $$
For repulsive $\lambda<0$ one obtains 
 $\ds\lim_{|r|\to 0}\lim_{L\to\infty} \{G(\beta,L,r,\eta)-G(\beta,L,r,0)\}=0\>. $  
\bigskip
\noindent{\bf Proof:} By a substitution of variables, one has 
$$\eqalignno{ \int& e^{-\U(\psi+\eta,\bar\psi+\bar\eta)}d\mu_C(\psi,\bar\psi)=
  \int e^{-\U(\psi+\eta,\bar\psi+\bar\eta)}\pro_{k\sigma} 
   \ts {\kappa\over a_k} 
  \> e^{-{1\over\kappa}\sum_{k\sigma}a_k\bar\psi_{k\sigma}\psi_{k\sigma}} 
  \ds \pro_{k\sigma}d\psi_{k\sigma}
   d\bar\psi_{k\sigma} \cr
 &=e^{-{1\over\kappa}\sum_{k\sigma}a_k\bar\eta_{k\sigma}\eta_{k\sigma}} 
   \pro_{k\sigma} 
  { \ts {\kappa\over a_k} }  \int e^{-\U(\psi,\bar\psi)} 
  e^{{1\over\kappa}\sum_{k\sigma}a_k[\bar\eta_{k\sigma}\psi_{k\sigma}+
   \bar\psi_{k\sigma}\eta_{k\sigma}-\bar\psi_{k\sigma}\psi_{k\sigma}]} 
  \ds \pro_{k\sigma}d\psi_{k\sigma}
   d\bar\psi_{k\sigma} \cr}$$
Let $\phi=u+iv,\;\bar\phi=u-iv$, $\gamma_r=\lambda^{1\over2}\phi-ir$, 
 $\bar\gamma_r=\lambda^{1\over2}\bar\phi+i\bar r$. Then 
$$\eqalignno{ e^{-\U(\psi,\bar\psi)} &={\ts {\kappa\over\pi}} \int_{\Bbb R^2} 
   e^{{1\over\kappa}\sum_k\left[ i\gamma_r
    \psi_{k\up}\psi_{-k\down}+i\bar\gamma_r
   \bar\psi_{k\up}\bar\psi_{-k\down}\right]} e^{-\kappa|\phi|^2} dudv \cr}$$
and 
$$\eqalignno{    \int& e^{-\U(\psi,\bar\psi)} 
  e^{{1\over\kappa}\sum_{k\sigma}a_k[\bar\eta_{k\sigma}\psi_{k\sigma}+
   \bar\psi_{k\sigma}\eta_{k\sigma}-\bar\psi_{k\sigma}\psi_{k\sigma}]} 
  \ds\pro_{k\sigma} 
   {\ts {\kappa\over a_k}} \pro_{k\sigma}d\psi_{k\sigma}
   d\bar\psi_{k\sigma} \cr
 &= \int \int e^{{1\over\kappa}\sum_k 
    \left[ i\gamma_r\psi_{k\up}\psi_{-k\down}+i\bar\gamma_r
   \bar\psi_{k\up}\bar\psi_{-k\down}\right]} e^{{1\over\kappa}\sum_k\left[ 
   a_k(\bar\psi_{k\up}\eta_{k\up}+\bar\eta_{k\up}\psi_{k\up})+
    a_{-k}(\bar\psi_{-k\down}\eta_{-k\down}+
     \bar\eta_{-k\down}\psi_{-k\down})\right]} 
   \times \cr
 &\phantom{mmmmm} e^{-{1\over\kappa}
    \sum_k\left[ a_k\bar\psi_{k\up}\psi_{k\up} 
    +a_{-k}\bar\psi_{-k\down}\psi_{-k\down}\right] }
   \pro_{k}  {\ts {\kappa^2\over a_ka_{-k}} } \pro_k d\psi_{k\up} 
   d\bar\psi_{k\up} d\psi_{-k\down}d\bar\psi_{k\down} \,
    {\ts {\kappa\over\pi}}\,e^{-\kappa|\phi|^2} dudv  \cr 
   &= \int\int \pro_{k}{\ts {a_k a_{-k}+\gamma_r\bar\gamma_r \over a_k a_{-k} }} 
   \> e^{  \1k 
    \sum_k \Bigl[ (\bar\psi_{k\up},\psi_{-k\down})\left( a_k\eta_{k\up} \atop 
    -a_{-k}\bar\eta_{-k\down}\right)+  (a_k\bar\eta_{k\up},
    -a_{-k}\eta_{-k\down} )      
      \left( {\psi_{k\up}\atop \bar\psi_{-k\down}}\right)
   \Bigr] } \times \cr
  & \phantom{m} \pro_k 
     {\ts {-\kappa^2 \over a_k a_{-k}+\gamma_r\bar\gamma_r  }} 
   e^{ -\1k\sum_{k}(\bar\psi_{k\up},\psi_{-k\down}) 
    \left( {a_k\atop i\gamma_r}\; {-i\bar\gamma_r \atop  -a_{-k}} \right) 
    \left( {\psi_{k\up}\atop \bar\psi_{-k\down}}\right) 
     }   \pro_{k}d\psi_{k\up}d\bar\psi_{k\up}
     d\bar\psi_{-k\down}d\psi_{-k\down}\,
   {\ts {\kappa\over\pi}}\,e^{-\kappa|\phi|^2} dudv   \cr
 &= \int \pro_{k}{\ts \left[ 1+{\gamma_r\bar\gamma_r \over a_k a_{-k} }\right] } 
     e^{ \1k\sum_k   (a_k\bar\eta_{k\up},
    -a_{-k}\eta_{-k\down} )
   \left( {a_k\atop i\gamma_r}\; {-i\bar\gamma_r \atop  -a_{-k}} \right)^{-1} 
    \left( a_k\eta_{k\up} \atop 
    -a_{-k}\bar\eta_{-k\down}\right) } 
     {\ts {\kappa\over\pi}}\,e^{-\kappa|\phi|^2} dudv    \cr}$$
Therefore the generating functional is given by 
$$\eqalignno{ G(\beta,L,\eta)= \log  {\ts{\kappa\over\pi}}\int &
    \pro_{k}{\ts \left[ 1+{\gamma_r\bar\gamma_r \over a_k a_{-k} }\right] } 
   e^{  -\1k\sum_k\left[ a_k\bar\eta_{k\up}\eta_{k\up}
    +a_{-k}\bar\eta_{-k\down}\eta_{-k\down}\right]  } \times \cr
 &\phantom{m }  e^{ \1k\sum_k   (a_k\bar\eta_{k\up},
    -a_{-k}\eta_{-k\down} ) {1\over a_ka_{-k}+\gamma_r\bar\gamma_r} 
   \left( {a_{-k}\atop i\gamma_r}\; {-i\bar\gamma_r \atop  -a_{k}} \right)
    \left( a_k\eta_{k\up} \atop 
    -a_{-k}\bar\eta_{-k\down}\right) } e^{-\kappa |\phi|^2} dudv \cr}$$
Since $\gamma_r=g\phi-ir$,
     $\bar\gamma_r=g\bar\phi+i\bar r$ one can 
 apply (II.9) 
$$\eqalignno{ \int F&\left( \sl \phi-i r,\sl \bar\phi+i\bar r\right)
     \> e^{-\kappa |\phi|^2} dudv =   \int F\left(e^{i\alpha}\sl \phi, 
    e^{-i\alpha}\sl\bar\phi \right) 
   e^{-{\kappa} \left(u^2+(v+{|r|\over\sl})^2\right)}
    dudv    \cr}$$
again to make a substitution of variables which proves part (a). 
\par
 The infinite volume limit  of the generating functional follows 
 from  (II.11,12)
$$\lim_{|r|\to 0}\lim_{\kappa\to\infty} {e^{-\kappa V_{\beta,r}(u,v)} \over 
    \int e^{-\kappa V_{\beta,r}(u,v)} dudv} =\cases{ \delta(u)
     \delta\left(v+{\ts {|\tr|\over g} }\right) &if $\lambda>0$ \cr
    \delta(u)\delta(v) &if $\lambda<0\;\blacksquare$ \cr}$$
%
%  CHAPTER II.2
%
\bigskip
\bigskip\goodbreak
\bigskip 
\noindent{\mittel II.2  Explicit Summation of Diagramms}
\bigskip
\bigskip
In this section we compute the partition function ($r_k=s_k=0$) 
$$Z=\int e^{-{\lambda\over\kappa}{1\over\kappa^2} 
    \sum_{k,p}\psi_{k\up}\psi_{-k\down}
      \bar\psi_{p\up} \bar\psi_{-p\down} } d\mu_C(\psi,\bar\psi)  $$
by explicit summation of diagramms. That is, we derive the formula 
$$Z= {2\kappa} \int e^{-\kappa\Bigl( \rho^2-{1\over L^d}\sum_\k
   {1\over\beta}  \log\left[{ \cosh({\beta\over2}\sqrt{e_\k^2+\lambda\rho^2})
     \over \cosh{\beta\over2}e_\k} \right]^2 \Bigr) } \rho \>d\rho \eqno(\II.14)  $$
by direct summation of the perturbation series without using the identity 
$$ e^{ -{\ts{\lambda\over\kappa^3}}
    \sum_{k,p} \psi_{k\up}\psi_{-k\down}
    \bar\psi_{p,\up}\psi_{-p\down}  }= 
  \int e^{ 
    {i\phi\>g
     \1k \sum_{k} \psi_{k\up}\psi_{-k\down} +i\bar\phi \>
   g \1k
    \sum_{k}  \bar\psi_{k\up}\bar\psi_{-k\down}} 
      }  {\ts{\kappa\over\pi}} \>e^{-\kappa|\phi|^2}dudv   $$
One has 
$$\eqalignno{Z&= \int e^{-{\lambda\over\kappa^3}
      \sum_{k,p}\psi_{k\up}\psi_{-k\down}
      \bar\psi_{p\up} \bar\psi_{-p\down} } d\mu_C(\psi,\bar\psi)   \cr
  &= \sum_{n=0}^\infty { \left( -{\lambda\over\kappa^3}\right)^n\over n!} 
     \sum_{k_1,\cdots,k_n}\sum_{p_1,\cdots,p_n} \int 
   \psi_{k_1\up}\psi_{-k_1\down}  \cdots\psi_{k_n\up}\psi_{-k_n\down}
     \bar\psi_{p_1\up}
      \bar\psi_{-p_1\down}\cdots 
    \bar\psi_{p_n\up} \bar\psi_{-p_n\down} \>  d\mu_C     \cr
 &= \sum_{n=0}^\infty { \left( -{\lambda\over\kappa^3}\right)^n\over n!} 
     \sum_{k_1,\cdots,k_n}\sum_{p_1,\cdots,p_n}(-1)^{n^2}   \int \psi_{k_1\up}
    \bar\psi_{p_1\up}
   \cdots\psi_{k_n\up}\bar\psi_{p_n\up}\>  d\mu_C \times \cr 
  &\phantom{= \sum_{n=0}^\infty { \left( -{\lambda\over\kappa}\right)^n\over n!} 
     \sum_{k_1,\cdots,k_n}\sum_{p_1,\cdots,p_n} \int} 
    \int \psi_{-k_1\down}\bar\psi_{-p_1\down}
   \cdots\psi_{-k_n\down}
            \bar\psi_{-p_n\down}\>  d\mu_C   \cr
 &= \sum_{n=0}^\infty { \left( -{\lambda\over\kappa^3}\right)^n\over n!} 
     \sum_{k_1,\cdots,k_n}\sum_{p_1,\cdots,p_n} (-1)^{n}\int \psi_{k_1\up}
   \bar\psi_{p_1\up}
   \cdots\psi_{k_n\up}\bar\psi_{p_n\up}\>  d\mu_C \times \cr 
  &\phantom{= \sum_{n=0}^\infty { \left( -{\lambda\over\kappa}\right)^n\over n!} 
     \sum_{k_1,\cdots,k_n}\sum_{p_1,\cdots,p_n} \int} 
   \sum_{\pi\in S_n}\epsilon_\pi\>\kappa\delta_{k_1,p_{\pi1}} C(-k_1)\cdots
     \kappa \delta_{k_n,p_{\pi n}} C(-k_n)   \cr 
 &= \sum_{n=0}^\infty { \left( -{\lambda\over\kappa^2}\right)^n\over n!} 
     \sum_{k_1,\cdots,k_n}(-1)^{n} \int \psi_{k_1\up}\bar\psi_{k_1\up}
   \cdots\psi_{k_n\up}\bar\psi_{k_n\up}\>  d\mu_C  \>
     n! C(-k_1)\cdots   C(-k_n)   \cr 
 &= \sum_{n=0}^\infty {\ts  \left( -{\lambda\over\kappa^2}\right)^n } 
     \sum_{k_1,\cdots,k_n}(-1)^{n}
     C(-k_1)\cdots   C(-k_n) \sum_{\pi\in S_n}\epsilon_\pi 
    \kappa\delta_{k_1,k_{\pi 1}}C(k_1) 
   \cdots \kappa\delta_{k_n,k_{\pi n}} C(k_n)  \cr}$$
Say that the permutation $\pi$ is of type $t(\pi)=1^{b_1}\cdots n^{b_n}$ if the 
 decomposition into disjoint cycles contains $b_r$ $r$-cycles for  $1\le r\le n$. 
Necessarily one has $1b_1+\cdots +nb_n=n$. The number of permutations 
which have $b_r$ $r$-cycles for  $1\le r\le n$ is 
$${n!\over b_1!\cdots b_n!\>1^{b_1}\cdots n^{b_n} }$$
The sign of such a permutation is given by 
$$\epsilon_\pi= (-1)^{(1-1)b_1+(2-1)b_2+\cdots+(n-1)b_n}=
   (-1)^{n-\sum_{r=1}^n b_r} $$
Therefore one obtains 
$$\eqalignno{ Z&=\sum_{n=0}^\infty {\ts  \left( -{\lambda\over\kappa}\right)^n } 
     \sum_{k_1,\cdots,k_n}(-1)^{n}
     C(-k_1)\cdots   C(-k_n) \sum_{\pi\in S_n}\epsilon_\pi 
    \delta_{k_1,k_{\pi 1}}C(k_1) 
   \cdots \delta_{k_n,k_{\pi n}} C(k_n)  \cr
 &=\sum_{n=0}^\infty {\ts  \left( -{\lambda\over\kappa}\right)^n } 
     \sum_{k_1,\cdots,k_n}(-1)^{n}
     C(-k_1)\cdots   C(-k_n)\times \cr
 &\phantom{\sum_{n=0}^\infty {\ts  \left( -{\lambda\over\kappa}\right)^n } 
     \sum_{k_1,\cdots,k_n}} 
   \sum_{b_1,\cdots,b_n=0\atop 1b_1+\cdots +nb_n=n}^n 
    \sum_{\pi\in S_n\atop t(\pi)=1^{b_1}\cdots n^{b_n}} \epsilon_\pi 
    \delta_{k_1,k_{\pi 1}}C(k_1) 
   \cdots \delta_{k_n,k_{\pi n}} C(k_n)  \cr    
 &=\sum_{n=0}^\infty {\ts  \left( -{\lambda\over\kappa}\right)^n } 
   \sum_{b_1,\cdots,b_n=0\atop 1b_1+\cdots +nb_n=n}^n 
    {n!\over b_1!\cdots b_n!\>1^{b_1}\cdots n^{b_n} }\>
    (-1)^n (-1)^{n-\sum_{r=1}^n b_r} \times \cr
 &\phantom{=\sum_{n=0}^\infty {\ts  \left( -{\lambda\over\kappa}\right)^n } 
   \sum_{b_1,\cdots,b_n=0\atop 1b_1+\cdots +nb_n=n}^n} 
   \biggl\{ \sum_k[C(k)C(-k)]^1\biggr\}^{b_1}
   \cdots \biggl\{ \sum_k[C(k)C(-k)]^n\biggr\}^{b_n}  \cr
 &=\sum_{n=0}^\infty 
   \sum_{b_1,\cdots,b_n=0\atop 1b_1+\cdots +nb_n=n}^n 
    n! \>  \prod_{r=1}^n {1\over b_r!}\biggl\{ -{\ts{1\over r}}
   \sum_k\left[{\ts  \left( -{\lambda\over\kappa}\right) }
     C(k)C(-k)\right]^r\biggr\}^{b_r}  \cr}$$
The only factor which prevents us from an  explicit summation of 
 the above series is the $n!$. Therefore we substitute 
$$n!=\int_0^\infty e^{-x} x^n dx$$
and obtain 
$$\eqalignno{Z &=\int_0^\infty e^{-x}
     \sum_{n=0}^\infty 
   \sum_{b_1,\cdots,b_n=0\atop 1b_1+\cdots +nb_n=n}^n 
   \prod_{r=1}^n {1\over b_r!} \biggl\{ -{\ts{1\over r}}\sum_k
    \left[ {\ts -{\lambda\over\kappa}x}\> C(k)C(-k)\right]^r\biggr\}^{b_r}dx  \cr
 &=\int_0^\infty e^{-x}\prod_{r=1}^\infty \sum_{b_r=0}^\infty 
   {1\over b_r!} \biggl\{ -{\ts{1\over r}}\sum_k
    \left[ {\ts -{\lambda\over\kappa}x}\> C(k)C(-k)\right]^r\biggr\}^{b_r}dx  \cr
 &=\int_0^\infty e^{-x}\prod_{r=1}^\infty e^{ -{\ts{1\over r}}\sum_k
    \left[ {\ts -{\lambda\over\kappa}x}\> C(k)C(-k)\right]^r } dx \cr
 &=\int_0^\infty e^{-x} e^{ -\sum_k\sum_{r=1}^\infty {\ts{1\over r}}
    \left[ {\ts -{\lambda\over\kappa}x}\> C(k)C(-k)\right]^r } dx \cr
 &=\int_0^\infty e^{-x} e^{  \sum_k \log\left[ 1+
    {\ts {\lambda x\over\kappa}}\> C(k)C(-k)\right] } dx \cr
 &=2\kappa \int_0^\infty  e^{ -\kappa\Bigl(\rho^2 -{1\over\kappa} 
     \sum_k \log\left[ 1+ {\lambda \rho^2\over k_0^2+e_\k^2} \right]\Bigr) }
   \rho\,d\rho  \cr
 &=2\kappa \int_0^\infty  e^{ -\kappa\Bigl(\rho^2 -{1\over L^d} 
     \sum_\k{1\over\beta} 
    \log\left[ {\cosh({\beta\over2}\sqrt{e_\k^2+\lambda \rho^2}) \over 
     \cosh{\beta\over2}e_\k }  \right]^2\Bigr) }
   \rho\,d\rho  \cr}$$
which  coincides with (II.14). 
%
%
%  CHAPTER III
%
%
\bigskip
\bigskip
\bigskip
\noindent {\gross III. The Model with Forward, Exchange and BCS Interaction} 
\bigskip
\bigskip
Let $H=H_0+H_{int}$ where
$$H_0={\ts {1\over L^d}}\sum_{\sigma\in \{\uparrow,\downarrow\}} 
    \sum_{\k\in M} e_\k \,a_{\k\sigma}^+ 
    a_{\k\sigma} \eqno(\III.1)$$
and
$$H_{int}=  {\ts {1\over L^{3d}}} 
        \sum_{\sigma,\tau\in\{\uparrow,\downarrow\}} 
   \sum_{\k,\p,\q} U(\k-\p) \> a_{\k,\sigma}^+ a_{{\q}-\k,\tau}^+ 
   a_{\p,\sigma} a_{{\q}-\p,\tau} \eqno(\III.2) $$
Then 
$$Z={Tr\,e^{-\beta (H_0+H_{int})}\over Tr\,e^{-\beta H_0}}=
    \int e^{-\V(\psi,\bar\psi)}  d\mu_C(\psi,\bar\psi)\eqno(\III.3)$$
where the exponent in the fermionic integral is given by ($\kappa=\beta L^d$)
$$\V(\psi)={\ts {1\over \kappa^3}}\sum_{\sigma,\tau\in\{\up,\down\}}
    \sum_{k,p,q} U(\k-\p)\>\bar\psi_{k,\sigma}
   \bar\psi_{{q}-k,\tau}\psi_{p,\sigma} 
  \psi_{{q}-p,\tau}\eqno(\III.4)$$
and $k=(k_0,\k)\in{\pi\over\beta}(2\Bbb Z+1)\times M$. 
\par
The forward, exchange and BCS approximation is obtained by restricting the 
above sum to the following terms
$${\rm forward:}\;\;\;  \delta_{k,p}\>,\hskip 1cm  {\rm exchange:}
  \;\; \;\delta_{k,-p} \>,\hskip 1cm {\rm BCS:}\;\;\;
 \delta_{q,0} \>. \eqno(\III.5)$$
We consider the approximation 
 $\V(\psi,\bar\psi)\approx\U(\psi,\bar\psi)$ where 
$$ \U(\psi,\bar\psi)={\ts {1\over \kappa^3}}
   \sum_{\sigma,\tau\in\{\up,\down\}}
    \sum_{k,p,q} U(\k'-\p')\bigl[\delta_{k,p}+\delta_{k,-p}
  +\delta_{q,0} \bigr] \>\bar\psi_{k,\sigma}
   \bar\psi_{{q}-k,\tau}\psi_{p,\sigma} 
  \psi_{{q}-p,\tau}   \eqno(\III.6)$$
and 
$$U(\k'-\p')=\cases{ \ds {1\over2} \sum_{\ell=-n}^n \lambda_{|\ell|}
     e^{i\ell\varphi_\k}e^{-i\ell\varphi_\p}
    +{\lambda_0\over2} & if $d=2$ \cr
 \ds \sum_{\ell=0}^n \sum_{m=-\ell}^\ell \lambda_\ell
   \bar Y_{\ell m}\left({\ts{\k'}}\right)
     Y_{\ell m}\left({\ts{\p'}}\right) & if $d=3$ \cr} \;\;=:\;
   \sum_{l=0}^N\lambda_l\> y_l(\k')\>\bar y_l(\p')\eqno(\III.7) $$
This model can be solved explicitely. 
\bigskip
\noindent{\bf Theorem III.1:} Let $\U(\psi,\bar\psi)$ be given by (III.6) and let 
$$Z(\beta,L,\{s_{k,\sigma}\})=\int e^{-\U(\psi,\bar\psi)+{1\over\kappa}
    \sum_{k,\sigma} s_{k,\sigma}\bar\psi_{k,\sigma}\psi_{k,\sigma} }
    d\mu_C(\psi,\bar\psi)$$
Let $w\in\Bbb R$, $\xi^l_{\sigma\tau}=a^l_{\sigma\tau}+ib^l_{\sigma\tau}$, 
 $\phi^l_{\sigma\tau}=u^l_{\sigma\tau}+iv^l_{\sigma\tau}\in \Bbb C$ for 
 $0\le l\le N$, $(\sigma\tau)\in\{\up\up,\down\down,\up\down\}$ and define the 
 fields 
$$\Xi_{\k\up\down}=\sum_{l=0}^N (2\lambda_l)^{1\over2} 
    \xi_{\up\down}^l\,y_l(\k')\>,\;\;\;
  \bar\Xi_{\k\up\down}=\sum_{l=0}^N (2\lambda_l)^{1\over2} 
     \bar\xi_{\up\down}^l\, \bar y_l(\k')$$
$$ \Xi_{\k\sigma\sigma}=\sum_{l=0}^N \lambda_l^{1\over2} 
    \xi_{\sigma\sigma}^l\,y_l(\k')\>, 
  \;\;\;
  \bar\Xi_{\k\sigma\sigma}=\sum_{l=0}^N \lambda_l^{1\over2}
          \bar\xi_{\sigma\sigma}^l\,\bar y_l(\k')\>,$$
$$\Gamma_{\k\up}=U_0^{1\over2}w
      +i\Xi_{\k\up\up}+i\bar\Xi_{\k\up\up}\>,\;\;\;
  \Gamma_{\k\down}=U_0^{1\over2}w
      +i\Xi_{\k\down\down}+i\bar\Xi_{\k\down\down}\>,$$
$$\Phi_{\k\up\down}=\sum_{l=0}^N (2\lambda_l)^{1\over2} 
    \phi_{\up\down}^l\,y_l(\k')\>,\;\;\;
  \bar\Phi_{\k\up\down}=\sum_{l=0}^N (2\lambda_l)^{1\over2} 
     \bar\phi_{\up\down}^l\, \bar y_l(\k')$$
$$\Phi_{\k\sigma\sigma}=\sum_{l=0}^N \lambda_l^{1\over2} \phi_{\sigma\sigma}^l\,
  {\ts [y_l(\k')-y_l(-\k')] } \>,\;\;\;
  \bar\Phi_{\k\sigma\sigma}=\sum_{l=0}^N \lambda_l^{1\over2} 
     \bar\phi_{\sigma\sigma}^l\,  {\ts [\bar y_l(\k')-\bar y_l(-\k')]}\>. $$
where $U_0=U(\k={\bf 0})=\int d^d\x\,U(\x)$. 
For each $k=(k_0,\k)$, let $S_k$ be the $8\times 8$ skew symmetric matrix 
$$S_k=-i \pmatrix{0&-{1\over i}A_{k\up}&\bar\Phi_{\k\up\down}&
    0&\bar\Xi_{\k\up\down}&
   0&0&\bar\Phi_{\k\up\up} \cr 
  {1\over i}A_{k\up} &0&0&\Phi_{\k\up\down}&0&-\Xi_{\k\up\down}&
   \Phi_{\k\up\up}&0\cr
  -\bar\Phi_{\k\up\down}&0&0&-{1\over i}A_{-k\down}&0&
   -\bar\Phi_{\k\down\down}
    &\Xi_{-\k\up\down}&0 \cr
 0&-\Phi_{\k\up\down}&{1\over i}A_{-k\down}&0&-\Phi_{\k\down\down}&0
    &0&-\bar\Xi_{-\k\up\down}\cr
  -\bar\Xi_{\k\up\down}&0&0&\Phi_{\k\down\down}&0
    &{1\over i}A_{k\down}&-\Phi_{-\k\up\down}&0\cr
   0&\Xi_{\k\up\down}&\bar\Phi_{\k\down\down}&0&-{1\over i}A_{k\down}&0
     &0&-\bar\Phi_{-\k\up\down}\cr
 0&-\Phi_{\k\up\up}&-\Xi_{-\k\up\down}&0
     &\Phi_{-\k\up\down}&0&0&{1\over i}A_{-k\up}\cr
 -\bar\Phi_{\k\up\up}&0&0&\bar\Xi_{-\k\up\down}&
   0&\bar\Phi_{-\k\up\down}&-{1\over i}A_{-k\up}
  &0\cr}  $$
where  $A_{k\sigma}=a_k-s_{k\sigma}-\Gamma_{k\sigma}=
     ik_0-e_\k-s_{k\sigma}-\Gamma_{k\sigma}$ 
and let ${\rm Pf}S_k$ be the Pfaffian of $S_k$ given by Lemma III.2 below. Then:
$$Z(\beta,L,\{s_{k,\sigma}\})= 
  \int \prod_{k_0>0\atop\k\in M}
    \biggl\{ {\ts \left({1\over a_k a_{-k}}\right)^2 }
    {\rm Pf} S_k \biggr\} d\nu_\kappa(w,\xi,\phi)  \leqno {\bf a)}$$
where 
$$\eqalignno{ d\nu_\kappa(w,\xi,\phi)&=  
   {\ts  \left( {\kappa\over{4\pi}}\right)^{1\over2}  }
    \,e^{-{\kappa\over4}w^2}dw\, \pro_{l=0}^N \pro_{\sigma\tau }  \Bigl\{
  \ts \left({\kappa\over\pi}\right)^2 \, e^{-\kappa(|\phi_{\sigma\tau}^l|^2
    +|\xi_{\sigma\tau}^l|^2)}
    du_{\sigma\tau}^ldv_{\sigma\tau}^l da_{\sigma\tau}^l
    db_{\sigma\tau}^l   \Bigr\}\cr}$$
$$\la \bar\psi_{p\sigma}\psi_{p'\sigma}\ra=\beta L^d\delta_{p,p'} \, 
   {  \int { {\partial\over \partial s_{p\sigma}}{\rm Pf}S_p \over {\rm Pf}S_p} 
    e^{-\kappa V(w,\xi,\phi)}dw \pro d\xi d\phi \over 
    \int^{\phantom{I}} e^{-\kappa V(w,\xi,\phi)}dw \pro d\xi d\phi } \leqno {\bf b)} $$
where the effective potential $V$ is given by 
$$V(w,\xi,\phi)={\ts {1\over4}} w^2+\sum_{\sigma\tau} \sum_{l=0}^N 
  ( |\phi_{\sigma\tau}^l|^2+|\xi_{\sigma\tau}^l|^2)-{\ts {1\over\beta L^d} }
     \sum_{\k\in M}\log\pro_{k_0>0}\ts {{\rm Pf}S_k\over a_k^2a_{-k}^2}
   \eqno(\III.8)  $$
and $\pro d\xi d\phi =\ds\pro_{\sigma\tau}\pro_{l=0}^N  d\xi_{\sigma\tau}^l
      d\phi_{\sigma\tau}^l$. Here $\sigma\tau\in \{\up\up,\down\down,\up\down\}$. 
\bigskip\goodbreak
\noindent{\bf Remark:} The product ${\ds\pro_{k_0>0}} { {\rm Pf}S_k\over a_k^2 
  a_{-k}^2}$ in the effective potential (III.8) 
   where ${\rm Pf}S_k={\rm Pf}S_k(a_k,a_{-k})$ has to be computed 
 according to the rule 
$$ \prod_{k_0>0}{ { {\rm Pf}S_k\over a_k^2  a_{-k}^2}}=
   \lim_{\epsilon\to 0\atop \epsilon<0} \prod_{k_0>0}
   { { {\rm Pf}S_k(e^{ik_0\epsilon}a_k,e^{-ik_0\epsilon}a_{-k})
     \over a_k^2  a_{-k}^2}}$$
That there are cases where it is necessary to make the phase factors explicit 
can be seen in looking at the example given in the proof of Lemma IV.1.
%
%  PROOF 
%
\bigskip
\noindent{\bf Proof:} Since we assume $U(\p-\k)=U(\k-\p)$, one has 
$$\eqalignno{ \U(\psi,\bar\psi)
   &={\ts {1\over\kappa^3}}\sum_{\sigma,\tau}\sum_{k,p}
   U({\bf 0})\>\bar\psi_{k,\sigma}
   \bar\psi_{p,\tau}\psi_{k,\sigma} 
  \psi_{p,\tau} +{\ts {1\over \kappa^3}}\sum_{\sigma,\tau}
     \sum_{k,p}U(\k'-\p')\>\bar\psi_{p,\sigma}
   \bar\psi_{k,\tau}\psi_{k,\sigma} 
  \psi_{p,\tau}  \cr
 &\phantom{=}+{\ts {1\over \kappa^3}}\sum_{\sigma,\tau}
    \sum_{k,p}U(\k'-\p')\>\bar\psi_{p,\sigma}
   \bar\psi_{-p,\tau}\psi_{k,\sigma} 
  \psi_{-k,\tau}  \cr
&=-{\ts {1\over\kappa^2}{U_0\over\kappa}}
  \sum_{k,p} \Bigl(\bar\psi_{k\up}\psi_{k\up}+\bar\psi_{k\down}\psi_{k\down}\Bigr)
  \Bigl(\bar\psi_{p\up}\psi_{p\up}+\bar\psi_{p\down}\psi_{p\down}\Bigr) \cr
 &\phantom{=}+{\ts {1\over \kappa^3}}\sum_{k,p}U(\k'-\p')\>
  \Bigl\{ \bar\psi_{p\up}\psi_{p\up} 
   \bar\psi_{k\up}\psi_{k\up}+  
   \bar\psi_{p\down}\psi_{p\down} \bar\psi_{k\down}\psi_{k\down}  +
  2 \bar\psi_{p\up}\psi_{p\down} \bar\psi_{k\down}\psi_{k\up}   \Bigr\}  \cr
 &\phantom{=}+{\ts {1\over \kappa^3}} 
   \sum_{k,p}U(\k'-\p')\Bigl\{ 
 \bar\psi_{p\up}\bar\psi_{-p\up}\psi_{k\up}  \psi_{-k\up} +
 \bar\psi_{p\down}\bar\psi_{-p\down}\psi_{k\down}  \psi_{-k\down}   +
  2\bar\psi_{p\up}\bar\psi_{-p\down}\psi_{k\up}\psi_{-k\down}   \Bigr\} \cr }$$
We substitute $ U(\k'-\p')=\sum_{l=0}^N\lambda_l y_l(\k') \bar y_l(\p')$ and use 
the identities 
$$ e^{{1\over2}X^2}=\ts {1\over\sqrt{2\pi}} \int_{\Bbb R} e^{Xw}
      e^{-{1\over2} w^2} dw $$
$$ e^{-2XY}=\ts {1\over 2\pi} \int_{\Bbb R^2} e^{iX\phi+iY\bar\phi} e^{-{1\over2} 
     |\phi|^2} dudv  $$
to obtain 
$$\eqalignno{ e^{-\U(\psi,\bar\psi)}&=  \cr
 \int &\exp\biggl\{ {\ts
  \left( {2U_0\over \kappa}\right)^{1\over2} w\,{1\over\kappa} } 
  \sum_k\Bigl[ \bar\psi_{k\up}\psi_{k\up}+
   \bar\psi_{k\down}\psi_{k\down}\Bigr]\biggr\}
   \times  \cr
  & \exp\biggl\{ \sum_l\biggl[  
   {\ts  i\left( {\lambda_l\over2\kappa}\right)^{1\over2} \bar\xi^l_{\up\up}\, 
  {1\over\kappa}}\sum_k \bar y_l(\k')\,\bar\psi_{k\up} \psi_{k\up} +
   {\ts  i\left( {\lambda_l\over2\kappa}\right)^{1\over2} \xi^l_{\up\up}\, 
  {1\over\kappa}}\sum_k  y_l(\k')\,\bar\psi_{k\up} \psi_{k\up} \biggr]\biggr\} 
   \times \cr
 & \exp\biggl\{ \sum_l\biggl[  
   {\ts  i\left( {\lambda_l\over2\kappa}\right)^{1\over2} \bar\xi^l_{\down\down}\, 
  {1\over\kappa}}\sum_k \bar y_l(\k')\,\bar\psi_{k\down} \psi_{k\down} +
   {\ts  i\left( {\lambda_l\over2\kappa}\right)^{1\over2} \xi^l_{\down\down}\, 
  {1\over\kappa}}\sum_k  y_l(\k')\,\bar\psi_{k\down} \psi_{k\down} \biggr]\biggr\} 
   \times \cr
 & \exp\biggl\{ \sum_l\biggl[  
   {\ts  i\left( {\lambda_l\over\kappa}\right)^{1\over2} \bar\xi^l_{\up \down}\, 
  {1\over\kappa}}\sum_k \bar y_l(\k')\,\bar\psi_{k\up} \psi_{k\down} +
   {\ts  i\left( {\lambda_l\over \kappa}\right)^{1\over2} \xi^l_{\up\down}\, 
  {1\over\kappa}}\sum_k  y_l(\k')\,\bar\psi_{k\down} \psi_{k\up} \biggr]\biggr\} 
   \times \cr  
 & \exp\biggl\{ \sum_l\biggl[  
   {\ts  i\left( {\lambda_l\over2\kappa}\right)^{1\over2} \bar\phi^l_{\up\up}\, 
  {1\over\kappa}}\sum_k \bar y_l(\k')\,\bar\psi_{k\up} \bar\psi_{-k\up} +
   {\ts  i\left( {\lambda_l\over2\kappa}\right)^{1\over2} \phi^l_{\up\up}\, 
  {1\over\kappa}}\sum_k  y_l(\k')\,\psi_{k\up} \psi_{-k\up} \biggr]\biggr\} 
   \times \cr
 & \exp\biggl\{ \sum_l\biggl[  
   {\ts  i\left( {\lambda_l\over2\kappa}\right)^{1\over2} \bar\phi^l_{\down\down}\, 
  {1\over\kappa}}\sum_k \bar y_l(\k')\,\bar\psi_{k\down} \bar\psi_{-k\down} +
   {\ts  i\left( {\lambda_l\over2\kappa}\right)^{1\over2} \phi^l_{\down\down}\, 
  {1\over\kappa}}\sum_k  y_l(\k')\,\psi_{k\down} \psi_{-k\down} \biggr]\biggr\} 
   \times \cr
 & \exp\biggl\{ \sum_l\biggl[  
   {\ts  i\left( {\lambda_l\over\kappa}\right)^{1\over2} \bar\phi^l_{\up\down}\, 
  {1\over\kappa}}\sum_k \bar y_l(\k')\,\bar\psi_{k\up} \bar\psi_{-k\down} +
   {\ts  i\left( {\lambda_l\over\kappa}\right)^{1\over2} \phi^l_{\up\down}\, 
  {1\over\kappa}}\sum_k  y_l(\k')\,\psi_{k\up} \psi_{-k\down} \biggr]\biggr\} 
   d\nu(w,\xi,\phi) \cr}$$
where 
$$d\nu(w,\xi,\phi)=   {\ts {1\over \sqrt{2\pi}} }
    \,e^{-{1\over2}w^2}dw\, \pro_l \pro_{(\sigma\tau)\in\atop\{\up\up,\down\down,
   \up\down\} }
  \ts {1\over (2\pi)^2} \, e^{-{1\over2}(|\phi_{\sigma\tau}^l|^2
    +|\xi_{\sigma\tau}^l|^2)}
    du_{\sigma\tau}^ldv_{\sigma\tau}^l da_{\sigma\tau}^l
    db_{\sigma\tau}^l    $$
By a substitution of variables and collecting terms, one obtains 
$$\eqalignno{ &e^{-\U(\psi,\bar\psi)}=  \cr
 &\int \exp\biggl\{ {\ts {1\over\kappa}}\sum_k \biggl[ U_0^{1\over2}w+
  i\sum_l \lambda_l^{1\over2}\Bigl(\bar\xi_{\up\up}^l \,\bar y_l(\k')+
    \xi_{\up\up}^l \, y_l(\k')\Bigr) \biggr] 
     \bar\psi_{k\up}\psi_{k\up} \biggr\}\times\cr 
  &\phantom{\int} \exp\biggl\{ {\ts {1\over\kappa}}\sum_k \biggl[ U_0^{1\over2}w+
  i\sum_l \lambda_l^{1\over2}\Bigl(\bar\xi_{\down\down}^l \,\bar y_l(\k')+
    \xi_{\down\down}^l \, y_l(\k')\Bigr) \biggr] 
     \bar\psi_{k\down}\psi_{k\down} \biggr\}\times\cr 
  &\phantom{\int} \exp\biggl\{ \sum_l\biggl[  
   {\ts  i(2\lambda_l)^{1\over2} \bar\xi^l_{\up \down}\, 
  {1\over\kappa}}\sum_k \bar y_l(\k')\,\bar\psi_{k\up} \psi_{k\down} +
   {\ts  i(2\lambda_l)^{1\over2} \xi^l_{\up\down}\, 
  {1\over\kappa}}\sum_k  y_l(\k')\,\bar\psi_{k\down} \psi_{k\up} \biggr]\biggr\} 
   \times \cr  
 &\phantom{\int} \exp\biggl\{ \sum_l\biggl[  
   {\ts  i\lambda_l^{1\over2} \bar\phi^l_{\up\up}\, 
  {1\over\kappa}}\sum_k {\ts  {\bar y_l(\k')-\bar y_l(-\k')\over2} }
    \,\bar\psi_{k\up} \bar\psi_{-k\up} +
   {\ts  i\lambda_l^{1\over2} \phi^l_{\up\up}\, 
  {1\over\kappa}}\sum_k {\ts  {y_l(\k')-y_l(-\k')\over2}} 
      \,\psi_{k\up} \psi_{-k\up} \biggr]\biggr\} 
   \times \cr
 &\phantom{\int} \exp\biggl\{ \sum_l\biggl[  
   {\ts  i\lambda_l^{1\over2} \bar\phi^l_{\down\down}\, 
  {1\over\kappa}}\sum_k {\ts  {\bar y_l(\k')-\bar y_l(-\k')\over2} }
     \,\bar\psi_{k\down} \bar\psi_{-k\down} +
   {\ts  i\lambda_l^{1\over2} \phi^l_{\down\down}\, 
  {1\over\kappa}}\sum_k {\ts  {y_l(\k')-y_l(-\k')\over 2} }
     \,\psi_{k\down} \psi_{-k\down} \biggr]\biggr\} 
   \times \cr
 &\phantom{\int} \exp\biggl\{ \sum_l\biggl[  
   {\ts  i(2\lambda_l)^{1\over2} \bar\phi^l_{\up\down}\, 
  {1\over\kappa}}\sum_k \bar y_l(\k')\,\bar\psi_{k\up} \bar\psi_{-k\down} +
   {\ts  i(2\lambda_l)^{1\over2} \phi^l_{\up\down}\, 
  {1\over\kappa}}\sum_k  y_l(\k')\,\psi_{k\up} \psi_{-k\down} \biggr]\biggr\} 
   d\nu_\kappa(w,\xi,\phi) \cr}$$
where $ d\nu_\kappa(w,\xi,\phi)$ is defined in the statement of the theorem. 
Using the definition of the fields $\Xi$, $\Gamma$ and $\Phi$, the above
expression reads 
$$\eqalignno{ e^{-\U(\psi,\bar\psi)}&=  \int \exp\biggl\{ {\ts {1\over\kappa}} \sum_k 
  \Bigl[  \Gamma_{\k\up}\, \bar\psi_{k\up}\psi_{k\up}+
    \Gamma_{\k\down}\, \bar\psi_{k\down}\psi_{k\down}
  +i\bar\Xi_{\k\up\down}\bar\psi_{k\up}\psi_{k\down} 
   +i\Xi_{\k\up\down}\bar\psi_{k\down}\psi_{k\up}  \cr
 &\phantom{\int \exp}   
    + i\bar\Phi_{\k\up\down}\bar\psi_{k\up}\bar\psi_{-k\down}+
   i\Phi_{\k\up\down}\psi_{k\up}\psi_{-k\down} 
  +{\ts {i\over2}}
   \sum_\sigma\bigl( \bar\Phi_{\k\sigma\sigma}\bar\psi_{k\sigma}
  \bar\psi_{-k\sigma}+
   \Phi_{\k\sigma\sigma}\psi_{k\sigma}\psi_{-k\sigma}\bigr) \Bigr] \biggr\}
   d\nu_\kappa  \cr}$$
We now rewrite the exponent in order to perform the fermionic functional integral. 
Since, if the set of spatial momenta satisfy $\k\in M=-M$, 
$$\eqalignno{ \sum_k\Gamma_{k\up}\,\bar\psi_{k\up}\psi_{k\up}&=\sum_{k_0\in 
   {\pi\over\beta}(2\Bbb Z+1)}
    \sum_{\k\in M}\Gamma_{k\up}\,\bar\psi_{k\up}\psi_{k\up}  
  =\sum_{k_0>0}\sum_{\k\in M}\Bigl[ \Gamma_{k\up}\,\bar\psi_{k\up}\psi_{k\up}+
     \Gamma_{-k\up}\,\bar\psi_{-k\up}\psi_{-k\up}\Bigr]  \cr
  &={\ts {1\over2}} \sum_{k\atop k_0>0}
     \Bigl[ \Gamma_{k\up}\bigl(\bar\psi_{k\up}\psi_{k\up}-\psi_{k\up}\bar\psi_{k\up}
    \bigr)+ \Gamma_{-k\up}\bigl( \bar\psi_{-k\up}\psi_{-k\up}-
    \psi_{-k\up}\bar\psi_{-k\up}\bigr)  \Bigr]  \cr}$$
one obtains, using the antisymmetry of the $\Phi_{k\sigma\sigma}$, 
 $\bar\Phi_{k\sigma\sigma}$ 
$$\eqalignno{ \sum_k&\Bigl\{ \su_\sigma  \Gamma_{\k\sigma}\, 
  \bar\psi_{k\sigma}\psi_{k\sigma} 
   +i \bar\Phi_{\k\up\down}\bar\psi_{k\up}\bar\psi_{-k\down}+
   i\Phi_{\k\up\down}\psi_{k\up}\psi_{-k\down}  \cr
 &\phantom{m}  +i\bar\Xi_{\k\up\down}\bar\psi_{k\up}\psi_{k\down} 
   +i\Xi_{\k\up\down}\bar\psi_{k\down}\psi_{k\up} 
  +{\ts {i\over2}} 
   \su_\sigma\bigl( \bar\Phi_{\k\sigma\sigma}\bar\psi_{k\sigma}\bar\psi_{-k\sigma}+
   \Phi_{\k\sigma\sigma}\psi_{k\sigma}\psi_{-k\sigma}\bigr) \Bigr\}   \cr  
 ={\ts {1\over2}}  \sum_{k\atop k_0>0}&
       \biggl\{ \su_\sigma \Bigl[  \Gamma_{\k\sigma} 
  \bigl(\bar\psi_{k\sigma}\psi_{k\sigma}  -\psi_{k\sigma}\bar\psi_{k\sigma} \bigr)+ 
  \Gamma_{-\k\sigma} 
  \bigl(\bar\psi_{-k\sigma}\psi_{-k\sigma} -\psi_{-k\sigma}\bar\psi_{-k\sigma}
   \bigr) \Bigr] \cr
 &\phantom{m}   + i\bar\Phi_{\k\up\down}\bigl(\bar\psi_{k\up}\bar\psi_{-k\down}-
     \bar\psi_{-k\down}\bar\psi_{k\up} \bigr)+
  i \bar\Phi_{-\k\up\down}\bigl(\bar\psi_{-k\up}\bar\psi_{k\down}-
     \bar\psi_{k\down}\bar\psi_{-k\up} \bigr)  \cr
 &\phantom{m} + i \Phi_{\k\up\down}\bigl( \psi_{k\up}
   \psi_{-k\down}-\psi_{-k\down}
     \psi_{k\up}\bigr) +
   i\Phi_{-\k\up\down}\bigl( \psi_{-k\up}\psi_{k\down}-\psi_{k\down}
     \psi_{-k\up}\bigr)  \cr
 &\phantom{m}  +i\bar\Xi_{\k\up\down}
    \bigl( \bar\psi_{k\up}\psi_{k\down}- \psi_{k\down}
    \bar\psi_{k\up} \bigr)+
   i\bar\Xi_{-\k\up\down}\bigl( \bar\psi_{-k\up}\psi_{-k\down}- \psi_{-k\down}
    \bar\psi_{-k\up} \bigr)  \cr
 &\phantom{m}  +i\Xi_{\k\up\down}\bigl( \bar\psi_{k\down}\psi_{k\up}- \psi_{k\up}
    \bar\psi_{k\down}  \bigr)+
   i\Xi_{-\k\up\down}\bigl( \bar\psi_{-k\down}\psi_{-k\up}- \psi_{-k\up}
    \bar\psi_{-k\down}  \bigr)  \cr
  &\phantom{m} +i \su_\sigma\Bigl[ \bar\Phi_{\k\sigma\sigma}\bigl( 
    \bar\psi_{k\sigma}\bar\psi_{-k\sigma}
    -\bar\psi_{-k\sigma}\bar\psi_{k\sigma}\bigr) 
   +\Phi_{\k\sigma\sigma}\bigl( \psi_{k\sigma}\psi_{-k\sigma}-
   \psi_{-k\sigma}\psi_{k\sigma} \bigr)\Bigr] \biggr\}   \cr }$$
Since, if $a_k=ik_0-e_\k$ 
$$d\mu_C(\psi,\bar\psi)=\pro_{k,\sigma}{\ts  {\kappa\over a_k}}\; 
   e^{-{1\over \kappa} \sum_{k,\sigma}a_k \bar\psi_{k,\sigma} 
   \psi_{k,\sigma} } \pro_{k,\sigma} d\psi_{k,\sigma} d\bar\psi_{k,\sigma} $$ one 
obtains 
$$\eqalignno{ Z(\beta,L,\{s_{k,\sigma}\})&=\int e^{-\U(\psi,\bar\psi)+{1\over\kappa}
    \sum_{k,\sigma} s_{k,\sigma}\bar\psi_{k,\sigma}\psi_{k,\sigma} }
    d\mu_C(\psi,\bar\psi)  \cr
 &=\pro_{k,\sigma}{\ts  {\kappa\over a_k}}\int
   \int e^{-{1\over2}{1\over\kappa}\sum_{k_0>0}\sum_{\k} 
     \la \Psi_k,S_k \Psi_k\ra} \pro_{k,\sigma} d\psi_{k\sigma}d\bar\psi_{k\sigma}
    \>d\nu_\kappa(w,\xi,\phi)  \cr
 &= \int \prod_{k_0>0\atop\k\in M}
      \biggl\{ {\ts \left({\kappa^2\over a_k a_{-k}}\right)^2 }
    \int e^{-{1\over2}{1\over\kappa}\la \Psi_k,S_k \Psi_k\ra}
   d\bar\psi_{k\up}d\psi_{k\up} 
      d\bar\psi_{-k\down}d\psi_{-k\down} \times  \cr
 &\phantom{\int \prod_{k_0>0\atop \k\in M } \biggl\{ {\ts
       \left({\kappa^2\over a_k a_{-k}}\right)^2 }
    \int e^{-{1\over2}{1\over\kappa}\la \Psi_k,S_k \Psi_k\ra} }
    d\psi_{k\down} d\bar\psi_{k\down}d\psi_{-k\up}d\bar\psi_{-k\up} \biggr\} 
  d\nu_\kappa(w,\xi,\phi)  \cr
 &=\int \prod_{k_0>0\atop\k\in M}
    \biggl\{ {\ts \left({1\over a_k a_{-k}}\right)^2 }
    {\rm Pf} S_k \biggr\} d\nu_\kappa(w,\xi,\phi) \cr}$$  
where 
$$\Psi_k=\bigl(\bar\psi_{k\up},\psi_{k\up},\bar\psi_{-k\down},\psi_{-k\down}, 
    \psi_{k\down}, \bar\psi_{k\down},\psi_{-k\up},\bar\psi_{-k\up}\bigr)$$ 
and ${\rm Pf}S_k$ is the Pfaffian of the $8\times 8$ skew symmetric 
 matrix $S_k$ defined in the statement of the theorem.
 We used that
$$\pro_{k,\sigma} d\psi_{k\sigma}d\bar\psi_{k\sigma}
 =\pro_{k_0>0}\pro_{\k}\Bigr\{+d\bar\psi_{k\up}d\psi_{k\up} 
      d\bar\psi_{-k\down}d\psi_{-k\down} 
    d\psi_{k\down} d\bar\psi_{k\down}d\psi_{-k\up}d\bar\psi_{-k\up}\Bigr\}  $$
Part b) of the theorem follows from  
$$ \ts {1\over\kappa}\la \bar\psi_{p\sigma}\psi_{p\sigma}\ra=\ts 
  {\partial\over \partial s_{p\sigma}}_{|s=0} \log Z(\beta,L,\{s_{k\sigma}\}) 
  \eqno \blacksquare$$
\bigskip
\noindent{\bf Lemma III.2:} Let $S_k$ be the skew symmetric $8\times 8$ 
 matrix of Theorem III.1. Then the Pfaffian of $S_k$ is given by  
$$\eqalignno{ {\rm Pf} S_k=&\bigl( A_{k\up}A_{-k\down} +\bar\Phi_{\k\up\down} 
  \Phi_{\k\up\down}\bigr)\bigl(A_{-k\up}A_{k\down} +\bar\Phi_{-\k\up\down} 
  \Phi_{-\k\up\down}\bigr)  \cr
 & +A_{k\up}A_{-k\up}\Phi_{\k\down\down}\bar\Phi_{\k\down\down}
  +A_{k\down}A_{-k\down}\Phi_{\k\up\up}\bar\Phi_{\k\up\up} \cr
 &+\Phi_{\k\up\up}\Phi_{\k\down\down}
    \bar\Phi_{\k\up\down}\bar\Phi_{-\k\up\down}+
  \bar\Phi_{\k\up\up}\bar\Phi_{\k\down\down}
    \Phi_{\k\up\down}\Phi_{-\k\up\down}  \cr
  &+\Phi_{\k\up\up}\Phi_{\k\down\down}
    \bar\Phi_{\k\up\up}\bar\Phi_{\k\down\down}  \cr
 &+i\Xi_{\k\up\down}A_{-k\up}\bar\Phi_{\k\up\down}\Phi_{\k\down\down}
  +i\bar\Xi_{\k\up\down}A_{-k\up}\Phi_{\k\up\down}\bar\Phi_{\k\down\down} \cr
  &-i\Xi_{\k\up\down}A_{-k\down}\Phi_{-\k\up\down}\bar\Phi_{\k\up\up} 
  -i\bar\Xi_{\k\up\down}A_{-k\down}\bar\Phi_{-\k\up\down}\Phi_{\k\up\up} \cr
 &+i\Xi_{-\k\up\down}A_{k\down}\Phi_{\k\up\down}\bar\Phi_{\k\up\up} 
  +i\bar\Xi_{-\k\up\down}A_{k\down}\bar\Phi_{\k\up\down}\Phi_{\k\up\up} \cr
 &-i\Xi_{-\k\up\down}A_{k\up}\bar\Phi_{-\k\up\down}\Phi_{\k\down\down} 
  -i\bar\Xi_{-\k\up\down}A_{k\up}\Phi_{-\k\up\down}\bar\Phi_{\k\down\down} \cr
 &+\Xi_{\k\up\down}\bar\Xi_{\k\up\down}A_{-k\up}A_{-k\down}
   +\Xi_{-\k\up\down}\bar\Xi_{-\k\up\down}A_{k\up}A_{k\down} \cr
 &+\Xi_{\k\up\down}\bar\Xi_{-\k\up\down}
     \bar\Phi_{\k\up\down}\Phi_{-\k\up\down}
  +\Xi_{-\k\up\down}\bar\Xi_{\k\up\down}
     \bar\Phi_{-\k\up\down}\Phi_{\k\up\down}  \cr
  &-\Xi_{\k\up\down}\Xi_{-\k\up\down}\bar\Phi_{\k\up\up}\Phi_{\k\down\down}
   -\bar\Xi_{\k\up\down}\bar\Xi_{-\k\up\down}
     \Phi_{\k\up\up}\bar\Phi_{\k\down\down}  \cr
  &+\Xi_{\k\up\down}\bar\Xi_{\k\up\down}
     \Xi_{-\k\up\down}\bar\Xi_{-\k\up\down}  \cr}$$
\bigskip
\noindent{\bf Proof:} The Pfaffian of $S_k$ is given by the sum of all 
 contractions $\sum\pro\la \psi\psi\ra$ of the fields
$$\bar\psi_{k\up},\psi_{k\up},\bar\psi_{-k\down},\psi_{-k\down}, 
    \psi_{k\down}, \bar\psi_{k\down},\psi_{-k\up},\bar\psi_{-k\up}$$
where the value $\la \psi\psi\ra$ is given by the corresponding matrix element. 
That is, the Pfaffian can be evaluated by using Wick's Theorem or integration 
by parts. Since $S_k$ is an $8\times8$ matrix, one has ${\rm Pf}[-S_k]=
  {\rm Pf}S_k$ and 
$$\eqalignno{ {\rm Pf}S_k=&\la \bar\psi_{k\up}\psi_{k\up}\bar\psi_{-k\down}
   \psi_{-k\down} \psi_{k\down}\bar\psi_{k\down}\psi_{-k\up}
     \bar\psi_{-k\up} \ra_{S_k} \cr
 =&-A_{k\up}\la  \bar\psi_{-k\down}
   \psi_{-k\down} \psi_{k\down}\bar\psi_{k\down}\psi_{-k\up}
     \bar\psi_{-k\up} \ra_{S_k} 
  -i\bar\Phi_{\k\up\down} \la
   \psi_{k\up} \psi_{-k\down} \psi_{k\down}\bar\psi_{k\down}\psi_{-k\up}
     \bar\psi_{-k\up} \ra_{S_k}  \cr
  &-i\bar\Xi_{\k\up\down}\la \psi_{k\up}\bar\psi_{-k\down}
   \psi_{-k\down} \bar\psi_{k\down}\psi_{-k\up}
     \bar\psi_{-k\up} \ra_{S_k} 
       +i\bar\Phi_{\k\up\up} 
  \la  \psi_{k\up}\bar\psi_{-k\down}
   \psi_{-k\down} \psi_{k\down}\bar\psi_{k\down}\psi_{-k\up} \ra_{S_k}  \cr
 =&+A_{k\up}A_{-k\down}\la \psi_{k\down}\bar\psi_{k\down}\psi_{-k\up}
     \bar\psi_{-k\up} \ra_{S_k}+iA_{k\up}\bar\Phi_{\k\down\down} \la 
    \psi_{-k\down} \psi_{k\down}\psi_{-k\up}
     \bar\psi_{-k\up} \ra_{S_k} \cr
  &+iA_{k\up}\Xi_{-\k\up\down}\la 
    \psi_{-k\down} \psi_{k\down}\bar\psi_{k\down}
     \bar\psi_{-k\up} \ra_{S_k}      \cr
 &+\bar\Phi_{\k\up\down} \Phi_{\k\up\down} \la 
       \psi_{k\down}\bar\psi_{k\down}\psi_{-k\up}
     \bar\psi_{-k\up} \ra_{S_k} -\bar\Phi_{\k\up\down}\Xi_{\k\up\down} \la 
  \psi_{-k\down} \psi_{k\down}\psi_{-k\up}
     \bar\psi_{-k\up} \ra_{S_k} \cr
  &-\bar\Phi_{\k\up\down}\Phi_{\k\up\up}
     \la\psi_{-k\down} \psi_{k\down}\bar\psi_{k\down}
     \bar\psi_{-k\up} \ra_{S_k} \cr    
 &- \bar\Xi_{\k\up\down}
    \Phi_{\k\up\down}\la \bar\psi_{-k\down}
     \bar\psi_{k\down}\psi_{-k\up} \bar\psi_{-k\up} \ra_{S_k}
  - \bar\Xi_{\k\up\down}\Xi_{\k\up\down}\la \bar\psi_{-k\down}
   \psi_{-k\down} \psi_{-k\up} \bar\psi_{-k\up} \ra_{S_k} \cr
 &-\bar\Xi_{\k\up\down}\Phi_{\k\up\up}\la  \bar\psi_{-k\down}
   \psi_{-k\down} \bar\psi_{k\down} \bar\psi_{-k\up} \ra_{S_k}  \cr
 & +\bar\Phi_{\k\up\up}\Phi_{\k\up\down}\la \bar\psi_{-k\down}
      \psi_{k\down}\bar\psi_{k\down}\psi_{-k\up} \ra_{S_k} 
  -\bar\Phi_{\k\up\up}\Xi_{\k\up\down}\la \bar\psi_{-k\down}
   \psi_{-k\down} \psi_{k\down}\psi_{-k\up} \ra_{S_k} \cr
 &-\bar\Phi_{\k\up\up}\Phi_{\k\up\up} \la \bar\psi_{-k\down}
   \psi_{-k\down} \psi_{k\down}\bar\psi_{k\down} \ra_{S_k} \cr  
 =&+\bigl( A_{k\up}A_{-k\down}+\Phi_{\k\up\down}\bar\Phi_{\k\up\down}\bigr) 
      \la \psi_{k\down}\bar\psi_{k\down}\psi_{-k\up}
     \bar\psi_{-k\up} \ra_{S_k}  \cr
  &+\bigl(A_{k\up}i\Xi_{-\k\up\down}-\bar\Phi_{\k\up\down}\Phi_{\k\up\up}\bigr) 
   \la \psi_{-k\down} \psi_{k\down}\bar\psi_{k\down}
     \bar\psi_{-k\up} \ra_{S_k}  \cr
 &+\bigl( A_{k\up}i\bar\Phi_{\k\down\down}
      -\bar\Phi_{\k\up\down}\Xi_{\k\up\down}\bigr) \la 
    \psi_{-k\down} \psi_{k\down}\psi_{-k\up}
     \bar\psi_{-k\up} \ra_{S_k} \cr 
  &- \bar\Xi_{\k\up\down}
    \Phi_{\k\up\down}\la \bar\psi_{-k\down}
     \bar\psi_{k\down}\psi_{-k\up} \bar\psi_{-k\up} \ra_{S_k}
  - \bar\Xi_{\k\up\down}\Xi_{\k\up\down}               \la \bar\psi_{-k\down}
   \psi_{-k\down} \psi_{-k\up} \bar\psi_{-k\up} \ra_{S_k} \cr
 &-\bar\Xi_{\k\up\down}\Phi_{\k\up\up}\la  \bar\psi_{-k\down}
   \psi_{-k\down} \bar\psi_{k\down} \bar\psi_{-k\up} \ra_{S_k} 
  +\bar\Phi_{\k\up\up}\Phi_{\k\up\down}\la \bar\psi_{-k\down}
      \psi_{k\down}\bar\psi_{k\down}\psi_{-k\up} \ra_{S_k} \cr
 & -\bar\Phi_{\k\up\up}\Xi_{\k\up\down}\la \bar\psi_{-k\down}
   \psi_{-k\down} \psi_{k\down}\psi_{-k\up} \ra_{S_k}
  -\bar\Phi_{\k\up\up}\Phi_{\k\up\up} \la \bar\psi_{-k\down}
   \psi_{-k\down} \psi_{k\down}\bar\psi_{k\down} \ra_{S_k} \cr  
 =&+\bigl( A_{k\up}A_{-k\down}+\Phi_{\k\up\down}\bar\Phi_{\k\up\down}\bigr) 
   \bigl(A_{k\down}A_{-k\up}+\Phi_{-\k\up\down}\bar\Phi_{-\k\up\down}\bigr) \cr 
 &+\bigl(A_{k\up}i\Xi_{-\k\up\down}-\bar\Phi_{\k\up\down}\Phi_{\k\up\up}\bigr) 
  \bigl(-\Phi_{\k\down\down}\bar\Phi_{-\k\up\down}-i\bar\Xi_{-k\up\down} 
   A_{k\down}\bigr)  \cr
 &+\bigl( A_{k\up}i\bar\Phi_{\k\down\down}
      -\bar\Phi_{\k\up\down}\Xi_{\k\up\down}\bigr)\bigl( -i\Phi_{\k\down\down}
   A_{-k\up}-\bar\Xi_{-k\up\down}\Phi_{-\k\up\down}\Bigr)  \cr
 &- \bar\Xi_{\k\up\down}\Phi_{\k\up\down}\bigl(-i\bar\Phi_{\k\down\down}
  A_{-k\up}-\Xi_{-\k\up\down}\bar\Phi_{-\k\up\down}\bigr) \cr
 &- \bar\Xi_{\k\up\down}\Xi_{\k\up\down}\bigl( -A_{-k\down}A_{-k\up}
    -\Xi_{-\k\up\down}\bar\Xi_{-\k\up\down}\bigr) \cr
 &-\bar\Xi_{\k\up\down}\Phi_{\k\up\up}\bigl(A_{-k\down}i\bar\Phi_{-\k\up\down}
   +\bar\Phi_{\k\down\down}\bar\Xi_{-k\up\down}\bigr) \cr
  &+\bar\Phi_{\k\up\up}\Phi_{\k\up\down}\bigl( \bar\Phi_{\k\down\down} 
   \Phi_{-\k\up\down}+i\Xi_{-\k\up\down}A_{k\down} \bigr) \cr
 &-\bar\Phi_{\k\up\up}\Xi_{\k\up\down}\bigl( A_{-k\down}i\Phi_{-\k\up\down} 
   +\Xi_{-\k\up\down}\Phi_{\k\down\down} \bigr) \cr
 &-\bar\Phi_{\k\up\up}\Phi_{\k\up\up} \bigr( -A_{-k\down}A_{k\down} 
   -\bar\Phi_{\k\down\down}\Phi_{\k\down\down}\bigr) \cr}$$
By multiplying out the brackets one obtains the stated formula $\blacksquare$
\bigskip
Before we consider some special cases in section IV where the effective potential 
 and the two point functions are computed more explicitely, in the following 
 theorem  we write down the
integral representation for the generating functional of the connected amputated 
 Greens functions. 
\bigskip
\noindent{\bf Theorem III.3:} Let $\U(\psi,\bar\psi)$ be given by (III.6) and let $V$ 
 be the effective potential (III.8) Let 
$$G(\eta)=\log \int e^{-\U(\psi+\eta,\bar\psi+\bar\eta)} d\mu_C(\psi,\bar\psi)$$
be the generating functional for the connected amputated Greens functions. 
Then, if $\Eta k$ denotes the eight component vector 
$$\Eta k=(a_{k}\eta_{k\up},-a_{k}\bar\eta_{k\up},
  a_{-k}\eta_{-k\down},-a_{-k}\bar\eta_{-k\down},
  -a_{k}\bar\eta_{k\down},a_{k}\eta_{k\down},-a_{-k}\bar\eta_{-k\up},
    a_{-k}\eta_{-k\up})$$
and $S_k=S_k(w,\xi,\phi)$ is the $8\times 8$ matrix of Theorem III.1, 
one has the following integral representation
$$G(\eta)-G(0)=-{\ts{1\over\kappa}}\sum_{k\sigma} a_k\bar\eta_{k\sigma}
    \eta_{k\sigma}\>+\>  \log{ \int e^{-{1\over2}{1\over\kappa}
     \sum_k\la \Eta k,S_k^{-1} \Eta k\ra}  e^{-\kappa V(w,\xi,\phi)} dw\pro d\xi
    d\phi  \over  \int^{\phantom{I}} e^{-\kappa V(w,\xi,\phi)} dw\pro d\xi d\phi }  $$
\bigskip
\noindent{\bf Proof:}  By a substitution of Grassmann variables, 
$$\int e^{-\U(\psi+\eta,\bar\psi+\bar\eta)}d\mu_C=$$
$$   e^{-{1\over\kappa}\sum_{k\sigma}a_k\bar\eta_{k\sigma}\eta_{k\sigma}} 
  \int e^{-\U(\psi,\bar\psi)}\pro_{k\sigma}{\ts{\kappa\over a_k}}\> 
  e^{{1\over\kappa}\sum_{k\sigma} a_k[\bar\psi_{k\sigma}\eta_{k\sigma} + 
   \bar\eta_{k\sigma}\psi_{k\sigma}-
    \bar\psi_{k\sigma}\psi_{k\sigma}]} \pro_{k\sigma}d\psi_{k\sigma} d\bar 
   \psi_{k\sigma} $$
As in the proof of Theorem III.1, one has 
$$e^{-\U(\psi,\bar\psi)-{1\over\kappa}\sum_{k\sigma}a_k\bar\psi_{k\sigma} 
    \psi_{k\sigma}}= \int e^{-{1\over2}{1\over\kappa} \sum_{k_0>0}\sum_{\k} 
   \la \Psi_k,S_k \Psi_k\ra } d\nu_\kappa(w,\xi,\phi) $$
such that 
$$\eqalignno{ \int e^{-\U(\psi,\bar\psi)}&\pro_{k\sigma}{\ts{\kappa\over a_k}}\> 
  e^{{1\over\kappa}\sum_{k\sigma} a_k[\bar\psi_{k\sigma}\eta_{k\sigma} + 
   \bar\eta_{k\sigma}\psi_{k\sigma}-
    \bar\psi_{k\sigma}\psi_{k\sigma}]} \pro_{k\sigma}d\psi_{k\sigma} d\bar 
   \psi_{k\sigma}   \cr
 &=  \int\int  \pro_{k_0>0,\k}{\ts \left( { {\rm Pf}S_k\over a_k^2a_{-k}^2}\right)}\,
     e^{{1\over\kappa}\sum_{k\sigma} a_k[\bar\psi_{k\sigma}\eta_{k\sigma} + 
   \bar\eta_{k\sigma}\psi_{k\sigma}]} \times  \cr
 &\phantom{+ \int\int } 
        \pro_{k_0>0,\k} {\ts {\kappa^4\over 
    {\rm Pf}S_k }} \>
    e^{-{1\over2}{1\over\kappa} \sum_{k_0>0}\sum_{\k} 
   \la \Psi_k,S_k \Psi_k\ra } d\nu_\kappa(w,\xi,\phi) \pro_{k\sigma} 
   d\psi_{k\sigma} d\bar  \psi_{k\sigma}   \cr
 &=  \int\int  \pro_{k_0>0,\k}{\ts \left( { {\rm Pf}S_k\over a_k^2a_{-k}^2}\right)}\,
     e^{{1\over\kappa}\sum_{k_0>0,\k} \la \Psi_k,\Eta k\ra } \times  \cr
 &\phantom{+ \int\int } 
   \pro_{k_0>0,\k} {\ts {\kappa^4\over 
    {\rm Pf}S_k }} \>
    e^{-{1\over2}{1\over\kappa} \sum_{k_0>0}\sum_{\k} 
   \la \Psi_k,S_k \Psi_k\ra }  \pro_{k_0>0,\k}d\Psi_k\;d\nu_\kappa(w,\xi,\phi)
     \cr
 &=\int  \pro_{k_0>0,\k}{\ts \left( { {\rm Pf}S_k\over a_k^2a_{-k}^2}\right)}\,
   e^{-{1\over2}{1\over\kappa} \sum_{k_0>0,\k} \la \Eta k,S_k^{-1} \Eta k\ra} 
   d\nu_\kappa(w,\xi,\phi)  \cr}$$
where 
$$d\Psi_k=d\bar\psi_{k\up}d\psi_{k\up} 
      d\bar\psi_{-k\down}d\psi_{-k\down} 
    d\psi_{k\down} d\bar\psi_{k\down}d\psi_{-k\up}d\bar\psi_{-k\up} $$
By definition of $V$ 
$$\pro_{k_0>0,\k}{\ts \left( { {\rm Pf}S_k\over a_k^2a_{-k}^2}\right)}\,
    d\nu_\kappa(w,\xi,\phi) =e^{-\kappa V(w,\xi,\phi)} dw\pro d\xi d\phi$$
which proves the theorem $\blacksquare$ 
%
%
% CHAPTER IV
%
%
\bigskip
\bigskip
\bigskip
\noindent {\gross IV. Some Special Cases} 
\bigskip
\bigskip
\noindent{\mittel IV.1 A Delta Function Interaction with Forward, Exchange 
  and BCS Term} 
\bigskip
We consider the model 
$$Z(\beta,L,\{s_{k,\sigma}\})=\int e^{-\U(\psi,\bar\psi)+{1\over\kappa}
    \sum_{k,\sigma} s_{k,\sigma}\bar\psi_{k,\sigma}\psi_{k,\sigma} }
    d\mu_C(\psi,\bar\psi)  \eqno (\IV.1)$$
where 
$$  \U(\psi,\bar\psi)={\ts {\lambda\over \kappa^3}}
   \sum_{\sigma,\tau\in\{\up,\down\}}
    \sum_{k,p,q}  \bigl[\delta_{k,p}+\delta_{k,-p}
  +\delta_{q,0} \bigr] \>\bar\psi_{k,\sigma}
   \bar\psi_{{q}-k,\tau}\psi_{p,\sigma} 
  \psi_{{q}-p,\tau} \eqno(\IV.2)$$
In that case one has $N=0$ in Theorem III.1 and all fields are independent 
  of $\k$: 
$$\Xi_{\k\up\down}=(2\lambda)^{1\over2}\xi_{\up\down}\,,\;\;\;
   \bar\Xi_{\k\up\down}=(2\lambda)^{1\over2}\bar\xi_{\up\down}\,,\hskip 1cm
  \Xi_{\k\sigma\sigma}=\lambda^{1\over2} \xi_{\sigma\sigma}\,,\;\;\;
   \bar\Xi_{\k\sigma\sigma}=\lambda^{1\over2} \bar\xi_{\sigma\sigma}\,,$$
$$\Gamma_{\k\sigma}=\Gamma_\sigma 
   =\lambda^{1\over2}(w+i\xi_{\sigma\sigma}
     +i\bar\xi_{\sigma\sigma})=\ts 
    (2\lambda)^{1\over2}{w+2ia_{\sigma\sigma}\over \sqrt 2}$$
if $\xi_{\sigma\sigma}=a_{\sigma\sigma}+ib_{\sigma\sigma}$ and 
$$\Phi_{\k\up\down}=(2\lambda)^{1\over2}\phi_{\up\down}
  \,,\;\;\;\bar\Phi_{\k\up\down}=(2\lambda)^{1\over2}
   \bar\phi$$
and $\Phi_{\k\up\up}=\Phi_{\k\down\down}=0$.  
 Substituting $2\lambda$ by $\lambda$, the Pfaffian of $S_k$ becomes
$$\eqalignno{ \Pf S_k=\,&(A_{k\up}A_{-k\down}
   +\Phi_{\up\down}\bar\Phi_{\up\down})(A_{-k\up}A_{k\down}+
  \Phi_{\up\down}\bar\Phi_{\up\down}) +
   (A_{k\up}A_{k\down}+A_{-k\up}A_{-k\down}) 
   \Xi_{\up\down}\bar\Xi_{\up\down}  \cr
 &+2\Xi_{\up\down}\bar\Xi_{\up\down} \Phi_{\up\down}\bar\Phi_{\up\down}
   +(\Xi_{\up\down}\bar\Xi_{\up\down})^2   \cr
 =\,&(A_{k\up}A_{-k\down}+\lambda\rho^2)(A_{-k\up}A_{k\down}+
  \lambda\rho^2)+(A_{k\up}A_{k\down}+A_{-k\up}A_{-k\down}) \lambda x^2 \cr
  &+2\lambda^2x^2\rho^2+\lambda^2x^4 &(\IV.3) \cr}$$
where $\rho=|\phi|$, $x=|\xi_{\up\down}|$ and 
 $A_{k\sigma}=ik_0-e_\k-s_{k\sigma}-\Gamma_\sigma$. 
 We assume $e_\k=e_{-\k}$. 
The two point function $\la \bar\psi_{p\up}\psi_{p\up}\ra$ is given by 
$$\la \bar\psi_{p\up}\psi_{p'\up}\ra=\beta L^d \delta_{p,p'} \, 
     \int\ts { {\partial\over \partial s_{p\up}}{\rm Pf}S_p \over {\rm Pf}S_p} \>
    \delta_\kappa(\Gamma_\sigma ,x,\rho)\>dwda_{\up\up}da_{\down\down}
     xdx \rho d\rho \eqno (\IV.4)$$
where 
$$ \delta_\kappa(\Gamma_\sigma,x,\rho)={e^{-\kappa V(\Gamma_\sigma,x,\rho)} 
     \over 
    \int e^{-\kappa V(\Gamma_\sigma,x,\rho)}
    dwda_{\up\up}da_{\down\down}
     xdx \rho d\rho}   \eqno(\IV.5)  $$
is, if the volume or $\kappa=\beta L^d$ goes to infinity, a $\delta$-sequence
    which 
 forces the variables  $w,a_{\sigma\sigma},x$ and $\rho$ to take values where 
 the real part of the effective potential 
$$V(\Gamma_\sigma,x,\rho)=
    {\ts {1\over4} w^2+a_{\up\up}^2+a_{\down\down}^2+
   x^2+\rho^2-{1\over \beta L^d}} \sum_{\k\in M} 
    \log\pro_{k_0>0}\ts {{\rm Pf}S_k\over a_k^2a_{-k}^2}  $$
 has its global minimum. 
\par
A detailed analysis to find the global minimum, in particular for the 
  case discussed 
 in section IV.2, we plan to give in a forthcoming paper. Here we simply 
 write down the two point function under certain assumptions of what the 
   minimum  may be.
\medskip
Since $V$ is symmetric in $a_{\up\up}$ and $a_{\down\down}$, we 
 substitute (IV.4) by 
$$\la \bar\psi_{p\up}\psi_{p'\up}\ra=\beta L^d \delta_{p,p'} \, 
   {  \int F^{\up\up}_p(w,a,x,\rho)\> 
    e^{-\kappa V(w,a,x,\rho)}dwda
     xdx \rho d\rho \over 
    \int^{\phantom{I}} e^{-\kappa V(w,a,x,\rho)}
    dwda xdx \rho d\rho  } \eqno(\IV.6)  $$
where
$$V(w,a,x,\rho)={\ts {1\over4} w^2+a^2+
   x^2+\rho^2-{1\over \beta L^d}} \sum_{\k\in M} 
    \log\pro_{k_0>0}\ts {{\rm Pf}S_k\over a_k^2a_{-k}^2} \eqno (\IV.7) $$
and, for $s_{k\sigma}=0$, $A_{k\up}=A_{k\down}=ik_0-e_\k-\gamma$ where 
 $\gamma=\lambda^{1\over2}(w+\sqrt 2\, a)$. Then the two point
function is given by 
$$ F^{\up\up}_p(w,a ,x,\rho)= \ts
     { {\partial\over \partial s_{p\up}}{\rm Pf}S_p \over {\rm Pf}S_p}{|_{s_{p\sigma}
  =0}} =-{ A_{-p}(A_pA_{-p}+\lambda \rho^2)+A_p\lambda x^2\over 
      (A_{p}A_{-p}+\lambda\rho^2)^2 
   +(A_{p}^2+A_{-p}^2+2\lambda\rho^2) \lambda x^2 
    +\lambda^2x^4   } \eqno (\IV.8)  $$
To proceed one has to compute the effective potential, that is, the product in 
 (IV.7).
\bigskip
\noindent{\bf Lemma IV.1:} {\bf a)} Let $a_k=ik_0-e_\k$, $e_{-\k}=e_{\k}$ 
  and let $b,c,d$ be some
complex numbers. Then 
$$\eqalignno{ \prod_{k_0\in{\pi\over\beta}(2\Bbb Z+1)} 
  {\ts { a_ka_{-k}+ba_k+ca_{-k}+d\over a_ka_{-k} }}\>&:=\>\lim_{\epsilon\to 0\atop 
  \epsilon<0} \prod_{k_0\in{\pi\over\beta}(2\Bbb Z+1)} 
 {\ts  { a_ka_{-k}+be^{i\ep k_0}a_k+ce^{-i\ep k_0}a_{-k}+d\over a_ka_{-k} }} \cr
 &\>=\;\ts{\cosh{\beta\over2}\sqrt{e_\k^2+\omega_+}\over \cosh{\beta\over2}e_\k}\,
 \, {\cosh{\beta\over2}\sqrt{e_\k^2+\omega_-}\over \cosh{\beta\over2}e_\k}\>
    e^{{\beta\over2}(b+c)} &(\IV.9) \cr}$$
where $\omega_{\pm}={p\pm\sqrt{p^2-4q}\over2}$ with $p=(b+c-e_\k)^2-e_k^2 
  -4bc+2d$ and $q=4bce_\k^2-2(b+c)e_\k d+d^2$. 
In particular, for $b=c$ and $d=c^2$ 
$$\prod_{k_0\in{\pi\over\beta}(2\Bbb Z+1)} {\ts 
  \left( 1+{c\over a_k}\right)}\>:=\>\lim_{\epsilon\to 0\atop 
  \epsilon<0} \prod_{k_0\in{\pi\over\beta}(2\Bbb Z+1)} \ts 
  \left( 1+e^{-i\ep k_0}{c\over a_k}\right)\>=\>  {1+e^{-\beta(e_\k-c)}\over 
    1+e^{-\beta e_\k} } $$ 
{\bf b)} Let $A_k=a_k-\gamma$, $\gamma$ a complex number, and let 
$$\Pf S_k=\Pf S_k(a_k,a_{-k})=(A_{k}A_{-k}+\lambda\rho^2)^2 
   +(A_{k}^2+A_{-k}^2+2\lambda\rho^2) \lambda x^2 
    +\lambda^2x^4  $$
Then 
$$\eqalignno{ \prod_{k_0\in{\pi\over\beta}(2\Bbb Z+1)\atop k_0>0}
  {\ts  { \Pf S_k(a_k,a_{-k})\over 
   a_k^2 a_{-k}^2}}\>&:=\>\lim_{\epsilon\to 0\atop 
  \epsilon<0} \prod_{k_0\in{\pi\over\beta}(2\Bbb Z+1)\atop k_0>0} 
 \ts { \Pf S_k(e^{i\ep k_0}a_k,e^{-i\ep k_0} a_{-k})\over 
   a_k^2 a_{-k}^2}  \cr
 &\>= \ts { \sinh^2({\beta\over2}i\sqrt\lambda \,x)\,+\,\cosh^2({\beta\over2}
    \sqrt{(e_\k+\gamma)^2+\lambda\rho^2}\,) \over 
    \cosh^2{\beta\over2}e_\k  } \,e^{-\beta \gamma}    &(\IV.10)   \cr}$$
\bigskip
\noindent{\bf Proof:} We first explain why the definition of the product given in 
 the lemma is the right one. 
\par
Consider the quadratic perturbation 
$$H=H_0+\lambda N={\ts {1\over L^d}} \sum_{\k\sigma} e_\k \, a_{\k\sigma}^+ 
  a_{\k\sigma}+{\ts {\lambda\over L^d}}\sum_{\k\sigma} a_{\k\sigma}^+ 
  a_{\k\sigma}$$
One finds by explicit computation of the trace 
$$Z(\lambda,\beta)={\ts { Tr\, e^{-\beta(H_0+\lambda N)}\over Tr\, e^{-\beta H_0}}}
  =\pro_{\k\sigma}\ts {1+e^{-\beta(e_\k+\lambda)}\over 1+e^{-\beta e_\k}} $$
On the other hand, by explicit summation of the perturbation series or
equivalently, by performing the fermionic functional integral, one obtains 
$$\eqalignno{ Z(\lambda,\beta)&=\int e^{{\lambda\over\beta L^d} \sum_{k\sigma} 
  \bar\psi_{k\sigma}  \psi_{k\sigma} } d\mu_C \cr 
 &=\pro_{k\sigma} {\ts {\beta L^d\over ik_0-e_\k}} \int 
   e^{{\lambda\over\beta L^d} \sum_{k\sigma} 
  \bar\psi_{k\sigma}  \psi_{k\sigma} }
  e^{-{1\over \beta L^d}\sum_{k\sigma}(ik_0-e_\k)\bar\psi_{k\sigma} 
    \psi_{k\sigma}} \pro_{k\sigma} d\psi_{k\sigma} d\bar\psi_{k\sigma}  \cr
 &=\pro_{k\sigma} {\ts {ik_0-e_\k-\lambda\over ik_0-e_\k}}= 
   \pro_{\k\sigma} \pro_{k_0}\ts \left( 1-{\lambda\over a_k}\right)&(\IV.11) \cr}$$
In the perturbation expansion the product in the last equation results from 
$$\pro_{k\sigma}{\ts \left( 1-{\lambda\over a_k}\right)}= 
  e^{\sum_{k\sigma} \log\left( 1-{\lambda\over a_k}\right)} =
   e^{-\sum_{r=1}^\infty {\lambda^r\over r} \sum_{k\sigma}{1\over {a_k}^r}}$$
The term for $r=1$ requires special attention since $\sum_{k_0} {1\over 
  ik_0-e_\k}$ is not absolutely convergent. This sum enters the perturbation 
 series as the propagator 
$$C(x_0-x_0',\x-\x')=\ts {Tr\, e^{-\beta H_0} T\psi(x_0\x)\psi^+(x_0'\x')\over 
    Tr\,e^{-\beta H_0} }$$
evaluated at $x_0-x_0'=0$. Here $T\psi(x_0\x)\psi^+(x_0'\x')$ is
defined to be 
 $\psi(x_0\x)\psi^+(x_0'\x')$ if $x_0\ge x_0'$ and $-\psi^+(x_0'\x')\psi(x_0\x)$ 
 if $x_0<x_0'$. By explicit computation one finds that for $x_0\ne x_0'$ 
$$\eqalignno{ C(x_0-x_0',\x-\x')&={\ts {1\over L^d}}\sum_\k 
  e^{i\k(\x-\x')} e^{-e_\k(x_0-x_0')}\left[ \theta_\beta(-e_\k)\theta(x_0'-x_0)-
  \theta_\beta(e_\k)\theta(x_0-x_0')\right]  \cr
 &={\ts {1\over \beta L^d}} \sum_{\k,k_0} e^{i\k(\x-\x')-ik_0(x_0-x_0')}
   \ts {1\over ik_0-e_\k} \cr}$$
and for $x_0=x_0'$
$$\eqalignno{ C(0,\x-\x')&={\ts {1\over L^d}} \sum_\k e^{i\k(\x-\x')}
    \theta_\beta(-e_\k) 
 =\lim_{\ep\to 0\atop \ep<0} {\ts {1\over \beta  L^d}}\sum_{\k,k_0} 
   e^{i\k(\x-\x')-ik_0\ep} \ts {1\over ik_0-e_\k} \cr}$$
Here $\theta$ is a step function being 1 for positive arguments and 
 $\theta_\beta(-v)={e^{-\beta v}\over 1+e^{-\beta v}}$ is an approximate step 
function. Therefore the $k_0$-sums have to be evaluated according to 
$$\sum_{k_0}{\ts {1\over ik_0-e_\k}}={\sum_{k_0}}^\ep{\ts {1\over ik_0-e_\k}}
  :=\lim_{\ep\to0\atop \ep<0}\sum_{k_0} e^{-ik_0\ep}{\ts {1\over ik_0-e_\k}}=
  \beta \theta_\beta(-e_\k)=\beta\> \ts {e^{-\beta e_\k}\over 1+e^{-\beta e_\k}}$$
We emphasize this point because the following computation leads to a wrong 
 result 
$$\sum_{k_0}{\ts {1\over ik_0-e_\k}}={\sum_{k_0}}^{sym}{\ts {1\over ik_0-e_\k}}
  :={\ts{1\over2}} \sum_{k_0} \left[ {\ts {1\over ik_0-e_\k}}+
    {\ts {1\over -ik_0-e_\k}}\right] 
  =\sum_{k_0}{\ts {-e_\k\over k_0^2+e_\k^2}}=\ts -{\beta\over2}\>
     {1-e^{-\beta e_\k}\over 1+e^{-\beta e_\k}}$$
Therefore the product in (IV.11) is found to be 
$$\eqalignno{ \pro_{k_0}{\ts\left(1-{\lambda\over ik_0-e_\k}\right)}&=
  {\pro_{k_0}}^\ep {\ts\left(1-{\lambda\over ik_0-e_\k}\right)}=
  e^{-\sum_{r=1}^\infty{\lambda^r\over r}
         \sum_{k_0}^\ep{1\over (ik_0-e_\k)^r} } \cr
  &=e^{-\sum_{r=1}^\infty{\lambda^r\over r}\sum_{k_0}^{sym} 
   {1\over (ik_0-e_\k)^r} }
  \; e^{\sum_{k_0}^{sym}{\lambda\over ik_0-e_\k}- 
   \sum_{k_0}^{\ep}{\lambda\over ik_0-e_\k} }  \cr 
 &={\pro_{k_0}}^{sym} {\ts\left(1-{\lambda\over ik_0-e_\k}\right)}\; 
 e^{-{\beta\over2}\lambda}=\pro_{k_0>0} {\ts {k_0^2+(e_\k+\lambda)^2 \over 
   k_0^2+e_\k^2 }\;e^{-{\beta\over2}\lambda}} \cr
 &\buildrel (\II.6)\over = 
 \ts {\cosh{\beta\over2}(e_\k+\lambda)\over \cosh{\beta\over2}e_\k}\;
   e^{-{\beta\over2}\lambda} =\ts {1+e^{-\beta(e_\k+\lambda)}\over 
   1+e^{-\beta e_\k}} &(\IV.12) \cr}$$
which proves (IV.9) for $b=c$ and $d=c^2$.  
\par
In the general case one may proceed similarly 
  to obtain (using $a_k+a_{-k}=-2e_\k$) 
$$\eqalignno{ \pro_{k_0\in{\pi\over\beta}(2\Bbb Z+1)} &
  {\ts { a_ka_{-k}+b a_k+ca_{-k}+d\over a_ka_{-k} }} 
   = \lim_{\ep\to 0\atop \ep<0} 
    \pro_{ k_0} {\ts \left( 1+{be^{i\ep k_0}a_k+c
   e^{-i\ep k_0}a_{-k}+d\over a_ka_{-k} }\right)}=:{\pro_{ k_0}}^\ep \cdots\cr
 &={\pro_{ k_0}}^{ sym}  {\ts \left( 1+{ba_k+c
       a_{-k}+d\over a_ka_{-k} }\right)}\> e^{b(\sum_{k_0}^\ep-
   \sum_{k_0}^{ sym}){1\over a_{-k}} +c(\sum_{k_0}^\ep-
   \sum_{k_0}^{ sym}){1\over a_{k}} }  \cr
 &=\pro_{k_0>0} {\ts { (a_ka_{-k}+b a_k+ca_{-k}+d)
    (a_ka_{-k}+b a_{-k}+ca_{k}+d)\over (a_ka_{-k})^2 }} \>e^{{\beta\over2}(b+c)} \cr
 &=\pro_{k_0>0} {\ts { (a_ka_{-k})^2 
  +a_ka_{-k}(-2e_\k(b+c)+ 2d+b^2+c^2)+bc(a_k^2+a_{-k}^2)
   -2e_\k d(b+c)+d^2
   \over (a_ka_{-k})^2 }} \>e^{{\beta\over2}(b+c)} \cr}$$
Since $a_k^2+a_{-k}^2=-2k_0^2+2e_\k^2=-2a_ka_{-k}+4e_\k^2$ one obtains 
$$\eqalignno{ \mathop{{\pro}^{\ep}}_{k_0\in{\pi\over\beta}(2\Bbb Z+1)} 
  {\ts { a_ka_{-k}+b a_k+ca_{-k}+d\over a_ka_{-k} }} 
   &= \pro_{k_0>0} {\ts { (a_ka_{-k})^2+pa_ka_{-k}+q\over (a_ka_{-k})^2 }}
        \>e^{{\beta\over2}(b+c)} \cr
 &=\pro_{k_0>0}  \ts \left(1+{\omega_+\over a_ka_{-k}}\right) 
     \left(1+{\omega_-\over a_ka_{-k}}\right) \>e^{{\beta\over2}(b+c)}  \cr    } $$
where $p=-2e_\k(b+c)+ 2d+(b+c)^2-4bc$ and $q=4bce_\k^2-2e_\k d(b+c)+d^2$. 
 Application of Lemma II.2 proves part (a) of the lemma. 
\smallskip\noindent
{\bf Part b)} One has $A_k A_{-k}=a_k a_{-k} 
    -\gamma(a_k+a_{-k})+\gamma^2$ and 
$$A_k^2+A_{-k}^2=a_k^2+a_{-k}^2-2\gamma(a_k+a_{-k})+2\gamma^2= 
   2A_k A_{-k} +(a_k-a_{-k})^2$$
which gives 
$$\eqalignno{ \Pf S_k& =(A_kA_{-k}+\lambda\rho^2+\lambda x^2)^2
   +(a_k-a_{-k})^2\lambda x^2   \cr
 &=[A_kA_{-k}+\lambda\rho^2+\lambda x^2+i\lambda^{1\over2}x(a_k-a_{-k})]\> 
  [A_kA_{-k}+\lambda\rho^2+\lambda x^2-i\lambda^{1\over2}x(a_k-a_{-k})]
    \cr
 &=\bigl[ a_ka_{-k}-(\gamma-i\lambda^{1\over2}x)a_k
    -(\gamma+i\lambda^{1\over2}x)a_{-k}+\gamma^2 +\lambda\rho^2
   +\lambda x^2 \bigr]\times \cr
 &\phantom{mm}\bigl[ a_ka_{-k}-(\gamma+i\lambda^{1\over2}x)a_k
    -(\gamma-i\lambda^{1\over2}x)a_{-k}+\gamma^2 +\lambda\rho^2
   +\lambda x^2 \bigr]   \cr}$$
Therefore one obtains
$$\eqalignno{ \mathop{{\pro}^\ep}_{k_0>0} {\ts  { \Pf S_k(a_k,a_{-k})\over 
   a_k^2 a_{-k}^2}}&=\lim_{\ep\to 0\atop \ep<0}  \pro_{k_0}\ts {
  a_ka_{-k}-(\gamma-i\lambda^{1\over2}x)e^{ik_0\ep}a_k
    -(\gamma+i\lambda^{1\over2}x)e^{-ik_0\ep}a_{-k}+\gamma^2 +\lambda\rho^2
   +\lambda x^2 \over a_ka_{-k} }  \cr 
 &\buildrel (a)\over = 
  \ts{\cosh{\beta\over2}\sqrt{e_\k^2+\omega_+}\over \cosh{\beta\over2}e_\k}\,
  {\cosh{\beta\over2}\sqrt{e_\k^2+\omega_-}\over \cosh{\beta\over2}e_\k}\>
    e^{-\beta\gamma} \cr}$$
where 
$$p=(e_\k+2\gamma)^2-e_\k^2-4(\gamma^2+\lambda x^2)+2\gamma^2
 +2\lambda\rho^2+2\lambda x^2=4\gamma e_\k+2\gamma^2+2\lambda\rho^2
 -2\lambda x^2$$
$$q=4(\gamma^2+\lambda x^2)e_\k^2+4\gamma e_\k
 ( \gamma^2 +\lambda\rho^2 +\lambda x^2)+
  ( \gamma^2 +\lambda\rho^2 +\lambda x^2)^2=  (2\gamma e_\k+ 
  \gamma^2 +\lambda\rho^2 +\lambda x^2)^2 +4\lambda x^2e_\k^2$$
such that 
$$\ts {p^2\over4}-q= -4(2\gamma e_\k+  \gamma^2 +\lambda\rho^2)\lambda x^2 
   -4\lambda x^2 e_\k^2=-4\bigl[ (e_\k+\gamma)^2+\lambda\rho^2\bigr] 
   \lambda x^2 $$
and therefore 
$$\eqalignno{ e_\k^2+\omega_{\pm}&=(e_\k+\gamma)^2 +\lambda\rho^2
 -\lambda x^2\pm 2i\lambda^{1\over2} x
      \bigl[ (e_\k+\gamma)^2+\lambda\rho^2\bigr]^{1\over2} \cr
 &=\left\{ \bigl[ (e_\k+\gamma)^2+\lambda\rho^2\bigr]^{1\over2}\pm
       i\lambda^{1\over2} x \right\}^2 \cr}$$
which results in 
$$\eqalignno{ \mathop{{\pro}^\ep}_{k_0>0} {\ts  { \Pf S_k(a_k,a_{-k})\over 
   a_k^2 a_{-k}^2}}&= 
   \ts{\cosh{\beta\over2}\left\{[ (e_\k+\gamma)^2
     +\lambda\rho^2]^{1\over2}+
       i\lambda^{1\over2} x \right\}    \over \cosh{\beta\over2}e_\k}\,
  {\cosh{\beta\over2}\left\{ [ (e_\k+\gamma)^2+\lambda\rho^2]^{1\over2}-
       i\lambda^{1\over2} x \right\}   \over \cosh{\beta\over2}e_\k}\>
    e^{-\beta\gamma} \cr
 &= \ts { \sinh^2({\beta\over2}i\sqrt\lambda \,x)\,+\,\cosh^2({\beta\over2}
    \sqrt{(e_\k+\gamma)^2+\lambda\rho^2}\,) \over 
    \cosh^2{\beta\over2}e_\k  } \,e^{-\beta \gamma} &\blacksquare \cr}$$
\bigskip
\bigskip
Using the lemma above the effective potential (IV.7) becomes 
$$V(w,a,x,\rho)={\ts {1\over4}} w^2+a^2+
   x^2+\rho^2-\int_M \ts {d^d\k\over(2\pi)^d}{1\over\beta}
   \log\left[   { \sinh^2({\beta\over2}i\sqrt\lambda \,x)\,+\,\cosh^2({\beta\over2}
    \sqrt{(e_\k+\gamma)^2+\lambda\rho^2}\,) \over 
    \cosh^2{\beta\over2}e_\k  } \,e^{-\beta \gamma}  \right] \eqno (\IV.13)$$
To compute the two point function, one has to find the global minimum 
 of the real part of (IV.13).   
\medskip
For attractive $\lambda=g^2>0$ one has $\sinh^2({\beta\over2}i\sqrt\lambda \,x)
 =-\sin^2({\beta\over2}gx)$ and 
 it seems likely that 
  $\rho>0$ and $x=0$ at the global minimum.  
Furthermore it seems favorable to put the imaginary part of $\gamma=g(w+
 \sqrt 2\,ia)$ to 0. That  is 
 $a=0$ and $w$ may be non zero to shift the Fermi surface to  
 $e_{\k g}=e_\k+gw$.  
The two point 
function becomes in that case 
$$\lim_{\kappa\to\infty} {\ts{1\over\kappa}} \la \bar\psi_{k\up}\psi_{k'\up}\ra= \ts
   -{ A_{-k}(A_kA_{-k}+\lambda \rho^2)\over 
      (A_{k}A_{-k}+\lambda\rho^2)^2 }=  {ik_0+e_{\k g}
    \over k_0^2+e_{\k g}^2 
   +g^2\rho^2}\eqno (\IV.14) $$
\par
For repulsive $\lambda=-g^2<0$ $\sinh^2({\beta\over2}i\sqrt\lambda \,x)=
  \sinh^2({\beta\over2}g \,x)$ and 
the values $\rho=0$ and $x>0$ seem 
favourable. The imaginary part of $\gamma$ is now given by $w$ which again 
 we put equal to 0. The value of $a$ may be non  zero to shift $e_\k$ to 
 $e_{\k g}=e_\k+2^{1\over2}ga\equiv e_\k+\delta_g$. Under these assumptions 
 the two point function becomes 
$$\eqalignno{ \lim_{\kappa\to\infty} \ts {1\over\kappa} 
  \la \bar\psi_{k\up}\psi_{k\up}\ra&=\ts
    -{ A_kA_{-k}^2+A_k\lambda x^2\over 
      A_{k}^2A_{-k}^2+(A_{k}^2+A_{-k}^2) \lambda x^2 
     +\lambda^2x^4   } = -{A_k\over A_k^2-g^2x^2}   \cr
 &=-\ts {1\over2}\left[ 
   {1\over ik_0-e_{\k g}-gx}+ {1\over ik_0-e_{\k g}+gx}\right] &(\IV.15) \cr} $$
which, at zero temperature,  results in a momentum distribution 
$$\eqalignno{ n_\k&=\lim_{L\to\infty} 
   \la a_\k^+a_\k\ra_{\beta,L} =\lim_{\ep\to 0\atop \ep<0} 
    \int_{\Bbb R}\ts {dk_0\over 2\pi} 
   \ts {1\over2}\left[ 
   {1\over e^{ik_0\ep}( ik_0-e_{\k })-\delta_g-gx}
   + {1\over e^{ik_0\ep}( ik_0-e_{\k })-\delta_g+gx}\right] \cr
  &=\ts {1\over2}\Bigl[
    \chi(e_{\k g}-gx<0)+\chi(e_{\k g}+gx<0)\Bigr]
  =\cases{ 1&if $e_{\k g}<-gx$ \cr {1\over2} &if $-gx< e_{\k g}< gx$\cr 
   0&if $gx<e_{\k g}$ \cr}  &(\IV.16)
   \cr}$$
%
%  CHAPTER IV.2
%
\bigskip
\bigskip\goodbreak
\noindent{\mittel IV.2 BCS with Non Zero Angular Momentum Terms} 
\bigskip
We consider the model 
$$Z(\beta,L,\{s_{k,\sigma}\})=\int e^{-\U(\psi,\bar\psi)+{1\over\kappa}
    \sum_{k,\sigma} s_{k,\sigma}\bar\psi_{k,\sigma}\psi_{k,\sigma} }
    d\mu_C(\psi,\bar\psi)\eqno (\IV.17)$$
with $e_\k={\k^2\over2m}-\mu$, $C(k)={1\over ik_0-e_\k}$ and 
$$  \U(\psi,\bar\psi)={\ts {1\over \kappa^2}}\!\!\!
   \sum_{\sigma,\tau\in\{\up,\down\}}
    \sum_{k,p} {\ts {1\over \kappa}} U(\k'-\p') \>\bar\psi_{k,\sigma}
   \bar\psi_{-k,\tau}\psi_{p,\sigma} 
  \psi_{-p,\tau} \eqno(\IV.18)$$
The electron electron interaction is given by (III.7)
$$U(\k'-\p')=\cases{ \ds {1\over2} \sum_{\ell=-n}^n \lambda_{|\ell|}
     e^{i\ell\varphi_\k}e^{-i\ell\varphi_\p}
    +{\lambda_0\over2} & if $d=2$ \cr
 \ds \sum_{\ell=0}^n \sum_{m=-\ell}^\ell \lambda_\ell
   Y_{\ell m}\left({\ts{\k'}}\right)
     \bar Y_{\ell m}\left({\ts{\p'}}\right) & if $d=3$ \cr} \;\;=:\;
   \sum_{l=0}^N\lambda_l\> y_l(\k')\>\bar y_l(\p') $$
where $\k'=\sqrt{2m\mu}{\k\over |\k|}$ is the projection of $\k$ onto the 
 Fermi surface $e_\k=0$. For simplicity, we write $Y_{\ell m}(\k')$ instead 
 of $Y_{\ell m}\left({\k'\over k_F}\right)$. 
\par
To write down the effective potential and the two point functions in this case, 
we first have to compute the Pfaffian of the matrix $S_k$ of Theorem III.1. 
Since we consider only a BCS interaction, the $\Xi$ and $\Gamma$ fields 
 are zero. With Lemma III.2 one obtains 
$$\eqalignno{ {\rm Pf} S_k=&\bigl( a_{k\up}a_{-k\down} +\bar\Phi_{\k\up\down} 
  \Phi_{\k\up\down}\bigr)\bigl(a_{-k\up}a_{k\down} +\bar\Phi_{-\k\up\down} 
  \Phi_{-\k\up\down}\bigr)  \cr
 & +a_{k\up}a_{-k\up}\Phi_{\k\down\down}\bar\Phi_{\k\down\down}
  +a_{k\down}a_{-k\down}\Phi_{\k\up\up}\bar\Phi_{\k\up\up} \cr
 &+\Phi_{\k\up\up}\Phi_{\k\down\down}
    \bar\Phi_{\k\up\down}\bar\Phi_{-\k\up\down}+
  \bar\Phi_{\k\up\up}\bar\Phi_{\k\down\down}
    \Phi_{\k\up\down}\Phi_{-\k\up\down}
   +\Phi_{\k\up\up}\Phi_{\k\down\down}
    \bar\Phi_{\k\up\up}\bar\Phi_{\k\down\down}  \cr}$$
where  $a_{k\sigma}=a_k-s_{k\sigma}$. In particular, for $s_{k\sigma}=0$ 
$$ {\rm Pf} S_k=(a_ka_{-k}+\Omega_\k^+)(a_ka_{-k}+\Omega_\k^-)
  \eqno   $$
where $\Omega_\k^\pm$ are the solutions of the quadratic equation 
$$\eqalignno{ \Omega^2&-\Bigl(\bar\Phi_{\k\up\down} \Phi_{\k\up\down}+
    \bar\Phi_{-\k\up\down}  \Phi_{-\k\up\down}+
   \Phi_{\k\up\up}\bar\Phi_{\k\up\up}+
   \Phi_{\k\down\down}\bar\Phi_{\k\down\down}\Bigr)\Omega+ 
  \Phi_{\k\up\down}\bar\Phi_{\k\up\down}
   \Phi_{-\k\up\down}\bar\Phi_{-\k\up\down} \cr
  &+\Phi_{\k\up\up}\Phi_{\k\down\down}
    \bar\Phi_{\k\up\down}\bar\Phi_{-\k\up\down}+
  \bar\Phi_{\k\up\up}\bar\Phi_{\k\down\down}
    \Phi_{\k\up\down}\Phi_{-\k\up\down} 
   +\Phi_{\k\up\up}\Phi_{\k\down\down}
    \bar\Phi_{\k\up\up}\bar\Phi_{\k\down\down}=0 &(\I.8) \cr}$$
Using Lemma II.2 again to compute the product over $k_0$, 
  the effective potential becomes 
$$\eqalignno{ V(\phi)&=\sum_{\sigma\tau\in 
  \{\up\up,\down\down,\up\down\}} \sum_{l=0}^N 
    |\phi_{\sigma\tau}^l|^2 -{\ts {1\over\beta L^d} }
     \sum_{\k\in M}\log\pro_{k_0>0}\ts \left\{ {a_k a_{-k}+\Omega_\k^+\over 
    a_ka_{-k} } {a_k a_{-k}+\Omega_\k^-\over 
    a_ka_{-k} } \right\}    \cr
 &=  \sum_{l=0}^N\left(  |\phi_{\up\down}^l|^2+ |\phi_{\up\up}^l|^2+
    |\phi_{\down\down}^l|^2\right)   -\sum_{\epsilon\in\{+,-\}} 
    \int_M \ts 
   {d^d\k\over (2\pi)^d}\>{1\over\beta} 
    \log\left[ {\cosh({\beta\over2}\sqrt{ e_\k^2+\Omega^\epsilon_\k }) 
    \over \cosh {\beta\over 2} e_\k} \right]\; \;\;\;\;\;\;&(\IV.19)   \cr} $$
\par
In the following we consider the case of an even and an odd interaction 
 $U(\k'-\p')$. For even $U$, the sum in (III.7) goes only over even angular 
 momenta and one has $y_l(-\k')=y_l(\k')$. That is, $\Phi_{\k\up\up}= 
 \Phi_{\k\down\down}=0$ and $\Phi_{-\k\up\down}=\Phi_{\k\up\down}$. 
The quadratic equation  becomes 
$$\Omega^2 -2\bar\Phi_{\k\up\down} \Phi_{\k\up\down} \Omega 
   + \bigl( \bar\Phi_{\k\up\down} \Phi_{\k\up\down} \bigr)^2 =0$$
which gives $\Omega_\k^\pm=\bar\Phi_{\k\up\down} \Phi_{\k\up\down}$ and 
$$V(\phi_{\up\down})=\sum_{l=0}^N |\phi_{\up\down}^l|^2- \int_M \ts 
   {d^d\k\over (2\pi)^d}\>{2\over\beta} 
    \log\left[ {\cosh({\beta\over2}\sqrt{ e_\k^2+
     \bar\Phi_{\k\up\down} \Phi_{\k\up\down} }) 
    \over \cosh {\beta\over 2} e_\k} \right]  \eqno (\IV.20)$$
The two point function is given by 
$$\eqalignno{ {\ts{1\over\kappa}} \la \bar\psi_{p\sigma}\psi_{p\sigma}\ra& = 
    \int {\ts{1\over\kappa}}\bar\psi_{p\sigma}\psi_{p\sigma}\, e^{-\U(\psi,\bar\psi)} 
    d\mu_C(\psi,\bar\psi)  \cr
   &= -{\int 
    {a_{-p}\over a_pa_{-p}+\bar\Phi_{\k\up\down} \Phi_{\k\up\down} }\, 
    e^{-\kappa V(\phi_{\up\down})} \ds \pro_{l=0}^N du_{\up\down}^l
       dv_{\up\down}^l \over 
   \int e^{-\kappa V(\phi_{\up\down})} \ds \pro_{l=0}^N du_{\up\down}^l
       dv_{\up\down}^l } &(\IV.21)  \cr}$$
\par
For a pure odd interaction the sum in (III.7) contains only  odd values of $l$ and 
 one  has $\Phi_{-\k\sigma\tau}=-\Phi_{\k\sigma\tau}$. Furthermore, since $V$ 
 is symmetric with respect to $\phi_{\up\up}^l\leftrightarrow 
  \phi_{\down\down}^l$, one may 
  substitute $\phi_{\up\up}^l=\phi_{\down\down}^l=\phi^l$, $\Phi_{\k\up\up}=
  \Phi_{\k\down\down}=\Phi_\k= 
  \sum_l \lambda_l^{1\over2} y_l(\k')\phi^l$  and the integral 
 over $du_{\up\up}^ldv_{\up\up}^ldu_{\down\down}^ldv_{\down\down}^l$ 
  by an integral  over $du^ldv^l$.  The quadratic equation 
becomes in that case 
$$\eqalignno{ \Omega^2&-2\Bigl(\bar\Phi_{\k\up\down} \Phi_{\k\up\down}+
   \Phi_{\k}\bar\Phi_{\k} \Bigr)\Omega +
  \bar\Phi_{\k\up\down}^2 \Phi_{\k\up\down}^2  
     -\Phi_{\k}^2 \bar\Phi_{\k\up\down}^2-
  \bar\Phi_{\k}^2 \Phi_{\k\up\down}^2 
   +\Phi_{\k}^2
    \bar\Phi_{\k}^2=0 \cr}$$
which gives 
 $\Omega^\pm_\k=(\Phi_{\k\up\down}\pm\Phi_\k)
     (\bar\Phi_{\k\up\down}\pm\bar\Phi_\k)     $ 
The two point function reads in this case 
$${\ts{1\over\kappa}}\la \bar\psi_{p\sigma}\psi_{p\sigma}\ra=  
   { \int^{\phantom{I}} F_{\p}(\phi)\>
   e^{-\kappa V(\phi)}\ds 
  \pro_{l=0}^N du_{\up\down}^l dv_{\up\down}^l du^ldv^l \over 
   \int e^{-\kappa V(\phi)} \ds 
  \pro_{l=0}^N du_{\up\down}^l dv_{\up\down}^l du^ldv^l } \eqno (\IV.22)$$
where 
$$F_{\p}(\phi)=-\ts { a_{-p} [ a_p a_{-p}+ \Phi_{\p\up\down} 
     \bar\Phi_{\p\up\down}+\Phi_{\p}\bar\Phi_{\p}] 
  \over (a_pa_{-p}+\Omega_\p^+)(a_pa_{-p}+\Omega_\p^-) } \eqno (\IV.23)$$
\par
A detailed analysis to compute the infinite volume limit of (IV.21,22) 
  we plan to give 
 in a forthcoming paper. In the following, concerning the two dimensional case, 
we only report on a result which is still under investigation of A. Schuette. 
In three dimensions, we argue in Theorem IV.2 below using  only  symmetry
considerations that, if $e(\k)$ has $SO(3)$ symmetry, then the 
 $\la a_{\k\sigma}^+ a_{\k\sigma}\ra$ expectations  also have to 
 have $SO(3)$ symmetry. 
\smallskip
Consider first the two dimensional case. An analysis done by Albrecht 
 Schuette from ETH Z\"urich indicates the following result:
\par
Let $U(\k'-\p')=\sum_{\ell=-n}^n \lambda_\ell e^{i\ell(\vp_\k-\vp_\p)}$, 
 suppose that $\lambda_m>0$ is attractive and $\lambda_m>\lambda_\ell$ for 
 all $\ell\ne m$. Then 
$$\lim_{\kappa\to\infty} {\ts{1\over\kappa}}\la \bar\psi_{p\sigma} \psi_{p\sigma} 
   \ra_{\beta,L}=\ts {ik_0+e_\p\over p_0^2+e_\p^2+\lambda_m\rho_m^2}
   \eqno (\IV.24)$$
where $\rho_m$ is determined by the BCS equation 
$$1-{\ts {\lambda_m\over 4\pi}} \int d|\k| |\k| \ts {\tanh( {\beta\over2}\sqrt{ e_\k^2+ 
     \lambda_m \rho_m^2}) \over \sqrt{e_\k^2+\lambda\rho_m^2} }=0$$
\par
The form of the two point function (IV.24) can still be obtained by applying the 
 standard  mean field 
formalism [AB], [BW]. However, the situation is different in 3 dimensions. 
Before we state the corresponding theorem, we shortly recall the mean field 
 equations [AB], [BW]:
\par
The $\la a^+ a\ra$ expectations are given by 
$$\la a_{\k\sigma}^+ a_{\k\sigma} \ra=\ts {1\over2}\left( 1-e_\k 
   \Bigl[ {\tanh({\beta\over2}\sqrt{ e_\k^2+\Delta_\k^*\Delta_\k})\over 
      \sqrt{ e_\k^2+\Delta_\k^*\Delta_\k}} \Bigr]_{\sigma\sigma}\right)
   \eqno (\IV.25 )$$
 where the $2\times 2$ matrix $\Delta_\k$, $\Delta_\k^T=-\Delta_{-\k}$, is 
 a solution of the gap equation 
$$\Delta_\p=\int_M \ts{d^d\k\over (2\pi)^d}\, U(\p'-\k')\, \Delta_\k 
   {\tanh({\beta\over2}\sqrt{ e_\k^2+\Delta_\k^*\Delta_\k})\over 
     2 \sqrt{ e_\k^2+\Delta_\k^*\Delta_\k}}  \eqno (\IV.26)$$
\par
In 2 dimensions, if one substitutes $U(\p'-\k')$ by a single attractive term 
 $\lambda_\ell e^{i\ell(\vp_\p-\vp_\k)}$, then (IV.26) has the unitary isotropic 
 solution $\Delta_\k=\Delta\left( {0\atop -e^{i\ell\vp_\k}}\; 
  {e^{i\ell\vp_\k}\atop 0} \right)$ for even $\ell$ and 
  $\Delta_\k=\Delta\left( {\cos\ell\vp_\k \atop \sin\ell\vp_\k}\; 
  {\sin\ell\vp_\k\atop -\cos\ell\vp_\k} \right)$ if $\ell$ is odd. In that case 
 $\Delta_\k^*\Delta_\k=|\Delta|^2 Id$ and (IV.25) coincides with (IV.24). 
\par
In 3 dimensions, it has been proven by Feldman, Kn\"orrer and Trubowitz [FKT] 
 that for all $\ell \ge 2$ (IV.26) does not have unitary isotropic 
  ($\Delta_\k^*\Delta_\k=const\,Id$) solutions. In view of that result, the symmetry 
 considerations  
 below indicate that in 3 dimensions for $\ell\ge 2$ the standard mean field 
 approach may be misleading since one would no longer expect $SO(3)$ 
 invariance for the $\la a_{\k\sigma}^+a_{\k\sigma}\ra$ expectations according to 
 (IV.25). But this is indeed the case. 
\bigskip
\noindent{\bf Theorem IV.2:} Let $\ell$ be even, $\lambda_\ell>0$ be attractive 
and let 
$$\U(\psi,\bar\psi)={\ts {\lambda_\ell\over L^{3d}} }\sum_{k,p} 
  \sum_{m=-\ell}^\ell \bar Y_{\ell m} 
   (\k') Y_{\ell m} (\p')\, \bar\psi_{k\up}\bar\psi_{-k\down}
    \psi_{p\up}\psi_{-p\down} $$
Then, if $e_{R\k}=e_\k$ for all $R\in SO(3)$, one has 
$$\eqalignno{ \lim_{\kappa\to\infty} {\ts{1\over\kappa}}
    \la \bar\psi_{p\sigma} \psi_{p\sigma}  \ra_{\beta,L}&= 
   \lim_{\kappa\to\infty} \int  {\ts{1\over\kappa}}\bar\psi_{p\sigma} \psi_{p\sigma}
   \, e^{-\U(\psi,\bar\psi)} d\mu_C(\psi,\bar\psi) \cr
  &=  \int_{S^2}\ts {ip_0+e_\p\over p_0^2+e_\p^2+
   \lambda_\ell \rho_0^2 | \su_m  \alpha_m^0 Y_{\ell m}(\x)|^2} 
   \, {d\Omega (\x)\over 4\pi}   \cr}$$
where $\rho_0\ge 0$ and $\alpha^0\in \Bbb C^{2\ell +1}$, 
 $\su_m |\alpha_m^0|^2=1$, are values at the global minimum (which 
 is degenerated) of 
$$ V(\rho,\alpha)=\rho^2-\int_M\ts  {d^3\k\over (2\pi)^3}\, {1\over\beta} \log \left[
   { \cosh ( {\beta\over2}\sqrt{ e_\k^2+
   \lambda_\ell \rho^2 | \su_m \alpha_m Y_{\ell m}(\k')|^2} ) \over 
   \cosh {\beta\over2} e_\k } \right]^2 $$
In particular, at zero temperature the momentum distribution $n_\p$ is given by 
$$n_\p=\lim_{L\to\infty} \la a_{\p\sigma}^+ a_{\p\sigma}\ra_{\beta,L}=
   \int_{S^2} \ts {1\over2}\left( 1- {e_\p\over \sqrt{ e_\p^2+|\Delta(\x)|^2} } \right)
   {d\Omega(\x)\over 4\pi}$$
and has $SO(3)$ symmetry. Here $\Delta(\x)=\lambda_\ell^{1\over2} \rho_0 
   \su_m  \alpha_m^0 Y_{\ell m}(\x)$. 
\bigskip
\medskip
\noindent{\bf Proof:}  Substituting $u_{\up\down}^l$ by $u_m$ and 
  $u_{\up\down}^l$ by $-v_m$ in (IV.21), 
   one has to compute the infinite volume limit of 
$$ {\ts{1\over\kappa}}\la \bar\psi_{p\sigma}\psi_{p\sigma}\ra= 
 -{\int_{\Bbb R^{4\ell+2}} 
    {a_{-p}\over a_pa_{-p}+ |\Phi_{\p}|^2 }\, 
    e^{-\kappa V(\phi)}  \pro_{m=-\ell}^\ell du_m
       dv_m \over 
   \int_{\Bbb R^{4\ell+2}} e^{-\kappa V(\phi)}  \pro_{m=-\ell}^\ell du_m
       dv_m } \eqno (\IV.27) $$
where 
$$V(\phi)=\sum_{m=-\ell}^\ell |\phi_m|^2-
    \int_M\ts  {d^3\k\over (2\pi)^3}\, {1\over\beta} \log \left[
   { \cosh ( {\beta\over2}\sqrt{ e_\k^2+ |\Phi_\k|^2} ) \over 
   \cosh {\beta\over2} e_\k } \right]^2$$ 
and 
$$\Phi_\k=\lambda_\ell^{1\over2} 
      \sum_{m=-\ell}^\ell \bar \phi_m Y_{\ell m}(\k')$$
Let $U\!(\!R)$ be the unitary representation of $SO(3)$ given by 
$$ Y_{\ell m}(R\k')=\sum_{m'} U\!(\!R)_{mm'} Y_{\ell m'}(\k')$$
and let $\sum_m (\overline{U\phi})_m Y_{\ell m}(\k')=: (U\Phi)_{\k}$. Then 
 for all $R\in SO(3)$ one has $\bigl(U\!(\!R)\Phi\bigr)_{\k}=\Phi_{R^{-1}\k}$ and 
$$\eqalignno{ V\bigl(U\!(\!R)\phi\bigr)&= \sum_m  |[U\!(\!R)\phi]_m|^2-
    \int_M\ts  {d^3\k\over (2\pi)^3}\, {1\over\beta} \log \left[
   { \cosh ( {\beta\over2}\sqrt{ e_\k^2+ |[U\Phi]_\k|^2} ) \over 
   \cosh {\beta\over2} e_\k } \right]^2  \cr
 &=\sum_m  |\phi_m|^2-
    \int_M\ts  {d^3\k\over (2\pi)^3}\, {1\over\beta} \log \left[
   { \cosh ( {\beta\over2}\sqrt{ e_\k^2+ |\Phi_{R^{-1}\k}|^2} ) \over 
   \cosh {\beta\over2} e_\k } \right]^2  \cr
 &=\sum_m  |\phi_m|^2-
    \int_M\ts  {d^3\k\over (2\pi)^3}\, {1\over\beta} \log \left[
   { \cosh ( {\beta\over2}\sqrt{ e_\k^2+ |\Phi_{\k}|^2} ) \over 
   \cosh {\beta\over2} e_\k } \right]^2 \>=\> V(\phi) &(\IV.28) \cr}$$
Let $S^{4\ell+1}=\{\phi\in \Bbb C^{2\ell +1}\>|\> \sum_m |\phi_m|^2=1\}$. Since 
 $U\!(\!R)$ leaves $S^{4\ell+1}$ invariant, $S^{4\ell+1}$ can be
   written as the union 
 of disjoint orbits, 
$$S^{4\ell+1}=\cup_{[\alpha]\in \O} [\alpha]$$
where $[\alpha]=\{ U\!(\!R)\alpha\>|\> R\in SO(3)\}$ is the orbit of $\alpha\in 
  S^{4\ell+1}$ under the action of $U\!(\!R)$ and $\O$  
 is the set of all orbits. If one chooses a fixed representant $\alpha$ in 
 each orbit $[\alpha]$, that is, if one chooses a fixed section $\sigma:\O\to
S^{4\ell+1}$, $[\alpha]\to\sigma_{[\alpha]}$  with $[\sigma_{[\alpha]}]=[\alpha]$, 
  every $\phi\in \Bbb C^{2\ell +1}$ can be uniquely 
 written as 
$$\phi=\rho\, U\!(\!R)\sigma_{[\alpha]}\>,\;\;\;\; \rho=\|\phi\|\ge 0,\;\;
   \ts \alpha={\phi\over 
   \|\phi\|},\;\;
   \;[\alpha]\in\O \;\hbox{and}\;  R\in SO(3)/I_{[\alpha]}$$
where $I_{[\alpha]}=I_{[\alpha]}^\sigma=\{S\!\in \!SO(3)\>|\>U\!(\!S)\sigma_{[\alpha]}
  =\sigma_{[\alpha]}\}$ 
 is the isotropy subgroup of 
  $\sigma_{[\alpha]}$. 
Let 
$$\ts \int_{\Bbb R^{4\ell +2}} \pro_m du_m dv_m\>g(\phi)=
 \int_{\Bbb R^+} D\!\rho \int_\O D[\alpha] \int_{[\alpha]} D\!R \>
    g\bigl( \rho\, U\!(\!R)\sigma_{[\alpha]}  \bigr)$$
be the integral in (IV.27)  over $\Bbb R^{4\ell +2}$ in the new coordinates. That is, 
 for example, $D\!\rho=\rho^{4\ell +1}d\rho$. In the new coordinates  
$$  |\Phi_\p|^2   
   =  \lambda_\ell
   \rho^2 \Bigl|\sum_m \bigl(\,\overline{U\!(\!R)\sigma_{[\alpha]} }\,\bigr)_m 
    Y_{\ell m}(\p')\Bigr|^2  
    =\lambda_\ell
   \rho^2 \Bigl|\sum_m \bar\sigma_{[\alpha],m} Y_{\ell m}\left(R^{-1}\p'\right)
    \Bigr|^2   $$
such that 
$$\ts {-a_{-p}\over a_pa_{-p}+|\Phi_\p|^2}={-a_{-p}\over a_pa_{-p}+
 \lambda_\ell
   \rho^2 |\su_m \bar\sigma_{[\alpha],m} Y_{\ell m}\left(R^{-1}\p'\right)|^2   }
   \equiv f(\rho,[\alpha],R^{-1}\p)$$
Since $ V(\phi)=      
    V(\rho,[\alpha])$    is independent of $R$, one obtains 
$$\eqalignno{  {\ts{1\over\kappa}}\la \bar\psi_{p\sigma}\psi_{p\sigma}\ra&=  
 {\int   
    {-a_{-p}\over a_pa_{-p}+ |\Phi_{\p}|^2 }\, 
    e^{-\kappa V(\phi)} \ds \pro_{m=-\ell}^\ell du_m
       dv_m \over  \int e^{-\kappa V(\phi)} \ds \pro_{m=-\ell}^\ell du_m
       dv_m } \cr
  & =  {\int_{\Bbb R^+}D\rho \int_{\O} D[\alpha] \int_{[\alpha] } D\!R \; 
    f(\rho,[\alpha],R^{-1}\p)\; e^{-\kappa V(\rho,[\alpha])}
       \over 
   \int_{\Bbb R^+}D\rho \int_{\O} D[\alpha] \int_{[\alpha] } D\!R \;
     e^{-\kappa V(\rho,[\alpha])}   } \cr
 &=  { \int_{\Bbb R^+}D\rho \int_\O D[\alpha]\;{\rm vol}([\alpha]) \;
    { \int_{[\alpha] }D\!R\>
     f(\rho,[\alpha],R^{-1}\p)\over \int_{[\alpha] }D\!R  }\;
       e^{-\kappa V(\rho,[\alpha])}    \over 
   \int_{\Bbb R^+}D\rho \int_\O D[\alpha]\;{\rm vol}([\alpha]) \;
      e^{-\kappa V(\rho,[\alpha])} } &(\IV.29) \cr}$$
It is plausible to assume that at the global minimum of $V(\rho,[\alpha])$ $\rho$ is 
uniquely determined, say $\rho_0$. Let $\O_{\rm min}\subset\O$ be the set of all 
 orbits at which $V(\rho_0,[\alpha])$ takes its global minimum. Then in the infinite 
 volume limit (\IV.29) becomes 
$$\eqalignno{ \lim_{\kappa\to\infty} 
   {\ts{1\over\kappa}}\la \bar\psi_{p\sigma}\psi_{p\sigma}\ra&=  
 { \int_{\O_{\rm min}} D[\alpha]\;{\rm vol}([\alpha]) \;
    { \int_{[\alpha] }D\!R\>
     f(\rho_0,[\alpha],R^{-1}\p)\over \int_{[\alpha] }D\!R  }   \over 
   \int_{\O_{\rm min}} D[\alpha]\;{\rm vol}([\alpha])   } &(\IV.30) \cr}$$
Consider the quotient of integrals in the numerator of (IV.30). Since 
$$\eqalignno{  f\left(\rho,[\alpha],R^{-1}\p\right)&=f\Bigl(
   \rho^2 \bigl|\su_m \bar\sigma_{[\alpha],m} Y_{\ell m}\left(R^{-1}\p\right)
    \bigr|^2 \Bigr)  \cr
 &=f\Bigl( \rho^2 \bigl|\su_m \bigr(\overline{U(S)\sigma_{[\alpha]}} \bigr)_m
     Y_{\ell m}\left(R^{-1}\p\right)   \bigr|^2 \Bigr)  \cr
 &=f\Bigl( \rho^2 \bigl|\su_m  \bar\sigma_{[\alpha],m}
     Y_{\ell m}\left((RS)^{-1}\p\right)   \bigr|^2 \Bigr) =
     f\left(\rho,[\alpha],(RS)^{-1}\p\right) \cr}$$
for all $S\in I_{[\alpha]}$, one has, since $[\alpha]\simeq SO(3)/I_{[\alpha]}$ 
$$\eqalignno{  { \int_{[\alpha] }D\!R\>
     f(\rho_0,[\alpha],R^{-1}\p)\over \int_{[\alpha] }D\!R  } &= 
   { \int_{SO(3)/I_{[\alpha]} }D\!R\>
     f(\rho_0,[\alpha],R^{-1}\p) \int_{I_{[\alpha]}} DS 
     \over \int_{SO(3)/I_{[\alpha]} }D\!R \int_{I_{[\alpha]}} DS } \cr
 &= { \int_{SO(3)/I_{[\alpha]} }D\!R\int_{I_{[\alpha]}}DS\>
     f(\rho_0,[\alpha],(RS)^{-1}\p)   
     \over \int_{SO(3)/I_{[\alpha]} }\int_{I_{[\alpha]}} D\!R\> DS } \cr
 &={ \int_{SO(3)} DR\> f(\rho_0,[\alpha],R^{-1}\p)  \over \int_{SO(3)} D\!R} \cr
 &={ \int_{S^2} d\Omega(\x) \int_{SO(3)_{\x\to\p}} D\!R\>
      f(\rho_0,[\alpha],R^{-1}\p)  \over
     \int_{S^2} d\Omega(\x)  \int_{SO(3)_{\x\to\p}} D\!R} \cr
  &={ \int_{S^2} d\Omega(\x) \>  f(\rho_0,[\alpha],\x) \int_{SO(3)_{\x\to\p}} D\!R\>
      \over \int_{S^2} d\Omega(\x)  \int_{SO(3)_{\x\to\p}} D\!R} &(\IV.31) \cr}$$
 where 
   $SO(3)_{\x\to\p}=\{R\!\in \!SO(3)\>|\> R\x=\p\}$.  If one assumes 
 that $D\!R$ has the usual
invariance properties of the Haar measure, then $\int_{SO(3)_{\x\to\p}} D\!R$ 
does not depend on $\x$ and it cancels out in (IV.31).  Then (IV.30) 
  gives 
$$\eqalignno{ \lim_{\kappa\to\infty} 
   {\ts{1\over\kappa}}\la \bar\psi_{p\sigma}\psi_{p\sigma}\ra&=  
 { \int_{\O_{\rm min}} D[\alpha]\;{\rm vol}([\alpha]) \;
    { \int_{S^2 }d\Omega(\x)\>
     f(\rho,[\alpha],\x)\over \int_{S^2 }d\Omega(\x)  }   \over 
   \int_{\O_{\rm min}} D[\alpha]\;{\rm vol}([\alpha])   } &(\IV.32) \cr}$$
Now, since the effective potential, which is constant on $O_{\rm min}$, 
   may be written as 
$$V(\rho,[\alpha])=\int_{S^2}\ts {d\Omega(\x)\over4\pi} \> G\Bigl( 
   \rho^2 \bigl|\su_m \bar\sigma_{[\alpha],m} Y_{\ell m}\left(\x\right)\bigr|^2
   \Bigr)$$
with $G(X)=\rho^2-\int\ts{dk\, k^2\over 2\pi^2}\> 
   \log\left[ {\cosh({\beta\over2}\sqrt{e_k^2+\lambda_\ell X}) \over 
   \cosh{\beta\over2}e_k}\right] $, it is plausible to assume that also 
$$ { \int_{S^2 }d\Omega(\x)\>
     f(\rho_0,[\alpha],\x)\over \int_{S^2 }d\Omega(\x)  } = 
  \int_{S^2}\ts {d\Omega(\x)\over4\pi} \ts {ip_0+e_\p\over p_0^2+e_\p^2+
   \lambda_\ell
   \rho_0^2 |\su_m \bar\sigma_{[\alpha],m} 
      Y_{\ell m}(x)|^2}$$
is constant on $O_{\rm min}$. In that case also the integrals over $O_{\rm min}$ 
 in (IV.32) cancel out and the theorem is proven $\blacksquare$  
\bigskip
\bigskip
\bigskip
\vfill\eject
\noindent{\gross References}
\bigskip
\item{[AB]} P.W. Anderson, W.F. Brinkman, {\it Theory of Anisotropic 
    Superfluidity} in {\it Basic Notions of Condensed Matter Physics} by P.W. 
    Anderson, Benjamin/Cummings, 1984. 
\item{[BW]} R. Balian, N.R. Werthamer, {\it Superconductivity with Pairs in a 
    Relative p Wave}, Phys. Rev. 131, p.1553-1564, 1963.
\item{[FKT]} J. Feldman, H. Kn\"orrer, E. Trubowitz, {\it A Remark on 
    Anisotropic Superconducting States}, Helv. Phys. Acta 64, p.695-699, 1991. 
\item{[FT]} J. Feldman, E. Trubowitz, {\it Perturbation Theory for Many Fermion 
   Systems}, Helv. Phys. Acta 63, 1990, p.156-260; 
  {\it The Flow of an Electron-Phonon 
    System to the Superconducting State}, Helv. Phys. Acta 64, p.214-357, 1991. 
\item{[H]} E.R. Hanson, {\it A Table of Series and Products}, Prentice-Hall, 1975, 
              Sec. 89.5. 
\end