%Paper: 
%From: mh@hep.physik.uni-muenchen.de (Marcus Hutter)
%Date: Mon, 9 Jan 95 17:11:33 MET

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%     This file is only the document-frame with various     %%
%%     include-statements including                          %%
%%     every Chapter, Appendix and Picture.                  %%
%%     These  extra .tex  and .ps files are contained in     %%
%%     the uufile.                                           %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%     Instantons in QCD     -      Marcus Hutter 28.12.94   %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%-----------------------------%
%       Document-Style        %
%-----------------------------%

\documentstyle[12pt]{article}
\parskip=1.5ex plus 1ex minus 1ex
\parindent=0ex
\pagestyle{headings}
\setcounter{tocdepth}{4}
\topmargin=0cm
\oddsidemargin=0cm
\evensidemargin=0cm
\textwidth=16cm
\textheight=23cm
\unitlength=1mm
\sloppy
%\makeindex

%-----------------------------%
%      Compiler-Switches      %
%-----------------------------%

\input epsf                   % to include post-script pictures
%\epsfverbosetrue
\newif\ifall\alltrue
%\allfalse                     % compile only parts
%\newif\ifprivate\privatetrue
%\def\private#1{{\it private: #1}}      % print private comments
\def\private#1{}             % not print private comments
%\nofiles  % no .aux .toc ... files
%-----------------------------%
%      Macro-Definitions      %
%-----------------------------%

\def\ff{\Longrightarrow}
\def\gdw{\Longleftrightarrow}
\def\toinfty#1{\stackrel{#1\to\infty}{\longrightarrow}}
\def\gtapprox{\buildrel{\lower.7ex\hbox{$>$}}\over
                       {\lower.7ex\hbox{$\sim$}}}
\def\nq{\hspace{-1em}}
\def\look{\(\uparrow\)}
\def\cpref#1{\ref{#1} auf Seite \pageref{#1}}
\def\ignore#1{}
\def\deltabar{{\delta\!\!\!^-}}
\def\hbar{h\!\!\!\!^{-}\,}
\def\dbar{d\!\!^{-}\!}
\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\def\beqn{\begin{displaymath}}
\def\eeqn{\end{displaymath}}
\def\bqa{\begin{eqnarray}}
\def\eqa{\end{eqnarray}}
\def\bqan{\begin{eqnarray*}}
\def\eqan{\end{eqnarray*}}
%\def\slash{\makebox[0pt][l]{/}}

\def\boxfig#1#2{
  \begin{figure} \fboxsep=8mm
  \framebox[\textwidth]{\centerline{#1}}
  \vspace{-2ex} \caption{#2}
  \end{figure} }

\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                        Title-Page                            %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\ifall

\hspace*{13cm}LMU 95-01

\hspace*{13cm}December 1994

\begin{center}       \vspace*{3cm}
  {\LARGE         Instantons			     \\
                     and		             \\
           Meson Correlation Functions               \\[5mm]
                   in QCD
  }                                                  \\[3cm]
  {\bf Marcus Hutter}                             \\[2cm]
  {\it Sektion Physik der Universit\"at M\"unchen}\\
  {\it Theoretische Physik}                       \\
  {\it Theresienstr. 37 $\quad$ 80333 M\"unchen} \\[2cm]
\end{center}

\begin{abstract}
Various QCD correlators are calculated in the instanton
liquid model in zeromode approximation and $1/N_c$ expansion.
Previous works are extended by including dynamical
quark loops. In contrast to the original "perturbative" $1/N_c$
expansion not all quark loops are suppressed. In the flavor
singlet meson correlators a chain of quark bubbles survives the
$N_c\to\infty$ limit causing a massive $\eta^\prime$ in the pseudoscalar
correlator while keeping massless pions in the triplet
correlator. The correlators are plotted
and meson masses and couplings are obtained from a spectral
fit. They are compared to the values obtained
from numerical studies of the instanton liquid and
to experimental results.

\private{s/t$\to$1/3, 1,5$\to$ S,P,V,A,T}

\end{abstract}
\newpage

%\listoffigures
%\listoftables
%\clearpage

\fi
\tableofcontents

%%%%%%%%%%%%%%%% Chapter 1 - ? %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\section{Introduction}
\input instanto.t1

%\section{One Instanton Formulas}
%%%\input instanto.t2

%\section{Theory of Instanton Liquid}
\input instanto.t3

%\section{Gluon Propagator}
%%%\input instanto.t4

%\section{Light Quark Propagator}
\input instanto.t5

%\section{Four \& Six Quark 't Hooft like Interaction}
\input instanto.t6

%\section{Correlators of Light Mesons}
\input instanto.t7

%\section{Axial Anomaly \& Quark Spin}
%%%\input instanto.t8

%\section{Conclusions}
\input instanto.t9

%\section{Private comments} \hfill
\private{}

\begin{appendix}
%\section{Zeromode Formulas}
\input instanto.ta

%\section{Numerical Evaluation of Integrals}
\input instanto.tb

%{References}
\input instanto.tr

%\section{Figures}
\input instanto.tf

\end{appendix}
\end{document}

%--------------------------------------------------------------
%   EPSF.TEX macro file:
%   Written by Tomas Rokicki of Radical Eye Software, 29 Mar 1989.
%   Revised by Don Knuth, 3 Jan 1990.
%   Revised by Tomas Rokicki to accept bounding boxes with no
%      space after the colon, 18 Jul 1990.
%
%   TeX macros to include an Encapsulated PostScript graphic.
%   Works by finding the bounding box comment,
%   calculating the correct scale values, and inserting a vbox
%   of the appropriate size at the current position in the TeX document.
%
%   To use with the center environment of LaTeX, preface the \epsffile
%   call with a \leavevmode.  (LaTeX should probably supply this itself
%   for the center environment.)
%
%   To use, simply say
%   \input epsf           % somewhere early on in your TeX file
%   \epsfbox{filename.ps} % where you want to insert a vbox for a figure
%
%   Alternatively, you can type
%
%   \epsfbox[0 0 30 50]{filename.ps} % to supply your own BB
%
%   which will not read in the file, and will instead use the bounding
%   box you specify.
%
%   The effect will be to typeset the figure as a TeX box, at the
%   point of your \epsfbox command. By default, the graphic will have its
%   `natural' width (namely the width of its bounding box, as described
%   in filename.ps). The TeX box will have depth zero.
%
%   You can enlarge or reduce the figure by saying
%     \epsfxsize=<dimen> \epsfbox{filename.ps}
%   (or
%     \epsfysize=<dimen> \epsfbox{filename.ps})
%   instead. Then the width of the TeX box will be \epsfxsize and its
%   height will be scaled proportionately (or the height will be
%   \epsfysize and its width will be scaled proportiontally).  The
%   width (and height) is restored to zero after each use.
%
%   A more general facility for sizing is available by defining the
%   \epsfsize macro.    Normally you can redefine this macro
%   to do almost anything.  The first parameter is the natural x size of
%   the PostScript graphic, the second parameter is the natural y size
%   of the PostScript graphic.  It must return the xsize to use, or 0 if
%   natural scaling is to be used.  Common uses include:
%
%      \epsfxsize  % just leave the old value alone
%      0pt         % use the natural sizes
%      #1          % use the natural sizes
%      \hsize      % scale to full width
%      0.5#1       % scale to 50% of natural size
%      \ifnum#1>\hsize\hsize\else#1\fi  % smaller of natural, hsize
%
%   If you want TeX to report the size of the figure (as a message
%   on your terminal when it processes each figure), say `\epsfverbosetrue'.
%
\newread\epsffilein    % file to \read
\newif\ifepsffileok    % continue looking for the bounding box?
\newif\ifepsfbbfound   % success?
\newif\ifepsfverbose   % report what you're making?
\newdimen\epsfxsize    % horizontal size after scaling
\newdimen\epsfysize    % vertical size after scaling
\newdimen\epsftsize    % horizontal size before scaling
\newdimen\epsfrsize    % vertical size before scaling
\newdimen\epsftmp      % register for arithmetic manipulation
\newdimen\pspoints     % conversion factor
%
\pspoints=1bp          % Adobe points are `big'
\epsfxsize=0pt         % Default value, means `use natural size'
\epsfysize=0pt         % ditto
%
\def\epsfbox#1{\global\def\epsfllx{72}\global\def\epsflly{72}%
   \global\def\epsfurx{540}\global\def\epsfury{720}%
   \def\lbracket{[}\def\testit{#1}\ifx\testit\lbracket
   \let\next=\epsfgetlitbb\else\let\next=\epsfnormal\fi\next{#1}}%
%
\def\epsfgetlitbb#1#2 #3 #4 #5]#6{\epsfgrab #2 #3 #4 #5 .\\%
   \epsfsetgraph{#6}}%
%
\def\epsfnormal#1{\epsfgetbb{#1}\epsfsetgraph{#1}}%
%
\def\epsfgetbb#1{%
%
%   The first thing we need to do is to open the
%   PostScript file, if possible.
%
\openin\epsffilein=#1
\ifeof\epsffilein\errmessage{I couldn't open #1, will ignore it}\else
%
%   Okay, we got it. Now we'll scan lines until we find one that doesn't
%   start with %. We're looking for the bounding box comment.
%
   {\epsffileoktrue \chardef\other=12
    \def\do##1{\catcode`##1=\other}\dospecials \catcode`\ =10
    \loop
       \read\epsffilein to \epsffileline
       \ifeof\epsffilein\epsffileokfalse\else
%
%   We check to see if the first character is a % sign;
%   if not, we stop reading (unless the line was entirely blank);
%   if so, we look further and stop only if the line begins with
%   `%%BoundingBox:'.
%
          \expandafter\epsfaux\epsffileline:. \\%
       \fi
   \ifepsffileok\repeat
   \ifepsfbbfound\else
    \ifepsfverbose\message{No bounding box comment in #1; using defaults}\fi\fi
   }\closein\epsffilein\fi}%
%
%   Now we have to calculate the scale and offset values to use.
%   First we compute the natural sizes.
%
\def\epsfsetgraph#1{%
   \epsfrsize=\epsfury\pspoints
   \advance\epsfrsize by-\epsflly\pspoints
   \epsftsize=\epsfurx\pspoints
   \advance\epsftsize by-\epsfllx\pspoints
%
%   If `epsfxsize' is 0, we default to the natural size of the picture.
%   Otherwise we scale the graph to be \epsfxsize wide.
%
   \epsfxsize\epsfsize\epsftsize\epsfrsize
   \ifnum\epsfxsize=0 \ifnum\epsfysize=0
      \epsfxsize=\epsftsize \epsfysize=\epsfrsize
%
%   We have a sticky problem here:  TeX doesn't do floating point arithmetic!
%   Our goal is to compute y = rx/t. The following loop does this reasonably
%   fast, with an error of at most about 16 sp (about 1/4000 pt).
% 
     \else\epsftmp=\epsftsize \divide\epsftmp\epsfrsize
       \epsfxsize=\epsfysize \multiply\epsfxsize\epsftmp
       \multiply\epsftmp\epsfrsize \advance\epsftsize-\epsftmp
       \epsftmp=\epsfysize
       \loop \advance\epsftsize\epsftsize \divide\epsftmp 2
       \ifnum\epsftmp>0
          \ifnum\epsftsize<\epsfrsize\else
             \advance\epsftsize-\epsfrsize \advance\epsfxsize\epsftmp \fi
       \repeat
     \fi
   \else\epsftmp=\epsfrsize \divide\epsftmp\epsftsize
     \epsfysize=\epsfxsize \multiply\epsfysize\epsftmp   
     \multiply\epsftmp\epsftsize \advance\epsfrsize-\epsftmp
     \epsftmp=\epsfxsize
     \loop \advance\epsfrsize\epsfrsize \divide\epsftmp 2
     \ifnum\epsftmp>0
        \ifnum\epsfrsize<\epsftsize\else
           \advance\epsfrsize-\epsftsize \advance\epsfysize\epsftmp \fi
     \repeat     
   \fi
%
%  Finally, we make the vbox and stick in a \special that dvips can parse.
%
   \ifepsfverbose\message{#1: width=\the\epsfxsize, height=\the\epsfysize}\fi
   \epsftmp=10\epsfxsize \divide\epsftmp\pspoints
   \vbox to\epsfysize{\vfil\hbox to\epsfxsize{%
      \special{PSfile=#1 llx=\epsfllx\space lly=\epsflly\space
          urx=\epsfurx\space ury=\epsfury\space rwi=\number\epsftmp}%
      \hfil}}%
\epsfxsize=0pt\epsfysize=0pt}%

%
%   We still need to define the tricky \epsfaux macro. This requires
%   a couple of magic constants for comparison purposes.
%
{\catcode`\%=12 \global\let\epsfpercent=%\global\def\epsfbblit{%BoundingBox}}%
%
%   So we're ready to check for `%BoundingBox:' and to grab the
%   values if they are found.
%
\long\def\epsfaux#1#2:#3\\{\ifx#1\epsfpercent
   \def\testit{#2}\ifx\testit\epsfbblit
      \epsfgrab #3 . . . \\%
      \epsffileokfalse
      \global\epsfbbfoundtrue
   \fi\else\ifx#1\par\else\epsffileokfalse\fi\fi}%
%
%   Here we grab the values and stuff them in the appropriate definitions.
%
\def\epsfgrab #1 #2 #3 #4 #5\\{%
   \global\def\epsfllx{#1}\ifx\epsfllx\empty
      \epsfgrab #2 #3 #4 #5 .\\\else
   \global\def\epsflly{#2}%
   \global\def\epsfurx{#3}\global\def\epsfury{#4}\fi}%
%
%   We default the epsfsize macro.
%
\def\epsfsize#1#2{\epsfxsize}
%
%   Finally, another definition for compatibility with older macros.
%
\let\epsffile=\epsfbox

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}\label{Ch1} \hfill
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The fundamental theories which describe the observed interactions
are all gauge theories.
They describe gravitation, electroweak and strong
interaction. In the quantized version the forces between
particles (fermions) are mediated by gauge bosons.
QCD describes the strong interaction between quarks and gluons.
It is a SU(3) gauge theory with the Lagrangian\footnote{
Throughout the whole work the euclidian formulation of QCD is used
 \cite{SVZ}. }
%This is the origin of some $i^\prime s$  or minus signs which may be
%unfamiliar to you}
\beq
  {\cal L} = {1\over 4g^2}G_{\mu\nu}^aG^{\mu\nu}_a +
    \sum_{i=1}^{N_f} \bar\psi_i(iD\!\!\!/+im_i)\psi.
\eeq
Taking into account only the light quarks $u$, $d$ and sometimes $s$
and setting their mass to zero one arrives at a theory with
only one parameter, the gauge coupling constant $g$,
which is actually no parameter
because of dimensional transmutation. Despite the fact that
QCD looks such simple (e.g. compared to electroweak interaction)
it is very hard to solve this theory due to nonperturbative effects.
To clarify the role of instantons in QCD let me first give a list of
the most important methods
used to tackle QCD starting with very general methods applicable
to any quantum field theory (QFT) and ending with more specific
approaches:

\begin{itemize}
 \item {\it Axiomatic field theory: }
   Wightman/Oswalder\&Schrader have stated a set of
   Minkowskian/Euclidian axioms for
   vacuum correlators a general QFT should respect (analyticity,
   regularity, Lorentz invariance, locality, \ldots).
   It is clear that the theorems derived from these axioms
   have to be very general because no Lagrangian is used \cite{Gli}.
 \item{\it Haag-Ruelle/LSZ Theory: }
   S-Matrix elements are related to vacuum correlators. %\cite{LSZ}
   The S-Matrix
   is the central object for particle phenomenology containing such
   important informations as cross sections, form factors,
   structure functions, \ldots. For theoretical studies 
   vacuum correlators are more suitable
   because they avoid the need of explicitly
   constructing the Hilbert space which is an extremely
   complicated task beyond perturbation theory.
 \item{\it Quantization: }
   One might think that quantization should not appear in a list
   of methods for solving QFT because it is the method to
   obtain and define a QFT. On the other hand, besides canonical
   quantization there are other ways of quantizing a theory.
   The most popular is the path integral quantization \cite{Fey}.
   Usually in textbooks for particle physics 
   it is only used as an abbreviation to derive
   theorems more quickly. In general (beyond perturbation theory), different
   quantization methods lead to different physical and mathematical
   insights and different methods to solve the theory. Variants of
   the path integral quantization
   are the random walk quantization used in lattice theories and
   stochastic quantization.
 \item{\it (Broken) Symmetry: }
   Every degree of freedom like spin, flavor and color is the
   possible origin of (approximate) symmetries
   like $SU(2N_f)$ or subgroups and $SU(3)$ gauge invariance.
   Conserved currents and Ward identities \cite{War}
   can be obtained. In the case of light quarks,
   one further has approximate chiral symmetry leading to
   PCAC, axial Ward identities, current algebra theorems,
   soft pion physics,\ldots. %%\cite{...}
   Furthermore, QCD with massless quarks
   possesses an anomalously broken scale invariance which is the
   origin of the huge field of renormalization group techniques \cite{CSZ}.
 \item{\it Perturbation theory: }
   Due to asymptotic freedom the coupling constant $g$ decreases for
   high energies and perturbation theory in $g$ is applicable. %%\cite{...}
   QCD can thus be solved for processes which involve only momenta of say
   more than 1 GeV. The small distance behaviour of vacuum correlators
   is thus calculable ($x\leq 0.2$ fm).
 \item{\it Operator Product Expansion (OPE): }
   An improvement of perturbation theory is to separate the small
   distance physics from large distance effects. The former is contained
   in the so-called Wilson coefficients calculated perturbatively.
   The latter nonperturbative effects are contained in a few vacuum
   or hadron expectation
   values of {\sl local} operators which have to be determined from
   phenomenology or uncertain assumptions \cite{SVZ}.
   Vacuum correlators can be obtained
   up to distances of $x\leq 0.3\ldots 0.5$ fm.
 \item{\it QCD Sum rules: }
   QCD sum rules are widely used to determine hadron masses and couplings.
   The general method is to assume the existence of certain hadrons and
   take a resonance+continuum ansatz in the Minkowskian region
   for some correlator. The Euclidian correlator is calculated
   from theory (OPE, lattice, instantons). Via dispersion relations
   one may match both in some Euclidian window by fitting the hadron
   parameters which leads to a prediction for them. %\cite{SVZ}
 \item{\it Effective Theories: }
   One may construct effective Lagrangians containing mesons
   and/or baryons more or less motivated by QCD or history or
   other physical branches.
   Their parameters are determined from experiment or QCD as far
   as possible.
\end{itemize}

A variety of phenomena have been explained qualitatively and
calculated quantitatively with the methods listed above, but the large
distance behaviour of QCD is still unsolved. There are
at least two problems belonging to this domain: chiral symmetry
breaking and confinement. Up to now chiral symmetry breaking was
assumed and the consequences such as Golstone bosons
were discussed within this assumption.
In OPE one takes the nonzero values of the quark and other condensates
from experiment but has no possibility to predict them from theory.
The quark condensate is the order parameter of chiral symmetry and a nonzero
values indicates spontaneous breaking of this symmetry (SBCS).
Confinement has also to be assumed.

These two problems are not so unsolved as it might appear.
Up to now I have not mentioned two further approaches:

\begin{itemize}
 \item{\it Lattice QCD: }
   In principle the method is very simple. The continuum
   is replaced by a fine lattice covering a large
   but finite volume. The path integral is thus replaced by
   a finite number of integrals evaluated numerically. All vacuum
   correlators can be obtained for arbitrary Euclidian distances.
   Confinement and SBCS have been shown and other hadronic parameters
   are obtained. In practice, lattice
   calculations are much less straightforward than this scetchy
   description might suggest \cite{Cre}.
 \item{\it Instantons: }
   As in lattice QCD one evaluates the Euclidian path integrals
   but now in semiclassical approximation. In addition to the
   global minimum of the QCD action $A_\mu^a=0$ used by perturbation theory
   there are many other local minima called instantons which have to be
   taken into account. Inclusion of light quarks leads to an effective
   $2N_f$ quark vertex responsible for SBCS. %\cite{tHo}
   Although confinement cannot be explained,
   a lot of hadron parameters can nevertheless
   be calculated \cite{Shu,ShV} suggesting
   that confinement is not essential for the properties of hadrons.
\end{itemize}

In this article I want to give a quantitative and systematic
study of the implications of instantons to hadron properties.

Chapter \ref{Ch3} is an introduction to the semiclassical evaluation of
nontrivial integrals. After developing the method in the finite
dimensional case the partition function of QCD is considered
and the instanton liquid model is introduced.

In chapter \ref{Ch5} the propagator of a light quark is calculated.
The approximations which have to be made are stated and discussed.
It is shown that for one quark flavor the $1/N_c$ expansion is exact.
The constituent quark masses and the quark condensates are calculated
for $u$,$d$ and $s$ quarks.

The same approximations are used in chapter \ref{Ch6} to calculate
the 4 point functions. Special attention is payed to the singlet
correlator where a chain of quark loops contributes and is {\it not}
suppressed in the large-$N_c$ limit. Within one and the same
approximation we get Goldstone bosons in the pseudoscalar triplet
correlator but no massless singlet boson.

In chapter \ref{Ch7} meson correlators are discussed and plotted.
Employing a spectral ansatz it is possible to extract various meson masses
and couplings. They are compared to the values obtained
from extensive numerical studies of the instanton liquid \cite{ShV} and
to the experimental values.

\private{
Chapter \ref{Ch8} contains a discussion of constituent quark form factors
and their relevance to the proton spin problem \cite{Ash}
(in preparation).
}

Conclusions are given in chapter \ref{Ch9}.

%--------------------------------------------------------------


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theory of Instanton Liquid}\label{Ch3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%In the last chapter we have studied classical euclidian cromostatics.
%We have choosen as a cromostatic potential the one instanton field
%and studied in some analogy to electro/magneto statics features of this
%classical field theory.
%Of course the choice of the potential was not at all arbitrary.
%One way of motivating this choice was that it gives a significant
%contribution
%to the QCD partitioning function.
%In this chapter we will concentrate again to {\bf quantum} cromodynamics.

\private{description of what has and what will be done}

%--------------------------------------------------------------
\paragraph{Separating Gaussian from non-Gaussian degrees of freedom} \hfill
%--------------------------------------------------------------

In the path integral formulation solving QCD or any other QFT is equivalent
to calculating the partition function $Z$ including external source terms.
Of course this is not a
simple task because one has an infinite number of degrees of freedom.
One way to tackle this problem is to reduce the number of degrees
of freedom by integrating the (nearly) gaussian degrees
perturbatively leaving the non-Gaussian degrees, called collective
coordinates,
to be treated with other methods.

Let me first describe this method most generally
without refering to QFT. Our aim is to calculate
\beq\label{e31}
  Z=\int\!d^n\!x\,e^{-S[x]}
\eeq
where S is some real valued function depending on the $n$ dimensional vector
$x\in I\!\!R^n$. Let the integral be dominated by values of $x$ which lie
in the vicinity of a $k$ dimensional submanifold of $I\!\!R^n$ which may
be parametrisized by $x=f(\gamma)$, $\gamma\in I\!\!R^k$.
Vectors in the tangent
space of $f(\gamma)$ are called (approximate) zeromodes, because
$S[x(\gamma)]$ is (nearly) constant in this direction. Usually
$f(\gamma)$ represents some degenerate or approximate minima of $S$.
In addition $f(\gamma)$ may contain points which do not contribute
much to the functional integral, this will cause no error, but 
$f(\gamma)$ must not forget any significant points.
In figure \ref{fig31} a two dimensional example is shown containing a
river (one dimensional manifold) with steep mountains aside his
banks. So the integral will be dominated by the shaded area.
This area can be parametrized in the following way:
\beq\label{e32}
  x=f(\gamma)+y                     \quad,\quad
  x,y\in I\!\!R^n                   \quad,\quad
  \gamma\in I\!\!R^k		    \quad.
\eeq
To make this representation unique, we have to demand $k$ linear extra
conditions to $y$:
\beq\label{e33}
  y\!\cdot\! g_i(\gamma)=c_i \quad,\quad 1\leq i\leq k
\eeq
E.g., $c_i=0$ and
$g_i(\gamma)=\partial f(\gamma) / \partial \gamma_i $
would fix $y$ to be orthogonal to the "river" or equally stated:
It disallows fluctuations in the non-Gaussian zeromode direction.
Now every point $x$ (at least in the vicinity of the river) can
uniquely be described by $y$ satisfying the condition
(\ref{e33}) and by $\gamma$ via (\ref{e32}).
All we have to do now is to represent $Z$ in terms of the fluctuation
vector $y$ and the collective coordinates $\gamma$.
A convenient way is to introduce a Faddeev-Popov unit
\beq\label{e34}
  1=\int\!d^k\!\gamma\,d^n\!y
  \,\delta^k\big(y\!\cdot\! g_i(\gamma)-c_i\big)
  \,\delta^n\big(f(\gamma)+y-x\big)
  \,\Phi(x)
\eeq
which serves as a definition of $\Phi$.
Inserting this into (\ref{e31}) and integrating over $x$ yields
\beq\label{e35}
  Z=\int\! d^k\!\gamma\,d^n\!y
  \,\delta^k(y\!\cdot\! g_i(\gamma)-c_i)
  \,\Phi(f(\gamma)+y)
  \,e^{-S[f(\gamma)+y]}               \quad.
\eeq
The $y$-integration in (\ref{e34}) can trivially be performed:
\beq\label{e36}
  \Phi^{-1}(x) =
  \int\! d^k\!\gamma^\prime
  \,\delta^k\big( (x-f(\gamma^\prime))g_i(\gamma^\prime)-c_i \big)
  \quad.
\eeq
For $x=f(\gamma)+y$ the $\delta$-function in the
integral only contributes for $\gamma^\prime=\gamma$.
Thus we can expand the $\delta$-argument
up to linear order in $\gamma^\prime-\gamma$
\begin{eqnarray}\label{e37}
  \Phi^{-1}(f(\gamma)+y) & = &
  \int\! d^k\!\gamma^\prime
  \delta^k
    \left( \sum_j {\textstyle
      (y\!\cdot\!{\partial g_i(\gamma)\over\partial\gamma_j}
       -g_i(\gamma)\!\cdot\!{\partial f(\gamma)\over\partial\gamma_j} ) }
      (\gamma^\prime_j-\gamma_j)
    \right)
\\
  & = & \left| \mbox{det}_{ij}
    \left(
      y\!\cdot\!\partial_j g_i
      -g_i\!\cdot\! \partial_j f
    \right) \right|^{-1}
   \quad \footnotemark
\nonumber
\end{eqnarray}
\footnotetext{In the following we will omit the absolute bars; 
              when necessary they have to be reinstated. }
$$ \mbox{with}\quad y\!\cdot\! g_i(\gamma)=c_i \quad.$$
Inserting $\Phi$ into (\ref{e35}) we get
\beq\label{e38}
  Z=\int\! d^k\!\gamma\,d^n\!y
  \,\delta^k(y\!\cdot\! g_i(\gamma)-c_i)
  \,\mbox{det}_{ij}
    \left(
      y\!\cdot\!\partial_j g_i(\gamma)
      -g_i(\gamma)\!\cdot\!\partial_j f(\gamma)
    \right)
  \,e^{-S[f(\gamma)+y]} \quad.
\eeq
One may write $Z$ in a slightly different form, usually used in QFT,
because it is more suitable for semiclassical approximations.
$Z$ is independent of $c_i$ and therefore, although the r.h.s. of (\ref{e38})
explicitly contains $c_i$, it is actually independent of it.
We can smooth the $\delta$-function by a further multiplication with
\beq\label{e39}
  1=(2\pi\xi)^{-k/2}\int\! d^k\!c\,e^{-{1\over{2\xi}}\sum_i c_i^2}
  \quad.
\eeq
The determinant can be written in the form
\beq\label{e310}
  \mbox{det}A =\int\!d\eta\,d\bar\eta\,e^{\bar\eta A\eta}
\eeq
where $\eta$ are anticommuting grassmann variables (ghosts).
Inserting (\ref{e39}) and (\ref{e310}) into (\ref{e38}) and performing
the $c$-integration we finally get
\begin{eqnarray}\label{e311}
  Z &=& (2\pi\xi)^{-k/2}\int\!d^k\!\gamma\,d^n\!y\,d\eta\,d\bar\eta
    \,e^{ - S[f(\gamma)+y] -
        S_{gf}[y,\gamma] + S_{FPG}[y,\gamma,\eta,\bar\eta]}
\quad,\nonumber \\
  S_{gf} &=& {1\over 2\xi}y^T \left( \sum_i g_i(\gamma)
    g_i^{_T}(\gamma) \right) y
\quad,\\
  S_{FPG} &=& \sum_{ij} \bar\eta_i
    \left(
      y\!\cdot\!\partial_j g_i(\gamma)
      -g_i(\gamma)\!\cdot\!\partial_j f(\gamma)
    \right) \eta_j
\quad.\nonumber
\end{eqnarray}

\private{ 2D-Example, problems if f is not bijective,
          exact to all orders pert., semiclassical approx.}

%--------------------------------------------------------------
\paragraph{Effective QCD Lagrangian in a background field} \hfill
%--------------------------------------------------------------

Now it is time to return to QCD
\beq \label{e312}
  Z=\int\!DA_\mu\,e^{-S_{YM}[A]} \quad,\quad
  S_{YM}[A]=\int\!dx\,{1\over {4g^2}} G_{\mu\nu}^a G^{\mu\nu}_a
  \quad.
\eeq
The background configurations which (approximately) minimize $S_{YM}$
will be denoted by $\bar A_\mu(\gamma)$. A general gauge field
can be written in the form
\beq \label{e313}
  A_\mu = (\bar A_\mu(\gamma)+B_\mu)^\Omega \quad,\quad
  \gamma\in I\!\!R^k
\eeq
where $B_\mu$ are fluctuations around the background and
$A_\mu^\Omega$ is a gauge transformed field
\beq\label{e314}
  A_\mu^\Omega = S A_\mu S^\dagger + iS\partial_\mu S^\dagger
  \quad , \quad S=e^{i\Omega}\in SU(N_c) \quad.
\eeq
As in the finite dimensional case we have to make this representation
unique by introducing extra conditions,
\beq\label{e315}
  D_\mu(\bar A)B_\mu(x)=C(x) \quad,
\eeq
  to fix the gauge of the fluctuating field and
\beq \label{e316}
  \int\!d^4x\,\psi_\mu^i(x;\gamma)B_\mu(x) = c_i \quad,\quad
  1\leq i\leq k
\eeq
to avoid fluctuations in (approximately) zero mode direction.
The derivation of an effective action similar to (\ref{e311})
can now be performed in full analogy to the previous case
with only some notational complication. There is the following
correspondence:

\beq\label{e317}
  \begin{array}{lllllll}
    \mbox{finite example} & : & i   & x & y & \gamma_i           & g_i \\
    {\mbox{QCD}}            & : & i,x & A & B & \gamma_i,\Omega(x) &
\psi_\mu^i
  \end{array} \quad.
\eeq

The Faddeev-Popov unit has the form

\beq\label{e318}
  1=\int\!d^k\!\gamma\,D\Omega\,DB_\mu\,
    \delta\big( D_\mu(\bar A)B_\mu\big )
    \delta^k\big( \int\!\psi_\mu^iB_\mu d^4\!x \big)
    \delta\big( (\bar A_\mu+B_\mu)^\Omega-A_\mu \big)
    \Phi[A_\mu] \quad.
\eeq

The steps to get an expression for $\Phi$ are now:

Add primes to $\gamma,\Omega$ and $B$, insert $A=\bar A+B$,
linearize the last $\delta$-argument around $B_\mu^\prime=B_\mu$,
perform the functional $B_\mu^\prime$ integration and linearize the
remaining $\delta$ arguments around $\gamma^\prime=\gamma$ and
$\Omega^\prime=0$. Omitting the details of this calculation one gets
\cite{Dya}

\beq \label{e319}
  \Phi^{-1}(\bar A(\gamma)+B) =
  \int\!d^k\!\gamma^\prime\,D\Omega^\prime
  \delta^k(X_i)\,\delta(Y) \quad,
\eeq

\beq \label{e320}
  X_i=\int\!d^4\!x\,\sum_j
    \left(
      \psi_\mu^i(\gamma) { \partial\bar A_\mu \over\partial\gamma_j }
      - { \partial\psi^i_\mu(\gamma) \over \partial\gamma_j }B_\mu
    \right) (\gamma_j^\prime-\gamma_j) +
    \psi_\mu^i D_\mu(\bar A+B)\Omega^\prime \quad,
\eeq

\beq \label{e321}
  Y = \sum_j D_\mu(\bar A+B)
        { \partial\bar A_\mu \over\partial\gamma_j }
        (\gamma_j^\prime-\gamma_j) +
      D_\mu(\bar A)D_\mu(\bar A+B)\Omega^\prime \quad.
\eeq

From (\ref{e315}), (\ref{e316}), (\ref{e320}) and (\ref{e321}) one can
read off the form of the partition function $Z$:

\beq \label{e322}
  Z=N(\xi) \int\!d^k\!\gamma\,DB_\mu\,D\eta\,D\bar\eta
    \delta^k\big( \int\!\psi_\mu^iB_\mu d^4\!x \big)
    e^{-S_{QCD}[\bar A,B,\eta,\bar\eta]} \quad,
\eeq

\begin{eqnarray} \label{e3225}
  S_{QCD} &=& S_{YM}[\bar A+B]-S_{gf}[\bar A,B]+
        S_{FPG}[\bar A,B,\eta,\bar\eta]                  \quad,\nonumber  \\
  S_{YM} &=& \int\!d^4\!x
        {1\over 4g^2}G_{\mu\nu}^a G^{\mu\nu}_a(\bar A+B) \quad,\nonumber  \\
  S_{gf}&=& {1\over 2\xi} \int\!d^4\!x\,(D_\mu(\bar A)B_\mu)^2 \quad,     \\
  S_{FPG} &=& \sum_{ij}\bar\eta_i
    \left[ \int\!d^4\!x\,
        \psi_\mu^i(\gamma) { \partial\bar A_\mu \over\partial\gamma_j }
      - { \partial\psi^i_\mu(\gamma) \over \partial\gamma_j }B_\mu
    \right] \eta_j                                           \nonumber  \\
    &+&
    \sum_i\int\!d^4\!x\,\bar\eta_i\psi_\mu^i D_\mu(\bar A+B)\eta(x)
    +
    \int\!d^4\!x\sum_j \bar\eta(x) D_\mu(\bar A+B)
      { \partial\bar A_\mu \over\partial\gamma_j }          \nonumber  \\
    &+&
    \int\!d^4\!x\, \bar\eta(x) D_\mu(\bar A)D_\mu(\bar A+B) \eta(x)
    \quad.\nonumber
\end{eqnarray}
$S_{QCD}$ does not depend on the gauge parameter $\Omega$.
For this reason the $\Omega$ integration can be
absorbed in the normalization factor $N(\xi)$.
$\eta(x)$ are the usual ghost fields originating from the gauge fixing.
For every extra condition (\ref{e316}) one gets an additional ghost variable
$\eta_i$. For $\bar A=0$ and no extra condition ($k=0$) the action
given above just reduces to the usual QCD action including Faddeev-Popov
ghosts in
$R_\xi$ gauge
\beq
  S_{gf} = {1\over 2\xi} \int\!d^4\!x\,(\partial_\mu B_\mu)^2 \quad,\quad
  S_{FPG} = \int\!d^4\!x\, \bar\eta(x)\partial_\mu D_\mu (B) \eta(x)
  \quad.
\eeq

Note that the action (\ref{e322}) is still exact
with the non-harmonic degrees of freedom $\gamma_i$ now separated
from the hopefully more gaussian ones, $B_\mu$ and $\eta$.

For small coupling $g$ it is now possible to establish feynman rules
from (\ref{e3225}) in analogy to the case with no background. For this
one has to know the "free" gluon, ghost and quark propagator in a given
background $\bar A$. If $\bar A$ is a non constant
field even this is a very complicated task in contrast to usual
perturbation theory around $\bar A=0$. For a multi-instanton
configuration explicit expressions for the gluon and ghost propagator
are derived in \cite{Bro,Hut1}
%will be derived in chapter \ref{Ch4}
and for light quark propagators in
chapter \ref{Ch5}.

\private{ In the rest of this chapter we want to develop the
feynman rules of (\ref{e3225}) and give formal expressions for the
propagators in a multi pseudoparticle background. To a large extend
the rules and formulas are independend of whether
the pseudoparticles are instantons or some other configurations.
}
%--------------------------------------------------------------
\paragraph{The Semiclassical Limit} \hfill
%--------------------------------------------------------------

Before developing perturbation theory to all orders it is wise
to study the semiclassical limit where one keeps only terms
up to quadratic order in the fields. In QCD (and many other field theories)
this is equivalent to lowest order perturbation theory,
but around a very nontrivial background!

Up to now we have not specified $\psi^i_\mu$. A natural choice
would be $\psi^i_\mu= \partial\bar A_\mu / \partial\gamma_i$
to fix the fluctuations to be orthogonal to the zero modes.
Somewhat more convenient is to bring $\psi_\mu^i$ in background
gauge:
\beq
  \psi^i_\mu=\left( {\partial\bar A_\mu \over \partial\gamma_j} \right)
             ^\Omega
  \quad \mbox{with }\Omega\mbox{ such that} \quad
  D_\mu(\bar A)\psi_\mu^i=0 \quad.
\eeq
Furthermore we assume that $\bar A$ minimizes the gauge action
$S_{YM}$ which is true for widely separated instantons thus
neglecting linear terms $S_{QCD}$.

Up to quadratic order in the fields one has
\begin{eqnarray}
  S_{QCD} &=& S_{YM}[\bar A]
  + \int\!d^4\!x\, {1\over 2g^2}B_\mu K_{\mu\nu}(\bar A) B_\nu
  + \int\!d^4\!x\, \bar\eta(x) D^2(\bar A)\eta(x)          \nonumber  \\
  &+& \sum_{ij} \bar\eta_i
       \psi_\mu^i { \partial\bar A_\mu \over\partial\gamma_j }\eta_j
  + \int\!d^4\!x\sum_j \bar\eta(x)D_\mu(\bar A)
      { \partial\bar A_\mu \over\partial\gamma_j }\eta_j
  + O(\mbox{field}^3) \quad,
\end{eqnarray}
\beq
  K_{\mu\nu} = -D^2\delta_{\mu\nu}+2iG_{\mu\nu}+
     (1-{1\over\xi})D_\mu D_\nu \quad,\quad
  G_{\mu\nu}=F^c G_{\mu\nu}^c  \quad,\quad
  (F^c)_{ab}=if_{acb} \quad.
\eeq
Performing the integration over gauge fields and ghosts one gets
an effective action depending only on the collective coordinates
$\gamma_i$:
\beq
  Z=\int\!d^k\!\gamma\, e^{-S_{eff}[\gamma]}
\eeq
\beq \label{e330}
  e^{-S_{eff}[\gamma]} =
  \mbox{det}_{ij}
  \left(
     \psi_\mu^i(\gamma) { \partial\bar A_\mu \over\partial\gamma_j }
  \right)
  {\mbox{Det}(-D^2(\bar A)) \over
     (\mbox{Det}^\prime K_{\mu\nu}^\prime(\bar A))^{1/2} }
  e^{-S_{YM}[\bar A]}
\eeq
The $\delta$-function in (\ref{e322}) causes a restriction of the gauge
field
fluctuation to be orthogonal to $\psi_\mu^i$. $K_{\mu\nu}^\prime$ is
defined as $K_{\mu\nu}$ projected to the space orthogonal to
$\psi_\mu^i$, $\mbox{Det}^\prime$ takes into account all eigenvalues
of $K_{\mu\nu}^\prime$ except the $k$ zeromodes caused by the projection.

%--------------------------------------------------------------
\paragraph{The Instanton Gas Approximation} \hfill
%--------------------------------------------------------------

For $N$ well separated pseudoparticles $\bar A(x)=\sum A_I(x)$
the partition function $Z$ factorizes like
\beq
  Z = {1\over {N!}}\prod_{I=1}^N Z_I
\eeq
where $Z_I$ is the partition function in a one pseudoparticle background,
i.e.\ with $A_I$ inserted in (\ref{e330}) instead of $\bar A$.
It was a great deal to evaluate the functional determinants
in the background of one instanton\footnote{
To simplifiy notations we will treat instantons and anti-instantons on the
same footing. Both will be called instantons and are distinguished
by their topological charge $Q_I=\pm$ if necessary.}
\begin{eqnarray} \label{e332}
  A_{I\mu}^a(x) &=& O_I^{ab}\eta_{b\mu\nu}^I
                  { (x-z_i)_\nu \over (x-z_i)^2 }
                  { 2\rho^2 \over (x-z_I)^2+\rho^2 }
                     \quad,\\
\gamma_I=(z_I,O_I,\rho_I,Q_I)&=& {\mbox{(location, orientation, radius,
topological charge)}} \quad.\nonumber
\end{eqnarray}
To get finite results $Z_I$ has to be normalized to the $\bar A_\mu^a=0$
case, regularized and renormalized. The final result of the very
complicated calculation is \cite{tHo,Ber}
\begin{eqnarray}
  \left({Z_I\over Z_0}\right)_{reg} &=&
    {1\over 2}\sum_{Q_I=\pm} \int\!d^4\!z_I dO_I d\rho_I
    \, D(\rho_I) = V_4\int_0^\infty\!d\rho\,D(\rho)
  \quad,\nonumber\\
  D(\rho) &=& {1\over\rho^5}
            {4.6e^{-1.679N_c} \over\pi^2(N_c-1)!(N_c-2)! }
            S_0(\rho)^{2N_c}e^{-S_0(\rho) }
  \quad,\\
  S_0(\rho) &=& {8\pi^2\over g^2(\rho)} =
    b\ln{1\over\rho\Lambda} + {b^\prime\over b}\ln\ln{1\over\rho\Lambda}
    + O({1\over\ln{1\over\rho\Lambda}}) \quad,\quad
    \Lambda=\Lambda_{PV}
  \quad.\nonumber
\end{eqnarray}
\private{2 loop accuracy}

$D(\rho)$ is the instanton density,
$g(\rho)$ the running coupling constant, $b={11\over 3}N_c$ and
$b^\prime={17\over 3}N_c^2$.

%--------------------------------------------------------------
\paragraph{Quarks} \hfill
%--------------------------------------------------------------

Additional fields coupled in a gauge invariant way to the gluon field
can simply be incorporated by adding the appropriate lagrangian
with gauge field $A$ replaced by $\bar A +B$  and performing the
functional integration over the new fields.
So every quark contributes an extra factor
\beq
  \int\!D\Psi D\bar\Psi e^{-\int\!dx\,\bar\Psi(iD\!\!\!\!/+im)\Psi}
  = \mbox{Det}(iD\!\!\!\!/+im)
\eeq
to the partition function $Z$ where
\beq
  iD_\mu=i\partial_\mu + \bar A_\mu + B_\mu
\eeq
is the covariant derivative. In the semiclassical approximation $B_\mu$
can be set to zero. $D(\rho)$ has to be multiplied by the fermionic
factor
\beq
  F(m\rho)=\left\{
  \begin{array}{l@{\quad\mbox{for}\quad}l}
    1.34m\rho(1+m^2\rho^2\ln(m\rho)+\ldots) & m\rho\ll 1 \\
    1-{2\over 75m^2\rho^2}+\ldots           & m\rho\gg 1
  \end{array}\right.
\eeq
and $b$ and $b^\prime$ are now
\beq
  b = {11\over 3}N_c - {2\over 3}N_f \quad,\quad
  b^\prime = {17\over 3}N_c^2 - {13\over 3}N_cN_f
           + {1\over 2}{N_f\over N_c} \quad.
\eeq

%--------------------------------------------------------------
\paragraph{The Instanton Liquid Model} \hfill
%--------------------------------------------------------------

The probabiliy of small size instantons is low
because $D(\rho)$ vanishes rapidly for small distances.
On the other hand for large distances $D(\rho)$ blows up and soon
gets large. This is the origin of the infrared problem
which made a lot of people no longer believing
in instanton physics. Those who were not deterred by that have thought
of the following outcome \cite{Shu}.
For larger and larger distances, the
vacuum gets more and more filled with instantons of increasing size.
At some scale the instanton gas approximation breaks down and one
has to consider the interaction between instantons which might be
repulsive to stabilize the medium. The stabilization might occur
at distances at which a semiclassical treatment is still possible
and at densities at which the various instantons are still well
separated objects - say - not much deformed through their interaction.
So there is a narrow region of allowed values for the instanton radius.
This picture of the vacuum is called the instanton liqiud model.
The idea has been confirmed in the course of years by very
different approaches:
{}
\begin{itemize}\parskip=0ex\parsep=0ex\itemsep=0ex
\item Hardcore assumption \cite{Ilg}
\item Variational Approach \cite{Dya}
\item Numerical studies \cite{Shu}
\item Phenomenological success \cite{ShV}
\end{itemize}
{}
The picture has now become generally accepted
at least by those who believe in instanton physics
and it seems that the vacuum can be described by
effectively independent instantons of size $\rho=600\,\mbox{MeV}^{-1}$
and mean
distance $L_0=200\,$MeV. The integral instanton density is fixed
by the experimentally known gluon condensate \cite{SVZ2}:
\beq
  n = N/V_4 = 1/L_0^4
    = {1\over 32\pi^2}<G_{\mu\nu}^aG^{\mu\nu}_a>
    = (200\,\mbox{MeV})^4_{exp.}
\eeq
The ratio $L_0/\rho$ is estimated in different works to be
\beq
  (L_0/\rho)_{theor.} = 3.0 \ldots 3.2
\eeq

%--------------------------------------------------------------
%\paragraph{Feynman Rules ??} \hfill
%--------------------------------------------------------------

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Light Quark Propagator}\label{Ch5}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\unitlength=1mm
In this chapter the average quark propagator in the multiinstanton
background will be calculated. In the first section the instanton
background is treated as a classical external perturbation,
but the background field is not small (e.g. in the coupling $g$)
and therefore we have to sum up {\it all} feynman graphs. This
is possible in the case of one quark flavour within the so-called
zeromode approximation. The quark condensate and a constituent quark
mass are extracted from the quark propagator. 
In the last section it is shown that the case of
two or more quark flavours can be reduced to the one flavour case
in the limit of $N_c\to\infty$. The results obtained in the
one flavour case are therefore still valid when making this further
approximation.

%--------------------------------------------------------------
\paragraph{Perturbation Theory in the Multi-instanton Background}
\hfill
%--------------------------------------------------------------

It is well known how to calculate correlators in the presence
of an external classical gauge field at least as perturbation series
in powers of the external field $A^a_\mu(x)$. In the case of QCD
(or more accurately in classical chromodynamics) within the
instanton liquid model the external field
is a sum of well separated scatterers $A=\sum_I A_I$ called
instantons with a fixed radius $\rho$ and distributed randomly and
independently in Euclidian space. 

For a while we will restrict ourself to the  case of one quark flavor
and ignore gluon loops.
The Euclidian feynman rules have the following form:
\beq
\begin{picture}(57,22)
\put(16,17){\vector(-1,0){8}}
\put(8,17){\line(-1,0){8}}
\put(20,17){\makebox(0,0)[lc]
  {$\displaystyle =\quad {1\over p\!\!/+im}\quad=\quad S_0\quad,$}}
\put(13,2){\vector(-1,0){3}}
\put(10,2){\vector(-1,0){5}}
\put(5,2){\line(-1,0){2}}
\put(8,2){\line(0,1){5}}
\put(13,7){\makebox(0,0)[cc]{$A_I$}}
\put(20,2){\makebox(0,0)[lc]{$ \displaystyle =\quad A\!\!\!/_I\quad.$}}
\put(8,7){\makebox(0,0)[cc]{$\sf x$}}
\end{picture}
\eeq
In operator notation,
\beq
  \langle p|S_0|q\rangle = {1\over p\!\!/+im}\deltabar(p-q)
  \quad,\quad
  \langle x|A\!\!\!/_I|y\rangle = A\!\!\!/_I(x)\delta(x-y)
  \quad,
\eeq
\beqn
  \deltabar^d(\cdots):=(2\pi)^d\delta(\cdots) 
  \quad,\quad
  \int\!\dbar^d\!p:=\int\!{d^d\!p\over(2\pi)^d}
  \quad,
\eeqn
graphs are simply alternating chains of $S_0$ and $A_I$.
To average a graph over the instanton parameters one has to
perform the following integration for each instanton:
\beq
  \langle \ldots \rangle_I =
  {N\over V_4}\int d\gamma_I\ldots =
     {N\over 2}\sum_{Q_I=\pm}{1\over V_4}\int d^4\!z_I\int dO_I\ldots
  \quad.
\eeq
For example
\begin{center}
\special{em:linewidth 0.4pt}
\linethickness{0.4pt}
\begin{picture}(60,25)
\put(2,5){\makebox(0,0)[lc]{$\displaystyle\bigg\langle$}}
\put(45,5){\vector(-1,0){5}}
\put(40,5){\vector(-1,0){10}}
\put(30,5){\vector(-1,0){10}}
\put(20,5){\vector(-1,0){10}}
\put(10,5){\line(-1,0){5}}
\put(15,5){\line(3,4){5.33}}
\put(20.33,12){\line(2,-3){4.67}}
\put(35,5){\line(0,1){7}}
\put(25,12){\makebox(0,0)[cc]{$A_I$}}
\put(40,12){\makebox(0,0)[cc]{$A_J$}}
\put(48,5){\makebox(0,0)[lc]{$\displaystyle\bigg\rangle_{I\neq J}=$}}
\put(27,20){\oval(12,10)[]}
\put(27,15){\vector(1,0){1}}
\put(27,25){\vector(-1,0){1}}
\put(35,12){\line(-3,5){3}}
\put(22,17){\line(-2,-5){2}}
\put(20,12){\line(0,0){0}}
\put(20,12){\line(0,0){0}}
\put(20.25,12){\makebox(0,0)[cc]{\sf x}}
\put(35,12){\makebox(0,0)[cc]{\sf x}}
\end{picture}
\end{center}
\beq
  = -{N^2\over V_4^2}\int d\gamma_I d\gamma_J
  \langle p|S_0 A\!\!\!/_I S_0 A\!\!\!/_I S_0 A\!\!\!/_J S_0|q\rangle
  \mbox{Tr}(S_0 A\!\!\!/_J S_0 A\!\!\!/_I) \quad.
\eeq
The origin of the factor $N^2$ is the summation over all
pairs of different instantons $(I,J)$ yielding a factor
$N(N-1)\approx N^2$. The quark loop is the origin of the minus
sign and of the functional trace ''Tr''.

%--------------------------------------------------------------
\paragraph{Exact Scattering Amplitude in the one Instanton
Background}
\hfill
%--------------------------------------------------------------

Perturbation theory is suitable to study scattering processes.
To achieve chiral symmetry breaking or bound states one
has to sum up infinite series of a subclass of graphs or solve
Schwinger Dyson or Bethe Selpeter equations.

The first thing we can do is to sum up successive scatterings
at one instanton
\begin{center}
  \begin{picture}(96,7.85)
  \put(5,2.85){\vector(-1,0){3}}
  \put(2,2.85){\line(-1,0){2}}
  \put(8,2.85){\circle{6}}
  \put(16,2.85){\vector(-1,0){3}}
  \put(13,2.85){\line(-1,0){2}}
  \put(8,2.85){\makebox(0,0)[cc]{$V_I$}}
  \put(21,2.85){\makebox(0,0)[cc]{$:=$}}
  \put(36,2.85){\vector(-1,0){3}}
  \put(33,2.85){\vector(-1,0){5}}
  \put(28,2.85){\line(-1,0){2}}
  \put(31,2.85){\line(0,1){5}}
  \put(36,7.85){\makebox(0,0)[cc]{$A_I$}}
  \put(41,2.85){\makebox(0,0)[cc]{$+$}}
  \put(61,2.85){\vector(-1,0){3}}
  \put(58,2.85){\vector(-1,0){5}}
  \put(53,2.85){\vector(-1,0){5}}
  \put(48,2.85){\line(-1,0){2}}
  \put(51,2.85){\line(2,5){2}}
  \put(53,7.85){\line(2,-5){2}}
  \put(58,7.85){\makebox(0,0)[cc]{$A_I$}}
  \put(66,2.85){\makebox(0,0)[cc]{$+$}}
  \put(91,2.85){\vector(-1,0){3}}
  \put(88,2.85){\vector(-1,0){5}}
  \put(83,2.85){\vector(-1,0){5}}
  \put(78,2.85){\vector(-1,0){5}}
  \put(73,2.85){\line(-1,0){2}}
  \put(76,2.85){\line(1,1){5}}
  \put(81,7.85){\line(1,-1){5}}
  \put(81,2.85){\line(0,1){5}}
  \put(86,7.85){\makebox(0,0)[cc]{$A_I$}}
  \put(96,2.85){\makebox(0,0)[lc]{$+\ldots\quad,$}}
  \put(31,7.85){\makebox(0,0)[cc]{$\sf x$}}
  \put(53,7.85){\makebox(0,0)[cc]{$\sf x$}}
  \put(81,7.85){\makebox(0,0)[cc]{$\sf x$}}
  \end{picture}
 %\input ifig3.pic
\end{center}
\beq
  V_I := A\!\!\!/_I + A\!\!\!/_I S_0 A\!\!\!/_I + A\!\!\!/_I S_0
  A\!\!\!/_I S_0 A\!\!\!/_I + \ldots
      = S_0^{-1}(S_I-S_0)S_0^{-1}
\eeq
where
\beq
  S_I = (S_0^{-1}-A\!\!\!/_I)^{-1}
\eeq
is the quark propagator in the one instanton background.

%--------------------------------------------------------------
\paragraph{Zero Mode Approximation} \hfill
%--------------------------------------------------------------

The quark propagator in the one instanton background
can be represented in terms of the eigenvalues $\lambda_i$ and
eigenfunctions $\psi_i$ of the Dirac operator
\beq
  (i\partial\!\!\!/-A\!\!\!/_I)\psi_i=\lambda_i\psi_i \quad\ff\quad
  \langle x|S_I|y\rangle = \sum_i
    { \psi_i(x)\psi_i^\dagger(y) \over \lambda_i+im } \quad.
\eeq
There is one zero eigenvalue $iD\!\!\!/\psi_I=0$ 
(\look appendix \ref{Appa})
which makes the propagator singular
in the chiral limit:
\beq
  S_I={|\psi_I\rangle\langle\psi_I| \over im} + S_I^{NZM} \quad.
\eeq
In the so called zero mode approximation one replaces the non
zero mode part by the free propagator
\beq
\makebox(24,6){
\begin{picture}(24,13)
\put(11,5){\circle{6}}
\put(11,5){\makebox(0,0)[cc]{$V_I$}}
\put(22,5){\vector(-1,0){4}}
\put(18,5){\line(-1,0){4}}
\put(8,5){\vector(-1,0){4}}
\put(4,5){\line(-1,0){4}}
\end{picture}
} =
   S_I-S_0 \approx {|\psi_I\rangle\langle\psi_I| \over im} \quad.
\eeq
\private{wrong for $m\neq 0$}

Although this approximation is good for large as well as for small
momenta it may be bad for intermediate ones, but what is more important
is the fact that it is a wild approximation
and so might violate general theorems like Ward identities.
In contrast, all other approximations we make
are of systematic nature respecting all known symmetries of QCD.
\begin{itemize}\parskip=0ex\parsep=0ex\itemsep=0ex
  \item semiclassical approximation (systematic)
  \item multi-instanton background ("systematic")
  \item large $N_c$ expansion (systematic) (see next section)
  \item zero mode approximation (wild)
\end{itemize}
Note that every choice of a background gauge field is "systematic"
in the sense of respecting the symmetries of QCD as long as the
background satisfies these symmetries on average.

In the following sections we will see that the advantage of
the zeromode approximation is so great that we cannot disregard this
simplification.

\private{systematic in leading density but too trivial}

%--------------------------------------------------------------
\paragraph{Effective Vertex in the Multi-instanton Background}
\hfill
%--------------------------------------------------------------

Let us consider a quark line with two scatterings at $V_I$
and insert in between a number of instantons
which differ from $I$ and from all other instantons occuring
elsewhere in the graph. This enables us to average over these
enclosed instantons independently from the rest of the graph.
Summation over all possible insertions with at least one instanton
just yields the exact quark
propagator minus the free propagator.
Remember that direct repeated scattering at $A_I$
is already included in $V_I$.

Let us define
\beq\label{e544}
\begin{picture}(129,23)
\put(40,20){\vector(-1,0){3}}
\put(37,20){\line(-1,0){2}}
\put(32,20){\circle{6}}
\put(32,20){\makebox(0,0)[cc]{$V_I$}}
\put(29,20){\vector(-1,0){3}}
\put(26,20){\line(-1,0){2}}
\put(20,20){\makebox(0,0)[cc]{$:=$}}
\put(16,20){\vector(-1,0){3}}
\put(13,20){\line(-1,0){2}}
\put(8,20){\circle{6}}
\put(8,20){\makebox(0,0)[cc]{$M_I$}}
\put(5,20){\vector(-1,0){3}}
\put(2,20){\line(-1,0){2}}
\put(44,20){\makebox(0,0)[cc]{$+$}}
\put(56,20){\circle{6}}
\put(56,20){\makebox(0,0)[cc]{$V_I$}}
\put(53,20){\vector(-1,0){3}}
\put(50,20){\line(-1,0){2}}
\put(75,20){\vector(-1,0){3}}
\put(72,20){\line(-1,0){2}}
\put(67,20){\circle{6}}
\put(67,20){\makebox(0,0)[cc]{$V_I$}}
\put(61,21){\makebox(0,0)[cb]{$.$}}
\put(79,20){\makebox(0,0)[cc]{$+$}}
\put(91,20){\circle{6}}
\put(91,20){\makebox(0,0)[cc]{$V_I$}}
\put(88,20){\vector(-1,0){3}}
\put(85,20){\line(-1,0){2}}
\put(102,20){\circle{6}}
\put(102,20){\makebox(0,0)[cc]{$V_I$}}
\put(96,21){\makebox(0,0)[cb]{$.$}}
\put(121,20){\vector(-1,0){3}}
\put(118,20){\line(-1,0){2}}
\put(113,20){\circle{6}}
\put(113,20){\makebox(0,0)[cc]{$V_I$}}
\put(107,21){\makebox(0,0)[cb]{$.$}}
\put(124,20){\makebox(0,0)[lc]{$+\ldots$}}
\put(8,6){\makebox(0,0)[cb]{$.$}}
\put(20,5){\makebox(0,0)[cc]{$:=$}}
\put(44,5){\makebox(0,0)[cc]{$-$}}
\put(64,5){\vector(-1,0){8}}
\put(56,5){\line(-1,0){8}}
\put(68,5){\makebox(0,0)[lc]{$=\quad S-S_0$}}
\thicklines
\put(64,20){\line(-1,0){5}}
\put(61,20){\makebox(0,0)[cc]{$<$}}
\put(99,20){\line(-1,0){5}}
\put(96,20){\makebox(0,0)[cc]{$<$}}
\put(110,20){\line(-1,0){5}}
\put(107,20){\makebox(0,0)[cc]{$<$}}
\put(16,5){\line(-1,0){16}}
\put(8,5){\makebox(0,0)[cc]{$<$}}
\put(40,5){\line(-1,0){16}}
\put(32,5){\makebox(0,0)[cc]{$<$}}
\thinlines
\end{picture}
\eeq
\bqan
  M_I&=& V_I+V_I(S-S_0)V_I+V_I(S-S_0)V_I(S-S_0)V_I+\ldots \\
     &=& V_I+V_I(S-S_0)M_I
\eqan

This equation can be solved for $M_I$ with the following ansatz:

\beq
  M_I={1\over i\mu}S_0^{-1}|\psi_I\rangle\langle\psi_I|S_0^{-1}
\eeq
Inserting $M_I$ and $V_I$ into (\ref{e544}) we get
\beq
  {1\over i\mu}S_0^{-1}|\psi_I\rangle\langle\psi_I|S_0^{-1}
  = {1\over im}(1+{1\over i\mu}
         \langle\psi_I|S_0^{-1}(S-S_0)S_0^{-1}|\psi_I\rangle )
  S_0^{-1}|\psi_I\rangle\langle\psi_I|S_0^{-1}
\eeq
\beq\label{e550}
  \ff\quad
  \mu = m + i\langle\psi_I|S_0^{-1}(S-S_0)S_0^{-1}|\psi_I\rangle
  \quad.
\eeq
\private{$\mu$=eff. mass, $\mu=m-<\psi\psi>$}

%--------------------------------------------------------------
\paragraph{A Nice Cancellation} \hfill
%--------------------------------------------------------------

It is possible to arrange the graphs in
such a way that every $M_I$ occurs only once.
Consider a graph containing two scattering processes $M_I$
at the same instanton. The interesting part of the graph has the following
form
\beq
  \begin{picture}(35,10)(0,10)
  \put(2,5){\makebox(0,0)[cc]{$s$}}
  \put(5,5){\vector(1,0){4}}
  \put(9,5){\line(1,0){4}}
  \put(16,5){\circle{6}}
  \put(16,5){\makebox(0,0)[cc]{$M_I$}}
  \put(19,5){\vector(1,0){4}}
  \put(23,5){\line(1,0){4}}
  \put(16,17){\circle{6}}
  \put(16,17){\makebox(0,0)[cc]{$M_I$}}
  \put(27,17){\vector(-1,0){4}}
  \put(23,17){\line(-1,0){4}}
  \put(13,17){\vector(-1,0){4}}
  \put(9,17){\line(-1,0){4}}
  \put(2,17){\makebox(0,0)[cc]{$p$}}
  \put(30,17){\makebox(0,0)[cc]{$q$}}
  \put(30,5){\makebox(0,0)[cc]{$r$}}
  \put(16,12){\makebox(0,0)[cc]{$\vdots$}}
  \end{picture}
 %\input ifig4.pic
           =\quad
                \bigg[^{\nq\!\!\alpha_p}_{\nq i_p}
                { \psi_I(p)\psi_I^\dagger(q) \over i\mu }\bigg]
                ^{\alpha_q}_{i_q}
            \quad
                \bigg[^{\nq\!\!\alpha_r}_{\nq i_r}
                { \psi_I(r)\psi_I^\dagger(s) \over i\mu }\bigg]
                ^{\alpha_s}_{i_s}
\eeq
$\alpha_p$/$i_p$ are the color/Dirac indices at the quark leg
with momentum $p$.
In a physical process in addition to the graph containing the above subgraph
there
exists another graph with only two quark lines interchanged.
\beq
\makebox(35,0)
{\begin{picture}(36,20)
 \put(2,5){\makebox(0,0)[cc]{$s$}}
 \put(5,5){\vector(1,0){3}}
 \put(8,5){\line(1,0){2}}
 \put(10,5){\line(0,1){3}}
 \put(10,11){\circle{6}}
 \put(10,11){\makebox(0,0)[cc]{$M_I$}}
 \put(10,14){\line(0,1){3}}
 \put(10,17){\vector(-1,0){3}}
 \put(7,17){\line(-1,0){2}}
 \put(2,17){\makebox(0,0)[cc]{$p$}}
 \put(27,17){\vector(-1,0){3}}
 \put(24,17){\line(-1,0){2}}
 \put(22,17){\line(0,-1){3}}
 \put(22,11){\circle{6}}
 \put(22,11){\makebox(0,0)[cc]{$M_I$}}
 \put(22,8){\line(0,-1){3}}
 \put(22,5){\vector(1,0){3}}
 \put(25,5){\line(1,0){2}}
 \put(30,17){\makebox(0,0)[cc]{$q$}}
 \put(30,5){\makebox(0,0)[cc]{$r$}}
 \put(16,11){\makebox(0,0)[cc]{$\cdots$}}
 \end{picture}
%\input ifig5.pic
 }
        = - \quad
                \bigg[^{\nq\!\!\alpha_p}_{\nq i_p}
                { \psi_I(p)\psi_I^\dagger(s) \over i\mu }\bigg]
                ^{\alpha_s}_{i_s}
            \quad
                \bigg[^{\nq\!\!\alpha_r}_{\nq i_r}
                { \psi_I(r)\psi_I^\dagger(q) \over i\mu }\bigg]
                ^{\alpha_q}_{i_q}
\eeq
As usual, the interchange of two quark lines causes a minus sign in
the amplitude. Inspecting the two expressions, we see that they
coincide except for the sign, thus there exists a complete cancellation
\beq\label{e552}
  \begin{picture}(84,17.85)
  \put(0,2.85){\vector(1,0){4}}
  \put(4,2.85){\line(1,0){4}}
  \put(11,2.85){\circle{6}}
  \put(11,2.85){\makebox(0,0)[cc]{$M_I$}}
  \put(14,2.85){\vector(1,0){4}}
  \put(18,2.85){\line(1,0){4}}
  \put(11,14.85){\circle{6}}
  \put(11,14.85){\makebox(0,0)[cc]{$M_I$}}
  \put(22,14.85){\vector(-1,0){4}}
  \put(18,14.85){\line(-1,0){4}}
  \put(8,14.85){\vector(-1,0){4}}
  \put(4,14.85){\line(-1,0){4}}
  \put(42,2.85){\vector(1,0){3}}
  \put(45,2.85){\line(1,0){2}}
  \put(47,2.85){\line(0,1){3}}
  \put(47,8.85){\circle{6}}
  \put(47,8.85){\makebox(0,0)[cc]{$M_I$}}
  \put(47,11.85){\line(0,1){3}}
  \put(47,14.85){\vector(-1,0){3}}
  \put(44,14.85){\line(-1,0){2}}
  \put(64,14.85){\vector(-1,0){3}}
  \put(61,14.85){\line(-1,0){2}}
  \put(59,14.85){\line(0,-1){3}}
  \put(59,8.85){\circle{6}}
  \put(59,8.85){\makebox(0,0)[cc]{$M_I$}}
  \put(59,5.85){\line(0,-1){3}}
  \put(59,2.85){\vector(1,0){3}}
  \put(62,2.85){\line(1,0){2}}
  \put(32,8.85){\makebox(0,0)[cc]{$+$}}
  \put(74,8.85){\makebox(0,0)[cc]{$=$}}
  \put(84,8.85){\makebox(0,0)[cc]{$0$}}
  \put(53,8.85){\makebox(0,0)[cc]{$\cdots$}}
  \put(11,9.71){\makebox(0,0)[cc]{$\vdots$}}
\end{picture}
%\input ifig7.pic
\eeq
Whenever an $M_I$ occurs twice or more than twice in a graph
there exists another graph with opposite sign. Both contributions
cancel each other and can be ignored.
So a quark can scatter only once at every
instanton. This can be seen in another way: Because of Fermi statistics
every state can be occupied only once, and there is only one state
for each quark in the zero mode approximation, namely
the zeromode.

There are two equivalent descriptions of feynman graphs:
\begin{enumerate}
\item Draw all topologically distinct graphs with non-numerated vertices
  and assign a symmetry factor to each graph,
\item Draw all topologically distinct graphs with numerated vertices
  and assign a factor $1/V$, where $V$ is the number of vertices.
\end{enumerate}
If all possible graphs containing $M_I$ are allowed it is not difficult
to see within the second description that they can really be paired
as stated above.

%--------------------------------------------------------------
\paragraph{Renormalization of the Instanton-Density} \hfill
%--------------------------------------------------------------

Up to now the cancellation is incomplete because not all graphs are allowed.
Consider e.g.
\beqn
  %\input ifig9.pic
  \begin{picture}(85,30)
  \put(27,9){\circle{6}}
  \put(27,9){\makebox(0,0)[cc]{$M_I$}}
  \put(38,9){\vector(-1,0){4}}
  \put(34,9){\line(-1,0){4}}
  \put(8,9){\vector(-1,0){4}}
  \put(4,9){\line(-1,0){4}}
  \put(11,9){\circle{6}}
  \put(11,9){\makebox(0,0)[cc]{$M_I$}}
  \put(24,9){\vector(-1,0){5}}
  \put(19,9){\line(-1,0){5}}
  \put(19,1){\makebox(0,0)[cc]{not allowed}}
  \put(59,9){\circle{6}}
  \put(59,9){\makebox(0,0)[cc]{$M_I$}}
  \put(70,9){\vector(-1,0){4}}
  \put(66,9){\line(-1,0){4}}
  \put(56,9){\vector(-1,0){4}}
  \put(52,9){\line(-1,0){4}}
  \put(43,9){\makebox(0,0)[cc]{$+$}}
  \put(85,9){\makebox(0,0)[rc]{$=\quad 0$}}
  \put(59,16.3){\makebox(0,0)[cc]{$\vdots$}}
  \put(59,21){\circle{6}}
  \put(59,21){\makebox(0,0)[cc]{$M_I$}}
  \put(55,28){\vector(1,0){4}}
  \put(59,28){\line(1,0){4}}
  \put(59,1){\makebox(0,0)[cc]{not allowed}}
  \put(55.50,24.5){\oval(10,7)[l]}
  \put(62.50,24.5){\oval(10,7)[r]}
 \end{picture}
\eeqn
As in the case of $V_I$ both graphs are not allowed. Another example is
\beqn
 \begin{picture}(85,30)
 \put(27,9){\circle{6}}
 \put(27,9){\makebox(0,0)[cc]{$M_I$}}
 \put(38,9){\vector(-1,0){4}}
 \put(34,9){\line(-1,0){4}}
 \put(8,9){\vector(-1,0){4}}
 \put(4,9){\line(-1,0){4}}
 \put(11,9){\circle{6}}
 \put(11,9){\makebox(0,0)[cc]{$M_I$}}
 \put(19,1){\makebox(0,0)[cc]{not allowed}}
 \put(59,9){\circle{6}}
 \put(59,9){\makebox(0,0)[cc]{$M_I$}}
 \put(70,9){\vector(-1,0){4}}
 \put(66,9){\line(-1,0){4}}
 \put(56,9){\vector(-1,0){4}}
 \put(52,9){\line(-1,0){4}}
 \put(43,9){\makebox(0,0)[cc]{$+$}}
 \put(85,9){\makebox(0,0)[rc]{$=\quad 0$}}
 \put(59,16.3){\makebox(0,0)[cc]{$\vdots$}}
 \put(59,21){\circle{6}}
 \put(59,21){\makebox(0,0)[cc]{$M_I$}}
 \thicklines
 \put(62.50,24.5){\oval(10,7)[r]}
 \put(55.50,24.5){\oval(10,7)[l]}
 \put(55,28){\line(1,0){8}}
 \put(59,28){\makebox(0,0)[cc]{$>$}}
 \put(24,9){\line(-1,0){10}}
 \put(19,9){\makebox(0,0)[cc]{$<$}}
 \thinlines
 \put(59,1){\makebox(0,0)[cc]{most general tadpole}}
 \put(19,10){\makebox(0,0)[cc]{$\cdot$}}
 \put(59,29){\makebox(0,0)[cc]{$\cdot$}}
 \end{picture}
 %\input ifig10.pic
\eeqn
It is a general fact that all disallowed graphs can be paired
with other disallowed graphs or with tadpoles and vice versa.

What we have to do is to "disallow" all tadpole graphs.
Every $M_I$ can be surrounded by tadpoles which contribute
with a universal multiplicative factor which can be absorbed in a
redefinition of the instanton density $n_R$. Using this renormalized
density $n_R$ the pairing is now perfect and the statement
"every $M_I$ occurs only once" becomes true.

One further can show that in the presence of dynamical quarks
this renormalized density has to be identified with the 
gluon condensate instead of the "bare" density because the same
tadpoles contribute to the gluon condensate too. 

\private{relation between $n_R = n_G = $ gluon condensate $\neq n/N_c$}

%--------------------------------------------------------------
\paragraph{Selfconsistency Equation for the Quark Propagator}
\hfill
%--------------------------------------------------------------

Quark loops are no longer possible because they cannot be connected
to another part of the graph via a common instanton. All graphs
which can contribute to the propagator are chains of different $M_I's$.
\begin{center}
 \unitlength=0.9 mm
 \begin{picture}(165,8)
 \put(70,5){\vector(-1,0){3}}
 \put(67,5){\line(-1,0){2}}
 \put(62,5){\circle{6}}
 \put(62,5){\makebox(0,0)[cc]{$M_I$}}
 \put(59,5){\vector(-1,0){3}}
 \put(56,5){\line(-1,0){2}}
 \put(74,5){\makebox(0,0)[cc]{$+$}}
 \put(86,5){\circle{6}}
 \put(86,5){\makebox(0,0)[cc]{$M_I$}}
 \put(83,5){\vector(-1,0){3}}
 \put(80,5){\line(-1,0){2}}
 \put(105,5){\vector(-1,0){3}}
 \put(102,5){\line(-1,0){2}}
 \put(97,5){\circle{6}}
 \put(97,5){\makebox(0,0)[cc]{$M_J$}}
 \put(109,5){\makebox(0,0)[cc]{$+$}}
 \put(121,5){\circle{6}}
 \put(121,5){\makebox(0,0)[cc]{$M_I$}}
 \put(118,5){\vector(-1,0){3}}
 \put(115,5){\line(-1,0){2}}
 \put(132,5){\circle{6}}
 \put(132,5){\makebox(0,0)[cc]{$M_J$}}
 \put(151,5){\vector(-1,0){3}}
 \put(148,5){\line(-1,0){2}}
 \put(143,5){\circle{6}}
 \put(143,5){\makebox(0,0)[cc]{$M_K$}}
 \put(151,5){\makebox(0,0)[lc]{$\Bigg\rangle_{I\neq J\neq\ldots}$}}
 \put(23.5,5){\makebox(0,0)[cc]{$=\,\Bigg\langle$}}
 \put(44,5){\vector(-1,0){8}}
 \put(36,5){\line(-1,0){8}}
 \thicklines
 \put(16,5){\line(-1,0){16}}
 \put(8,5){\makebox(0,0)[cc]{$<$}}
 \thinlines
 \put(140,5){\vector(-1,0){3}}
 \put(137,5){\line(-1,0){2}}
 \put(129,5){\vector(-1,0){3}}
 \put(126,5){\line(-1,0){2}}
 \put(94,5){\vector(-1,0){3}}
 \put(91,5){\line(-1,0){2}}
 \put(49,5){\makebox(0,0)[cc]{$+$}}
 \end{picture}
%\input ifig11.pic
\unitlength= 1mm
\end{center}
The $M_I's$ can be averaged independently

\beq\label{e553}
  M(p) := i \langle M_I \rangle =
         {n_R \over 2\mu}p^2\varphi^{\prime 2}(p) \quad,
\eeq
where $\varphi^\prime$ is defined in appendix \ref{Appa}.
The resulting expression for the propagator now has the form
\beq\label{e554}
  S = S_0 + S_0{M\over i}S_0 + S_0{M\over i}S_0{M\over i}S_0
      +\ldots
    = (S_0^{-1}+M)^{-1}
\eeq
where $M=M(p)$ is the momentum dependent mass defined above. There is
just one thing to do: We have to solve the circular dependence
\begin{center}
 \begin{picture}(28,15.15)
 \put(1,9.14){\makebox(0,0)[cc]{$n_R$}}
 \put(14,9.14){\makebox(0,0)[cc]{$M$}}
 \put(22,4.14){\makebox(0,0)[cc]{$\mu$}}
 \put(22,14.14){\makebox(0,0)[cc]{$S$}}
 \put(16,10.14){\vector(1,1){4}}
 \put(22,12.14){\vector(0,-1){6}}
 \put(20,4.14){\vector(-1,1){4}}
 \put(13,15.15){\makebox(0,0)[cc]{$\scriptstyle (\ref{e553})$}}
 \put(28,9.14){\makebox(0,0)[cc]{$\scriptstyle (\ref{e550})$}}
 \put(13,2.99){\makebox(0,0)[cc]{$\scriptstyle (\ref{e554})$}}
 \put(5,9.03){\vector(1,0){6}}
 \end{picture}
%\input ifig12.pic
\end{center}
but $\mu$ is just a number, which makes the solution very simple.
Inserting (\ref{e553}) and (\ref{e554}) into (\ref{e550}) one gets the
following equation for $\mu$
\beq\label{e555}
  \mu = m + \int\dbar^4\!p\,
   {2\varphi^{\prime 2}(p)M_p \over p^2+(m+M_p)^2}
   (p^2+m(m+M_p))
\eeq
\private{$M_p$ \& $\mu$ definition such that not poles in $M_p$}

which may be solved numerically for different current masses $m$.

%--------------------------------------------------------------
\paragraph{Some Phenomenological Results} \hfill
%--------------------------------------------------------------

In the chiral limit (\ref{e555}) reduces to
\beq\label{e556}
  \mu^2 = n_R\int\dbar^4\!p\,
   {p^4\varphi^{\prime 4}(p) \over p^2+M_p^2} =
   \alpha n_R\rho^2 + O(n_R^2)
\eeq
$\mu^2$ is proportional to $n_R$ and thus $M$ is proportional to
$\sqrt{n_R}$ in contrast to a linear dependence on $n_R$ obtained
from a naive density expansion.

\private{ failure of density expansion $\sqrt{n_R+m}$ }

In the last expression the denominator
has been expanded in the density and
\beq
  \alpha = \rho^{-2}\int\dbar^4\!p\,
   p^2\varphi^{\prime 4}(p) = 6.6
\eeq
is a universal number.
For the standard values of $n_R$ and $\rho$ one gets
\bqa
  \mu_0^2 &=& 6.6n_R\rho^2 = (100\mbox{MeV})^2   \quad,\nonumber\\
  M(p=0)  &=& 7.7\rho\sqrt{n_R} = 300\mbox{MeV}  \quad.
\eqa
The exact solution of (\ref{e556}) which has been obtained numerically by
iteration,
differs from the leading density value by 15\%:
\beq
  M(0) = 345\mbox{MeV} \quad.
\eeq
The momentum dependence of the quark mass is shown in figure \ref{fig51}.
The mass $m+M(p)$ may be interpreted as the mass of a constituent
quark. At high energies it tends to the current mass, at low momentum
chiral symmety breaking occurs and the quark gets its constituent
mass M(0). Note that this is not a pole mass but a virtual mass at
zero momentum squared. 

Let us now take into account a small current mass $m$ formally of
the order $\sqrt{n_R}$. The selfconsistency equation now reads
\beq
  1={m\over\mu}+{\mu_0^2\over\mu^2}+O(n_R) \quad.
\eeq
Solving it for $\mu$ leads to
\beq
  \mu_0 \quad\leq\quad \mu={1\over 2}m+\sqrt{{1\over 4}m^2+\mu_0^2}
        \quad\leq\quad m+\mu_0
\eeq
For the strange quark $\mu$ is increased by a factor of 2:
\beq
  \mu(m_s=150\mbox{MeV}) = 200\mbox{MeV}
\eeq
It is interesting that $m_s+M(0)$ remains to be $300$MeV.
For zero momentum the increase of the current mass is just
compensated by an equal decrease of the dynamical mass $M(0)$.

From the propagator one can obtain the quark condensate
\beq
  \langle \bar\psi\psi\rangle :=
  \lim_{x\to 0}\mbox{tr}_{CD}(S(x)-S_0(x)) =
  N_c\int\!\dbar^4\!p\mbox{tr}_D(S(p)-S_0(p)) \quad.
\eeq
In leading order in the density one gets
\beq
  i\langle \bar\psi\psi\rangle = {n_RN_c\over\mu} =
  \langle G_{\mu\nu}^aG^{\mu\nu}_a\rangle /32\pi^2\mu \quad.
\eeq
This leads to the following condensates for $u$,$d$ and $s$ quarks:
\beq
  i\langle \bar uu\rangle = i\langle \bar dd\rangle =
    (250\mbox{MeV})^3 \quad,\quad
  \langle \bar ss\rangle = 0.5\langle\bar uu\rangle \quad.
\eeq
For heavy quarks there exists a similar relation
\beq
  i\langle \bar\psi\psi\rangle =
  \langle G_{\mu\nu}^aG^{\mu\nu}_a\rangle /48\pi^2m + O(m^{-3})
  \quad,
\eeq
which leads within 10\% to the same value for the strange quark
condensate. This nicely confirms 
the hypothesis that the strange quark can be treated as a
light quark as well as a heavy quark. This hypothesis is used in heavy 
to light quark matching formulas.

\private{discussion}

%--------------------------------------------------------------
\paragraph{Large $N_c$ expansion} \hfill
%--------------------------------------------------------------

%All higher connected quark correlators are zero in the one flavor
%case because there is no way to connect four or more vertices.

%Things change in the multiple flavor case or including gluon loops.
%These cases will be considered now.

Consider now the case of $N_f$ light quark flavors $u,d,s,\ldots$.
The discussion of the one flavor case in the previous sections
can be copied up to the pairing and cancelation of graphs
which contain more than one $M_I$ (\ref{e552}). This is still true in
the multiple flavor case but now both quark lines in (\ref{e552})
must have the same
flavor because $M_I$ always connects quarks of the same flavor.
So we have the theorem: "every $M_I$ occurs only once for each flavor".
From this point on the discussion of the one flavor case breaks
down because there are now graphs contributing to the
propagator containing quark loops. The simplest new contribution
has the form
\begin{center}
 \begin{picture}(38,18)
 \put(27,3){\circle{6}}
 \put(27,3){\makebox(0,0)[cc]{$M_J$}}
 \put(38,3){\vector(-1,0){4}}
 \put(34,3){\line(-1,0){4}}
 \put(8,3){\vector(-1,0){4}}
 \put(4,3){\line(-1,0){4}}
 \put(11,3){\circle{6}}
 \put(11,3){\makebox(0,0)[cc]{$M_I$}}
 \put(24,3){\vector(-1,0){5}}
 \put(19,3){\line(-1,0){5}}
 \put(11,9.86){\makebox(0,0)[cc]{$\vdots$}}
 \put(11,15){\circle{6}}
 \put(11,15){\makebox(0,0)[cc]{$M_I$}}
 \put(27,9.86){\makebox(0,0)[cc]{$\vdots$}}
 \put(27,15){\circle{6}}
 \put(27,15){\makebox(0,0)[cc]{$M_J$}}
 \put(4,5){\makebox(0,0)[cc]{$u$}}
 \put(19,5){\makebox(0,0)[cc]{$u$}}
 \put(34,5){\makebox(0,0)[cc]{$u$}}
 \put(25,17){\vector(-1,0){6}}
 \put(19,17){\line(-1,0){6}}
 \put(25,13){\line(0,0){0}}
 \put(13,13){\vector(1,0){6}}
 \put(19,13){\line(1,0){6}}
 \end{picture}
 %\input ifig14.pic
\end{center}
Is this contribution small in some sense ? Yes it is !
Quark loops are suppressed by a factor $1/N_c$. In perturbative
context this is extensively discussed in \cite{Wit2}, in instanton physics
it was first used by \cite{Dya}.
Although $1/3$ is not a very small number
the large $N_c$ expansion seems to be a good approximation in
various cases.
\private{$N_c\to\infty$ is formal in I-Physics because
         no I in $N_c=\infty$}

Consider a graph and add to it a new quark loop consisting of
\begin{center}
\begin{tabular}{l l}
  $N$ & new instantons        \\
  $S$ & instanton scatterings \\
\end{tabular}$\quad\quad S\geq N \quad.$
\end{center}
This multiplies the graph (see table \ref{tab2}) by a factor
$\mbox{loop} = O(N_c^{1+N-S})$.
The following cases are possible:
\begin{center}
\begin{tabular}{c l}
  $\underline{1+N-S}$ &                                      \\
  $=1$    & $\gdw M=N \gdw$ all instantons are new            \\
          & $\gdw$ the loop is disconnected                   \\
  $=0$    & $\gdw$ one old instanton $\gdw$ the loop is a tadpole \\
  $<0$    & the loop is supressed by at least one $1/N_c$     \\
\end{tabular}
\end{center}
Disconnected graphs are cancelled by the denominator and tadpoles
have been absorbed in $n_R$. So quark loops are indeed suppressed
in the limit $N_c\to\infty$. The same is true for gluon loops.
This can be seen by the same argument using the $N_c$ dependences
from table \ref{tab2}.

\begin{table}
  \begin{center}\begin{tabular}{|l|l|} \hline
  Parameters           & $n_R,\rho,N_c,(g_R)$     \\\hline\hline
  Instanton density    & $n = N/V_4 = n_R N_c \approx
                                           (200\mbox{MeV})^4$ \\
  Instanton radius     & $\rho \approx (600\mbox{MeV})^{-1}$  \\
  Number of colors     & $N_c=3$                              \\
  Coupling constant    & $g=g_R/N_c$
                         (not used in semicl. limit)          \\
  \hline
  Gluon condensate     & $\langle GG\rangle =
                          \langle G_{\mu\nu}^a G^{\mu\nu}_a \rangle
                          /32\pi^2 =: n_R N_c $               \\
  Quark condensate     & $\langle\bar\psi\psi\rangle \approx
                          0.39\,N_c\rho^{-1}\sqrt{n_R}
                          \approx (253\mbox{MeV})^3 $         \\
  Constituent mass     & $M(p)\sim\rho\sqrt{n_R}$             \\
  Quark mass           & $M_{quark}(0)\approx 7.7\,\rho n_R^{1/2}\approx
                          300\mbox{MeV}$                      \\
  Gluon mass           & $M_{gluon}(0)\approx 10.9\,\rho n_R^{1/2}\approx
                          420\mbox{MeV}$\cite{Hut1}           \\
  Ghost mass           & $M_{ghost}(0)\approx 7.7\,\rho n_R^{1/2}\approx
                          300\mbox{MeV}$\cite{Hut1}           \\
  Meson correlator     & $ \langle \bar\psi\Gamma\psi(x)\bar\psi
                         \Gamma\psi(0) \rangle_{trunc.} \sim N_c $ \\
  \hline
  Quark loop           & \makebox(20,0) {\unitlength=0.7mm
                         \begin{picture}(22,8)
                         \put(11,5){\circle{6}}
                         \put(12,8){\vector(-1,0){2}}
                         \end{picture}}
                         $\sim N_c$                 \\
  Instanton scattering & \makebox(20,0) {\unitlength=0.7mm
                         \begin{picture}(22,8)
                         \put(11,5){\circle{6}}
                         \put(11,5){\makebox(0,0)[cc]{$I$}}
                         \put(22,5){\line(-1,0){8}}
                         \put(8,5){\line(-1,0){8}}
                         \end{picture}}
                        $\sim N_c^{-1}\rho n_R^{-1/2}$ \\
  Instanton occurance  &\hspace{20mm} $\sim N_c n_R$    \\
  Gluon loop           & \makebox(20,0) {\unitlength=0.7mm
                         \begin{picture}(22,8)
                         \put(11,5){\circle{6}}
                         \put(12,8){\vector(-1,0){2}}
                         \end{picture}}
                       $\sim N_c^2-1$               \\
  Instanton scattering & \makebox(20,0) {\unitlength=0.7mm
                         \begin{picture}(22,8)
                         \put(11,5){\circle{6}}
                         \put(11,5){\makebox(0,0)[cc]{$I$}}
                         \put(22,5){\line(-1,0){8}}
                         \put(8,5){\line(-1,0){8}}
                         \end{picture}}
                      $\sim (N_c^2-1)^{-1}$             \\
  \hline
\end{tabular}\end{center}
\vspace{-3ex}
\caption{\label{tab2}
  Dependence of various quantities on the parameters of the
  instanton liquid model $n_R,\rho,N_c,(g_R)$.
}\end{table}

%--------------------------------------------------------------
%\paragraph{Analytical Properties} \hfill
%--------------------------------------------------------------

\private{analytical continuation,
  concluions, absorb gluon tadpole, $N_c$ counting arg. only
  true if one $M_I$ per color-loop}

%--------------------------------------------------------------

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Four Point Functions}\label{Ch6}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\unitlength=1mm
%--------------------------------------------------------------
\paragraph{Introduction} \hfill
%--------------------------------------------------------------

In the last section we have derived comprehensive Feynman rules within
the zero mode aprroximation:
\begin{center}
  \begin{picture}(60,18)
  \put(14,3){\circle{6}}
  \put(14,3){\makebox(0,0)[cc]{$M_I$}}
  \put(25,3){\vector(-1,0){4}}
  \put(21,3){\line(-1,0){4}}
  \put(11,3){\vector(-1,0){4}}
  \put(7,3){\line(-1,0){4}}
  \put(1,3){\makebox(0,0)[cc]{$p$}}
  \put(32,3){\makebox(0,0)[lc]{$\displaystyle =\; {1 \over {i\mu}}
p\!\!/\psi_I(p) \psi_I^\dagger(q)q\!\!/$}}
  \put(22,18){\vector(-1,0){8}}
  \put(14,18){\line(-1,0){8}}
  \put(32,18){\makebox(0,0)[lc]
      {$ \displaystyle =\; {1 \over {p\!\!/ + im}}\quad =\quad  S_0(p)$}}
  \put(27,3){\makebox(0,0)[cc]{$q$}}
  \end{picture}
  %\input ifig15.pic
\end{center}
A quark of a given flavor can scatter only once at a instanton $I$
via $M_I$. Tadpole graphs are not allowed, they are absorbed in
the renormalized instanton density $n_R$ which has to be used
when averaging over the instantons. In the large $N_c$ limit
dynamical quark loops are supressed and $\mu$ can be determined
by (\ref{e555}). In the chiral limit
\beq
   \mu^2=6.6 n_R\rho^2 + O(n_R^2) \quad.
\eeq
In this section we want to calculate 4 point functions in the
case of two quark flavors of equal mass in the limit $N_c\to\infty$
within the zero mode approximation:
\beq\label{e662}
  \deltabar(p-s+r-q)\Pi_{\Gamma\Gamma^\prime}(p,s,q,r) =
  \makebox(20,2){
     \unitlength=0.5mm
     \begin{picture}(32,19)
     \put(13,1){\framebox(8,18)[cc]{}}
     \thicklines
     \put(6,4){\line(1,0){7}}
     \put(13,16){\line(-1,0){7}}
     \put(28,16){\line(-1,0){7}}
     \put(21,4){\line(1,0){7}}
     \put(9,16){\makebox(0,0)[cc]{$\scriptstyle <$}}
     \put(24,16){\makebox(0,0)[cc]{$\scriptstyle <$}}
     \put(24,4){\makebox(0,0)[cc]{$\scriptstyle >$}}
     \put(9,4){\makebox(0,0)[cc]{$\scriptstyle >$}}
     \put(1,10){\makebox(0,0)[cc]{$\scriptstyle \Gamma$}}
     \put(4,16){\makebox(0,0)[cc]{$\scriptstyle p$}}
     \put(4,4){\makebox(0,0)[cc]{$\scriptstyle s$}}
     \put(30,16){\makebox(0,0)[cc]{$\scriptstyle q$}}
     \put(30,4){\makebox(0,0)[cc]{$\scriptstyle r$}}
     \put(32,10){\makebox(0,0)[cc]{$\scriptstyle \Gamma^\prime$}}
     \thinlines
     \end{picture}
    %\input ifig16.pic
    } =
\eeq
\beqn
  = -\int\!dxdydzdw\,e^{i(py-qz+rw-sx)}
    \langle 0|{\cal T} \bar\psi(x)\Gamma\psi(y)
    \bar\psi(z)\Gamma^\prime\psi(w)|0\rangle
  \quad.
\eeqn
The $\psi$ fields are $u$ or $d$ quark fields arbitrarily mixed.
Without restriction to generality we have taken the correlator
to be a color singlet \private{color decomposition}.
These 4 point functions
can be used to study meson correlators (see chapter \ref{Ch7})
\beqn
  \Pi(t)= \makebox(17,2)
  {\unitlength=0.5mm
   \begin{picture}(31,19)
   \put(14,1){\framebox(8,18)[cc]{$K$}}
   \thicklines
   \put(22,10){\oval(14,12)[r]}
   \put(14,10){\oval(14,12)[l]}
   \put(11,4){\makebox(0,0)[cc]{$\scriptstyle >$}}
   \put(11,15.14){\makebox(0,0)[cc]{$\scriptstyle <$}}
   \put(25,16){\makebox(0,0)[cc]{$\scriptstyle <$}}
   \put(25,4.86){\makebox(0,0)[cc]{$\scriptstyle >$}}
   \put(30,10){\makebox(0,0)[lc]{$\sim\!\!\!\!\prime$}}
   \put(6,10){\makebox(0,0)[rc]{$\sim\!\!\!\!\prime$}}
   \put(5,14.15){\makebox(0,0)[cc]{$\scriptstyle \Gamma$}}
   \put(31,14.15){\makebox(0,0)[cc]{$\scriptstyle \Gamma^\prime$}}
   \thinlines
   \end{picture}
   %\input ifig17.pic
   } = 
\eeqn
\beqn
  = \int\!(dpds)\int\!(dqdr)\Pi(p,s,q,r) =
  - \int\!dx\,e^{itx}
    \langle 0|{\cal T}j_\Gamma(x)j_{\Gamma^\prime}(0)|0\rangle \quad,
\eeqn
\beq\label{e666}
  j_\Gamma(x) = \bar\psi\Gamma\psi(x) \quad,\quad
  j_{\Gamma^\prime}(0) = \bar\psi\Gamma^\prime\psi(0) \quad,
\eeq
\beqn
  \int\!(dpds) = \int\!\dbar p\dbar s\,\deltabar(p-s-t) \quad,\quad
  \int\!(dqdr) = \int\!\dbar q\dbar r\,\deltabar(q-r-t) \quad,\quad
\eeqn
\beqn
  t=p-s=q-r \quad.
\eeqn
\private{Another application is the study of quark form factors
         (see chapter \ref{Ch8}). }

%--------------------------------------------------------------
\paragraph{Large $N_c$ approximation} \hfill
%--------------------------------------------------------------

The most general graph for the quark propagator is
a sequence of different instantons $M_I$, according to
the rules above.
Similarly the most general graph for the triplet correlator
$\langle(\bar ud)(\bar du)\rangle$
consists of two quark propagators each containing
every insanton only once. However, the $u$ and $d$ propagator may contain
common instantons, e.g.
\begin{center}
  \begin{picture}(95,17.85)
  \put(6,2.85){\makebox(0,0)[cc]{$s$}}
  \put(9,2.85){\vector(1,0){4}}
  \put(13,2.85){\line(1,0){4}}
  \put(20,2.85){\circle{6}}
  \put(20,2.85){\makebox(0,0)[cc]{$M_I$}}
  \put(23,2.85){\vector(1,0){4}}
  \put(27,2.85){\line(1,0){4}}
  \put(20,14.85){\circle{6}}
  \put(20,14.85){\makebox(0,0)[cc]{$M_I$}}
  \put(31,14.85){\vector(-1,0){4}}
  \put(27,14.85){\line(-1,0){4}}
  \put(17,14.85){\vector(-1,0){4}}
  \put(13,14.85){\line(-1,0){4}}
  \put(6,14.85){\makebox(0,0)[cc]{$p$}}
  \put(90,14.85){\makebox(0,0)[cc]{$q$}}
  \put(90,2.85){\makebox(0,0)[cc]{$r$}}
  \put(20,9.71){\makebox(0,0)[cc]{$\vdots$}}
  \put(34,2.85){\circle{6}}
  \put(34,2.85){\makebox(0,0)[cc]{$M_N$}}
  \put(37,2.85){\vector(1,0){4}}
  \put(41,2.85){\line(1,0){4}}
  \put(34,14.85){\circle{6}}
  \put(34,14.85){\makebox(0,0)[cc]{$M_J$}}
  \put(45,14.85){\vector(-1,0){4}}
  \put(41,14.85){\line(-1,0){4}}
  \put(48,2.85){\circle{6}}
  \put(48,2.85){\makebox(0,0)[cc]{$M_P$}}
  \put(51,2.85){\vector(1,0){4}}
  \put(55,2.85){\line(1,0){4}}
  \put(48,14.85){\circle{6}}
  \put(48,14.85){\makebox(0,0)[cc]{$M_K$}}
  \put(59,14.85){\vector(-1,0){4}}
  \put(55,14.85){\line(-1,0){4}}
  \put(62,2.85){\circle{6}}
  \put(62,2.85){\makebox(0,0)[cc]{$M_K$}}
  \put(65,2.85){\vector(1,0){4}}
  \put(69,2.85){\line(1,0){4}}
  \put(62,14.85){\circle{6}}
  \put(62,14.85){\makebox(0,0)[cc]{$M_L$}}
  \put(73,14.85){\vector(-1,0){4}}
  \put(69,14.85){\line(-1,0){4}}
  \put(76,2.85){\circle{6}}
  \put(76,2.85){\makebox(0,0)[cc]{$M_M$}}
  \put(79,2.85){\vector(1,0){4}}
  \put(83,2.85){\line(1,0){4}}
  \put(76,14.85){\circle{6}}
  \put(76,14.85){\makebox(0,0)[cc]{$M_M$}}
  \put(87,14.85){\vector(-1,0){4}}
  \put(83,14.85){\line(-1,0){4}}
  \put(76,9.71){\makebox(0,0)[cc]{$\vdots$}}
  \put(1,8.85){\makebox(0,0)[lc]{$\cdots$}}
  \put(95,8.85){\makebox(0,0)[rc]{$\cdots$}}
  \put(13,16.85){\makebox(0,0)[cc]{$u$}}
  \put(83,16.85){\makebox(0,0)[cc]{$u$}}
  \put(83,4.85){\makebox(0,0)[cc]{$d$}}
  \put(13,4.85){\makebox(0,0)[cc]{$d$}}
  \put(55,9.71){\makebox(0,0)[cc]{$\ddots$}}
  \end{picture}
  %\input ifig18.pic
\end{center}
It is always assumed that the left and right
hand sides of the graphs form color singlets.
Non-common instantons can be averaged like in the propagator case
to yield graphs of the form
\beq\label{e678}
  \begin{picture}(63,17.85)
  \thicklines
  \put(18,2.85){\circle{6}}
  \put(18,2.85){\makebox(0,0)[cc]{$M_I$}}
  \put(18,14.85){\circle{6}}
  \put(18,14.85){\makebox(0,0)[cc]{$M_I$}}
  \put(18,9.71){\makebox(0,0)[cc]{$\vdots$}}
  \put(15,14.85){\line(-1,0){8}}
  \put(11,14.85){\makebox(0,0)[cc]{$<$}}
  \put(7,2.85){\line(1,0){8}}
  \put(11,2.85){\makebox(0,0)[cc]{$>$}}
  \put(29,14.85){\line(-1,0){8}}
  \put(25,14.85){\makebox(0,0)[cc]{$<$}}
  \put(21,2.85){\line(1,0){8}}
  \put(25,2.85){\makebox(0,0)[cc]{$>$}}
  \put(46,2.85){\circle{6}}
  \put(46,2.85){\makebox(0,0)[cc]{$M_N$}}
  \put(46,14.85){\circle{6}}
  \put(46,14.85){\makebox(0,0)[cc]{$M_N$}}
  \put(46,9.71){\makebox(0,0)[cc]{$\vdots$}}
  \put(57,14.85){\line(-1,0){8}}
  \put(53,14.85){\makebox(0,0)[cc]{$<$}}
  \put(49,2.85){\line(1,0){8}}
  \put(53,2.85){\makebox(0,0)[cc]{$>$}}
  \put(31,2.85){\circle{6}}
  \put(31,2.85){\makebox(0,0)[cc]{$M_K$}}
  \put(31,14.85){\circle{6}}
  \put(31,14.85){\makebox(0,0)[cc]{$M_K$}}
  \put(31,9.71){\makebox(0,0)[cc]{$\vdots$}}
  \put(42,14.85){\line(-1,0){8}}
  \put(38,14.85){\makebox(0,0)[cc]{$<$}}
  \put(34,2.85){\line(1,0){8}}
  \put(38,2.85){\makebox(0,0)[cc]{$>$}}
  \put(11,17.85){\makebox(0,0)[cc]{$u$}}
  \put(11,5.85){\makebox(0,0)[cc]{$d$}}
  \put(53,17.85){\makebox(0,0)[cc]{$u$}}
  \put(53,5.85){\makebox(0,0)[cc]{$d$}}
  \put(63,8.85){\makebox(0,0)[rc]{$\cdots$}}
  \put(1,8.85){\makebox(0,0)[lc]{$\cdots$}}
  \thinlines
  \end{picture}
 %\input ifig19.pic
\eeq

where the thick line represents the full propagator
\begin{center}
 \unitlength=0.9 mm
 \begin{picture}(165,8)
 \put(70,5){\vector(-1,0){3}}
 \put(67,5){\line(-1,0){2}}
 \put(62,5){\circle{6}}
 \put(62,5){\makebox(0,0)[cc]{$M_I$}}
 \put(59,5){\vector(-1,0){3}}
 \put(56,5){\line(-1,0){2}}
 \put(74,5){\makebox(0,0)[cc]{$+$}}
 \put(86,5){\circle{6}}
 \put(86,5){\makebox(0,0)[cc]{$M_I$}}
 \put(83,5){\vector(-1,0){3}}
 \put(80,5){\line(-1,0){2}}
 \put(105,5){\vector(-1,0){3}}
 \put(102,5){\line(-1,0){2}}
 \put(97,5){\circle{6}}
 \put(97,5){\makebox(0,0)[cc]{$M_J$}}
 \put(109,5){\makebox(0,0)[cc]{$+$}}
 \put(121,5){\circle{6}}
 \put(121,5){\makebox(0,0)[cc]{$M_I$}}
 \put(118,5){\vector(-1,0){3}}
 \put(115,5){\line(-1,0){2}}
 \put(132,5){\circle{6}}
 \put(132,5){\makebox(0,0)[cc]{$M_J$}}
 \put(151,5){\vector(-1,0){3}}
 \put(148,5){\line(-1,0){2}}
 \put(143,5){\circle{6}}
 \put(143,5){\makebox(0,0)[cc]{$M_K$}}
 \put(151,5){\makebox(0,0)[lc]{$\Bigg\rangle_{I\neq J\neq\ldots}$}}
 \put(23.5,5){\makebox(0,0)[cc]{$=\,\Bigg\langle$}}
 \put(44,5){\vector(-1,0){8}}
 \put(36,5){\line(-1,0){8}}
 \thicklines
 \put(16,5){\line(-1,0){16}}
 \put(8,5){\makebox(0,0)[cc]{$<$}}
 \thinlines
 \put(140,5){\vector(-1,0){3}}
 \put(137,5){\line(-1,0){2}}
 \put(129,5){\vector(-1,0){3}}
 \put(126,5){\line(-1,0){2}}
 \put(94,5){\vector(-1,0){3}}
 \put(91,5){\line(-1,0){2}}
 \put(49,5){\makebox(0,0)[cc]{$+$}}
 \end{picture}
  \unitlength = 1mm
 %\input ifig11.pic
\end{center}
It can be shown that non planar diagrams like

\begin{center}
  \begin{picture}(36,17.85)
  \thicklines
  \put(11,2.85){\circle{6}}
  \put(11,2.85){\makebox(0,0)[cc]{$M_I$}}
  \put(11,14.85){\circle{6}}
  \put(11,14.85){\makebox(0,0)[cc]{$M_I$}}
  \put(8,14.85){\line(-1,0){8}}
  \put(4,14.85){\makebox(0,0)[cc]{$<$}}
  \put(0,2.85){\line(1,0){8}}
  \put(4,2.85){\makebox(0,0)[cc]{$>$}}
  \put(22,14.85){\line(-1,0){8}}
  \put(18,14.85){\makebox(0,0)[cc]{$<$}}
  \put(14,2.85){\line(1,0){8}}
  \put(18,2.85){\makebox(0,0)[cc]{$>$}}
  \put(25,2.85){\circle{6}}
  \put(25,2.85){\makebox(0,0)[cc]{$M_N$}}
  \put(25,14.85){\circle{6}}
  \put(25,14.85){\makebox(0,0)[cc]{$M_N$}}
  \put(36,14.85){\line(-1,0){8}}
  \put(32,14.85){\makebox(0,0)[cc]{$<$}}
  \put(28,2.85){\line(1,0){8}}
  \put(32,2.85){\makebox(0,0)[cc]{$>$}}
  \put(4,17.85){\makebox(0,0)[cc]{$u$}}
  \put(4,5.85){\makebox(0,0)[cc]{$d$}}
  \put(32,17.85){\makebox(0,0)[cc]{$u$}}
  \put(32,5.85){\makebox(0,0)[cc]{$d$}}
  \thinlines
  \put(15,11.18){\makebox(0,0)[cc]{$\cdot$}}
  \put(18,9){\makebox(0,0)[cc]{$\cdot$}}
  \put(21,6.88){\makebox(0,0)[cc]{$\cdot$}}
  \put(15,6.88){\makebox(0,0)[cc]{$\cdot$}}
  \put(21,11.18){\makebox(0,0)[cc]{$\cdot$}}
  \end{picture}
  %\input ifig20.pic
\end{center}
are suppressed by $1/N_c$, where again the left and right
hand sides of the graphs have to form color singlets.
So only the ladder diagrams shown in (\ref{e678}) contribute to the
triplet correlator.

\private{ That these graphs are indeed all leading
 order graphs can not be seen by the simple $N_c$ counting rules
 derived in section ?? because they are only applicable ... }

One might think that the mixed correlator
$\langle(\bar uu)(\bar dd)\rangle$ is zero because the graphs
necessarily include quark loops, or that only the two loop graphs
contribute --- but this is not the case!
Let us first state the result and then discuss it. Graphs
contributing to the mixed correlator are chains of quark bubbles
\beq\label{e679}
  \begin{picture}(87,19.57)
  \thicklines
  \put(14,4.57){\line(0,1){3}}
  \put(14,10.57){\circle{6}}
  \put(14,10.57){\makebox(0,0)[cc]{$M_I$}}
  \put(14,13.57){\line(0,1){3}}
  \put(26,16.57){\line(0,-1){3}}
  \put(26,10.57){\circle{6}}
  \put(26,10.57){\makebox(0,0)[cc]{$M_I$}}
  \put(26,7.57){\line(0,-1){3}}
  \put(20,10.57){\makebox(0,0)[cc]{$\cdots$}}
  \put(14,16.57){\line(-1,0){6}}
  \put(11,16.57){\makebox(0,0)[cc]{$<$}}
  \put(8,4.57){\line(1,0){6}}
  \put(11,4.57){\makebox(0,0)[cc]{$>$}}
  \put(26,16.57){\line(1,0){12}}
  \put(32,16.57){\makebox(0,0)[cc]{$<$}}
  \put(26,4.57){\line(1,0){12}}
  \put(32,4.57){\makebox(0,0)[cc]{$>$}}
  \put(38,4.57){\line(0,1){3}}
  \put(38,10.57){\circle{6}}
  \put(38,10.57){\makebox(0,0)[cc]{$M_I$}}
  \put(38,13.57){\line(0,1){3}}
  \put(50,16.57){\line(0,-1){3}}
  \put(50,10.57){\circle{6}}
  \put(50,10.57){\makebox(0,0)[cc]{$M_I$}}
  \put(50,7.57){\line(0,-1){3}}
  \put(44,10.57){\makebox(0,0)[cc]{$\cdots$}}
  \put(50,16.57){\line(1,0){12}}
  \put(56,16.57){\makebox(0,0)[cc]{$<$}}
  \put(50,4.57){\line(1,0){12}}
  \put(56,4.57){\makebox(0,0)[cc]{$>$}}
  \put(62,4.57){\line(0,1){3}}
  \put(62,10.57){\circle{6}}
  \put(62,10.57){\makebox(0,0)[cc]{$M_I$}}
  \put(62,13.57){\line(0,1){3}}
  \put(74,16.57){\line(0,-1){3}}
  \put(74,10.57){\circle{6}}
  \put(74,10.57){\makebox(0,0)[cc]{$M_I$}}
  \put(74,7.57){\line(0,-1){3}}
  \put(68,10.57){\makebox(0,0)[cc]{$\cdots$}}
  \put(80,16.57){\line(-1,0){6}}
  \put(77,16.57){\makebox(0,0)[cc]{$<$}}
  \put(74,4.57){\line(1,0){6}}
  \put(77,4.57){\makebox(0,0)[cc]{$>$}}
  \put(1,10.57){\makebox(0,0)[lc]{$\cdots$}}
  \put(87,10.57){\makebox(0,0)[rc]{$\cdots$}}
  \put(11,19.57){\makebox(0,0)[cc]{$u$}}
  \put(32,19.57){\makebox(0,0)[cc]{$d$}}
  \put(56,19.57){\makebox(0,0)[cc]{$u$}}
  \put(77,19.57){\makebox(0,0)[cc]{$d$}}
  \put(11,1.28){\makebox(0,0)[cc]{$u$}}
  \put(32,1.28){\makebox(0,0)[cc]{$d$}}
  \put(56,1.28){\makebox(0,0)[cc]{$u$}}
  \put(77,1.28){\makebox(0,0)[cc]{$d$}}
  \thinlines
  \end{picture}
 %\input ifig21.pic
\eeq
Application of the $N_c$ counting rules shows that this chain
is of order $1/N_c$. Taking the color trace at the left and right
hand side of the chain we see that the mixed correlator
is of order $N_c$. Using (\ref{e552}) it is clear that the triplet
correlator is of the same order.

What is wrong with the derivation of quark loop suppression
in the last chapter ? The main assumption was that every graph
containing a loop can be constructed from a graph not possessing
this loop by simply adding the loop. Eliminating a loop from the bubble
chain (\ref{e679}) yields a disconnected graph, but we only consider
connected 4 point functions. So the quark loop chain cannot be
constructed in a way needed to prove quark loop suppression.

In the case of meson correlators one can take another point of view.
The disconnected two loop contribution is of order $N_c^2$ but
(except for the scalar case) the contribution is zero. So
the bubble chain is a subleading graph of order $N_c$ and
nothing has been said about the form of subleading graphs.

Nevertheless, all connected graphs can be obtained starting
from (\ref{e679}) by adding further instantons and bubbles ---
but now $N_c$
counting rules tell us that every attempt results in a $1/N_c$
suppression. Therefore the bubble chain is the most general leading
order graph.

Nothing has to be changed for the correlator
$\langle(\bar dd)(\bar dd)\rangle$ except that chain (\ref{e679}) must
start with $d$.

To calculate the connected 4 point functions one must now
average and sum up the chains. Alternatively this can be represented
in recursive from usually called Bethe Salpeter equations:
\beqn
  \unitlength=1mm
  \linethickness{0.4pt}
  \begin{picture}(124,20)
  \thicklines
  \put(0,5){\line(1,0){7}}
  \put(3,5){\makebox(0,0)[cc]{$>$}}
  \put(7,17){\line(-1,0){7}}
  \put(3,17){\makebox(0,0)[cc]{$<$}}
  \put(7,2){\framebox(8,18)[cc]{$G$}}
  \put(3,20){\makebox(0,0)[cc]{$u$}}
  \put(3,1.79){\makebox(0,0)[cc]{$d$}}
  \put(15,5){\line(1,0){7}}
  \put(19,5){\makebox(0,0)[cc]{$>$}}
  \put(22,17){\line(-1,0){7}}
  \put(19,17){\makebox(0,0)[cc]{$<$}}
  \put(19,20){\makebox(0,0)[cc]{$u$}}
  \put(19,1.79){\makebox(0,0)[cc]{$d$}}
  \put(31,11){\makebox(0,0)[cc]{$=\,\bigg\langle$}}
  \put(51,5){\circle{6}}
  \put(51,5){\makebox(0,0)[cc]{$M_I$}}
  \put(51,17){\circle{6}}
  \put(51,17){\makebox(0,0)[cc]{$M_I$}}
  \put(51,11.94){\makebox(0,0)[cc]{$\vdots$}}
  \put(48,17){\line(-1,0){8}}
  \put(44,17){\makebox(0,0)[cc]{$<$}}
  \put(40,5){\line(1,0){8}}
  \put(44,5){\makebox(0,0)[cc]{$>$}}
  \put(62,17){\line(-1,0){8}}
  \put(58,17){\makebox(0,0)[cc]{$<$}}
  \put(54,5){\line(1,0){8}}
  \put(58,5){\makebox(0,0)[cc]{$>$}}
  \put(44,20){\makebox(0,0)[cc]{$u$}}
  \put(44,1.79){\makebox(0,0)[cc]{$d$}}
  \put(58,20){\makebox(0,0)[cc]{$u$}}
  \put(58,1.79){\makebox(0,0)[cc]{$d$}}
  \put(70,11){\makebox(0,0)[cc]{$+$}}
  \put(89,5){\circle{6}}
  \put(89,5){\makebox(0,0)[cc]{$M_I$}}
  \put(89,17){\circle{6}}
  \put(89,17){\makebox(0,0)[cc]{$M_I$}}
  \put(89,11.94){\makebox(0,0)[cc]{$\vdots$}}
  \put(86,17){\line(-1,0){8}}
  \put(82,17){\makebox(0,0)[cc]{$<$}}
  \put(78,5){\line(1,0){8}}
  \put(82,5){\makebox(0,0)[cc]{$>$}}
  \put(100,17){\line(-1,0){8}}
  \put(96,17){\makebox(0,0)[cc]{$<$}}
  \put(92,5){\line(1,0){8}}
  \put(96,5){\makebox(0,0)[cc]{$>$}}
  \put(82,20){\makebox(0,0)[cc]{$u$}}
  \put(82,1.79){\makebox(0,0)[cc]{$d$}}
  \put(96,20){\makebox(0,0)[cc]{$u$}}
  \put(96,1.79){\makebox(0,0)[cc]{$d$}}
  \put(100,2){\framebox(8,18)[cc]{$G$}}
  \put(108,5){\line(1,0){7}}
  \put(112,5){\makebox(0,0)[cc]{$>$}}
  \put(115,17){\line(-1,0){7}}
  \put(112,17){\makebox(0,0)[cc]{$<$}}
  \put(112,20){\makebox(0,0)[cc]{$u$}}
  \put(112,1.79){\makebox(0,0)[cc]{$d$}}
  \put(124,10){\makebox(0,0)[rc]{$\bigg\rangle_I$}}
  \thinlines
  \end{picture}
  %\input ifig221.pic
\eeqn

\beq\label{e680}
\unitlength=1mm
\linethickness{0.4pt}
\begin{picture}(124,20)
\thicklines
\put(0,5){\line(1,0){7}}
\put(3,5){\makebox(0,0)[cc]{$>$}}
\put(7,17){\line(-1,0){7}}
\put(3,17){\makebox(0,0)[cc]{$<$}}
\put(7,2){\framebox(8,18)[cc]{$H$}}
\put(3,20){\makebox(0,0)[cc]{$u$}}
\put(3,1.74){\makebox(0,0)[cc]{$u$}}
\put(15,5){\line(1,0){7}}
\put(19,5){\makebox(0,0)[cc]{$>$}}
\put(22,17){\line(-1,0){7}}
\put(19,17){\makebox(0,0)[cc]{$<$}}
\put(19,20){\makebox(0,0)[cc]{$d$}}
\put(19,1.74){\makebox(0,0)[cc]{$d$}}
\put(31,11){\makebox(0,0)[cc]{$=\,\bigg\langle$}}
\put(45,5){\line(0,1){3}}
\put(45,11){\circle{6}}
\put(45,11){\makebox(0,0)[cc]{$M_I$}}
\put(45,14){\line(0,1){3}}
\put(57,17){\line(0,-1){3}}
\put(57,11){\circle{6}}
\put(57,11){\makebox(0,0)[cc]{$M_I$}}
\put(57,8){\line(0,-1){3}}
\put(51,11){\makebox(0,0)[cc]{$\cdots$}}
\put(45,17){\line(-1,0){5}}
\put(42,17){\makebox(0,0)[cc]{$<$}}
\put(40,5){\line(1,0){5}}
\put(42,5){\makebox(0,0)[cc]{$>$}}
\put(42,20){\makebox(0,0)[cc]{$u$}}
\put(42,1.74){\makebox(0,0)[cc]{$u$}}
\put(62,17){\line(-1,0){5}}
\put(60,17){\makebox(0,0)[cc]{$<$}}
\put(57,5){\line(1,0){5}}
\put(60,5){\makebox(0,0)[cc]{$>$}}
\put(60,20){\makebox(0,0)[cc]{$d$}}
\put(60,1.74){\makebox(0,0)[cc]{$d$}}
\put(70,11){\makebox(0,0)[cc]{$+$}}
\put(83,5){\line(0,1){3}}
\put(83,11){\circle{6}}
\put(83,11){\makebox(0,0)[cc]{$M_I$}}
\put(83,14){\line(0,1){3}}
\put(95,17){\line(0,-1){3}}
\put(95,11){\circle{6}}
\put(95,11){\makebox(0,0)[cc]{$M_I$}}
\put(95,8){\line(0,-1){3}}
\put(89,11){\makebox(0,0)[cc]{$\cdots$}}
\put(83,17){\line(-1,0){5}}
\put(80,17){\makebox(0,0)[cc]{$<$}}
\put(78,5){\line(1,0){5}}
\put(80,5){\makebox(0,0)[cc]{$>$}}
\put(80,20){\makebox(0,0)[cc]{$u$}}
\put(80,1.74){\makebox(0,0)[cc]{$u$}}
\put(100,17){\line(-1,0){5}}
\put(97,17){\makebox(0,0)[cc]{$<$}}
\put(95,5){\line(1,0){5}}
\put(97,5){\makebox(0,0)[cc]{$>$}}
\put(97,20){\makebox(0,0)[cc]{$d$}}
\put(97,1.74){\makebox(0,0)[cc]{$d$}}
\put(100,2){\framebox(8,18)[cc]{$K$}}
\put(108,5){\line(1,0){7}}
\put(112,5){\makebox(0,0)[cc]{$>$}}
\put(115,17){\line(-1,0){7}}
\put(112,17){\makebox(0,0)[cc]{$<$}}
\put(112,20){\makebox(0,0)[cc]{$d$}}
\put(112,1.74){\makebox(0,0)[cc]{$d$}}
\put(124,10){\makebox(0,0)[rc]{$\bigg\rangle_I$}}
\thinlines
\end{picture}
%\input ifig222.pic
\eeq

\beqn
\unitlength=1mm
\linethickness{0.4pt}
\begin{picture}(124,19.99)
\thicklines
\put(0,4.99){\line(1,0){7}}
\put(3,4.99){\makebox(0,0)[cc]{$>$}}
\put(7,16.99){\line(-1,0){7}}
\put(3,16.99){\makebox(0,0)[cc]{$<$}}
\put(7,1.99){\framebox(8,18)[cc]{$K$}}
\put(3,19.99){\makebox(0,0)[cc]{$d$}}
\put(3,1.70){\makebox(0,0)[cc]{$d$}}
\put(15,4.99){\line(1,0){7}}
\put(19,4.99){\makebox(0,0)[cc]{$>$}}
\put(22,16.99){\line(-1,0){7}}
\put(19,16.99){\makebox(0,0)[cc]{$<$}}
\put(19,19.99){\makebox(0,0)[cc]{$d$}}
\put(19,1.70){\makebox(0,0)[cc]{$d$}}
\put(31,10.99){\makebox(0,0)[cc]{$=\quad$}}
\put(83,4.99){\line(0,1){3}}
\put(83,10.99){\circle{6}}
\put(83,10.99){\makebox(0,0)[cc]{$M_I$}}
\put(83,13.99){\line(0,1){3}}
\put(95,16.99){\line(0,-1){3}}
\put(95,10.99){\circle{6}}
\put(95,10.99){\makebox(0,0)[cc]{$M_I$}}
\put(95,7.99){\line(0,-1){3}}
\put(89,10.99){\makebox(0,0)[cc]{$\cdots$}}
\put(83,16.99){\line(-1,0){5}}
\put(80,16.99){\makebox(0,0)[cc]{$<$}}
\put(78,4.99){\line(1,0){5}}
\put(80,4.99){\makebox(0,0)[cc]{$>$}}
\put(80,19.99){\makebox(0,0)[cc]{$d$}}
\put(80,1.70){\makebox(0,0)[cc]{$d$}}
\put(100,16.99){\line(-1,0){5}}
\put(97,16.99){\makebox(0,0)[cc]{$<$}}
\put(95,4.99){\line(1,0){5}}
\put(97,4.99){\makebox(0,0)[cc]{$>$}}
\put(97,19.99){\makebox(0,0)[cc]{$u$}}
\put(97,1.70){\makebox(0,0)[cc]{$u$}}
\put(100,1.99){\framebox(8,18)[cc]{$H$}}
\put(108,4.99){\line(1,0){7}}
\put(112,4.99){\makebox(0,0)[cc]{$>$}}
\put(115,16.99){\line(-1,0){7}}
\put(112,16.99){\makebox(0,0)[cc]{$<$}}
\put(112,19.99){\makebox(0,0)[cc]{$d$}}
\put(112,1.70){\makebox(0,0)[cc]{$d$}}
\put(124,9.99){\makebox(0,0)[rc]{$\bigg\rangle_I$}}
\put(73,10.99){\makebox(0,0)[cc]{$\bigg\langle$}}
\thinlines
\end{picture}
%\input ifig223.pic
\eeqn

%--------------------------------------------------------------
\paragraph{Solution of the Bethe Salpeter Equations} \hfill
%--------------------------------------------------------------

Before solving the BS equations we have to construct the kernel.
The l.h.s.\ of the kernel always forms a color singlet
because of the restriction to color singlet correlators.

Contracting the color and Dirac indices on the l.h.s.\
and using the formulas of appendix \ref{Appa} one gets
\bqa
  \makebox(45,8)
  {\begin{picture}(39,23)
   \put(21,3){\circle{6}}
   \put(21,3){\makebox(0,0)[cc]{$M_I$}}
   \put(21,15){\circle{6}}
   \put(21,15){\makebox(0,0)[cc]{$M_I$}}
   \put(21,10.29){\makebox(0,0)[cc]{$\vdots$}}
   \put(18,15){\line(-1,0){8}}
   \put(14,15){\makebox(0,0)[cc]{$\prime$}}
   \put(10,3){\line(1,0){8}}
   \put(14,3){\makebox(0,0)[cc]{$\prime$}}
   \put(32,15){\line(-1,0){8}}
   \put(28,15){\makebox(0,0)[cc]{$\prime$}}
   \put(24,3){\line(1,0){8}}
   \put(28,3){\makebox(0,0)[cc]{$\prime$}}
   \put(1,9){\makebox(0,0)[cc]{$\bigg\langle\Gamma$}}
   \put(8,15){\makebox(0,0)[cc]{$p$}}
   \put(34,15){\makebox(0,0)[cc]{$q$}}
   \put(34,3){\makebox(0,0)[cc]{$r$}}
   \put(8,3){\makebox(0,0)[cc]{$s$}}
   \put(39,9){\makebox(0,0)[cc]{$\bigg\rangle_I$}}
   \end{picture}
 %\input ifig23.pic
  } =
  {1\over (i\mu)^2}
    \langle r\!\!/\psi_I(r)\psi_I^\dagger(s)s\!\!/\Gamma
            p\!\!/\psi_I(p)\psi_I^\dagger(q)q\!\!/\rangle_I = \\[0.5cm]
  = -{n_R\over\mu^2}
    p\varphi^\prime(p)q\varphi^\prime(q)r\varphi^\prime(r)s\varphi^\prime(s)
    \deltabar(p-s+r-q)\left\langle\mbox{tr}_D
      \left(\Gamma{1\pm\gamma_5\over 2}\right)
                  {1\pm\gamma_5\over 2}\right\rangle_\pm \quad.\nonumber
\eqa
The kernel can now be determined to be
\begin{center}
 \begin{picture}(130,18)
 \put(4,3){\line(1,0){7}}
 \put(7,3){\makebox(0,0)[cc]{$\prime$}}
 \put(11,15){\line(-1,0){7}}
 \put(7,15){\makebox(0,0)[cc]{$\prime$}}
 \put(11,0){\framebox(8,18)[cc]{$1$}}
 \put(19,3){\line(1,0){7}}
 \put(23,3){\makebox(0,0)[cc]{$\prime$}}
 \put(26,15){\line(-1,0){7}}
 \put(23,15){\makebox(0,0)[cc]{$\prime$}}
 \put(2,15){\makebox(0,0)[cc]{$p$}}
 \put(28,15){\makebox(0,0)[cc]{$q$}}
 \put(28,3){\makebox(0,0)[cc]{$r$}}
 \put(2,3){\makebox(0,0)[cc]{$s$}}
 \put(34,9){\makebox(0,0)[cc]{$:=$}}
 \put(57,3){\circle{6}}
 \put(57,3){\makebox(0,0)[cc]{$M_I$}}
 \put(57,15){\circle{6}}
 \put(57,15){\makebox(0,0)[cc]{$M_I$}}
 \put(57,10.29){\makebox(0,0)[cc]{$\vdots$}}
 \put(54,15){\line(-1,0){8}}
 \put(50,15){\makebox(0,0)[cc]{$\prime$}}
 \put(46,3){\line(1,0){8}}
 \put(50,3){\makebox(0,0)[cc]{$\prime$}}
 \put(68,15){\line(-1,0){8}}
 \put(64,15){\makebox(0,0)[cc]{$\prime$}}
 \put(60,3){\line(1,0){8}}
 \put(64,3){\makebox(0,0)[cc]{$\prime$}}
 \put(39,9){\makebox(0,0)[cc]{$\bigg\langle$}}
 \put(44,15){\makebox(0,0)[cc]{$p$}}
 \put(70,15){\makebox(0,0)[cc]{$q$}}
 \put(70,3){\makebox(0,0)[cc]{$r$}}
 \put(44,3){\makebox(0,0)[cc]{$s$}}
 \put(82,9){\makebox(0,0)[cc]{$\,\bigg\rangle_I=\,-\,\bigg\langle$}}
 \put(102,3){\line(0,1){3}}
 \put(102,9){\circle{6}}
 \put(102,9){\makebox(0,0)[cc]{$M_I$}}
 \put(102,12){\line(0,1){3}}
 \put(114,15){\line(0,-1){3}}
 \put(114,9){\circle{6}}
 \put(114,9){\makebox(0,0)[cc]{$M_I$}}
 \put(114,6){\line(0,-1){3}}
 \put(108,9){\makebox(0,0)[cc]{$\cdots$}}
 \put(102,15){\line(-1,0){5}}
 \put(99,15){\makebox(0,0)[cc]{$\prime$}}
 \put(97,3){\line(1,0){5}}
 \put(99,3){\makebox(0,0)[cc]{$\prime$}}
 \put(119,15){\line(-1,0){5}}
 \put(117,15){\makebox(0,0)[cc]{$\prime$}}
 \put(114,3){\line(1,0){5}}
 \put(117,3){\makebox(0,0)[cc]{$\prime$}}
 \put(130,9){\makebox(0,0)[cc]{$\bigg\rangle_I=$}}
 \put(95,15){\makebox(0,0)[cc]{$p$}}
 \put(121,15){\makebox(0,0)[cc]{$q$}}
 \put(121,3){\makebox(0,0)[cc]{$r$}}
 \put(95,3){\makebox(0,0)[cc]{$s$}}
 \end{picture}
 %\input ifig24.pic
\end{center}
\beq\label{e670}
  = -{1\over n_R N_c}\sqrt{M_pM_qM_rM_s}\deltabar(p-s+r-q)
  \big( \delta^{i_p}_{\,\,i_s}\delta^{i_r}_{\,\,i_q}  +
         \gamma^{i_p}_{^5\,\,i_s}\gamma^{i_r}_{^5\,\,i_q} \big)
  \delta^{\alpha_p}_{\,\,\alpha_s}\delta^{\alpha_r}_{\,\,\alpha_q}
  \quad.
\eeq

The result is just proportional to the nonlocal version of the 't Hooft
vertex between color singlet states.

The solutions of the BS equations have a very similar structure:

\bqa\label{e671}
  \makebox(28,11)
  {\begin{picture}(34,26)
   \put(4,3){\line(1,0){7}}
   \put(7,3){\makebox(0,0)[cc]{$\prime$}}
   \put(11,15){\line(-1,0){7}}
   \put(7,15){\makebox(0,0)[cc]{$\prime$}}
   \put(11,0){\framebox(8,18)[cc]{$A$}}
   \put(19,3){\line(1,0){7}}
   \put(23,3){\makebox(0,0)[cc]{$\prime$}}
   \put(26,15){\line(-1,0){7}}
   \put(23,15){\makebox(0,0)[cc]{$\prime$}}
   \put(2,15){\makebox(0,0)[cc]{$p$}}
   \put(28,15){\makebox(0,0)[cc]{$q$}}
   \put(28,3){\makebox(0,0)[cc]{$r$}}
   \put(2,3){\makebox(0,0)[cc]{$s$}}
   %\put(34,9){\makebox(0,0)[cc]{$=$}}
   \end{picture}
  %\input ifig25.pic
   }
  &:=& -{1\over n_R N_c}\sqrt{M_pM_qM_rM_s}\deltabar(p-s+r-q) \\
  && \big( A_0(t)\delta^{i_p}_{\,\,i_s}\delta^{i_r}_{\,\,i_q}  +
        A_5(t)\gamma^{i_p}_{^5\,\,i_s}\gamma^{i_r}_{^5\,\,i_q} \big)
  \delta^{\alpha_p}_{\,\,\alpha_s}\delta^{\alpha_r}_{\,\,\alpha_q}
  \quad.\nonumber
\eqa
$A_0$ and $A_5$ are scalar functions depending only on $t=p-s=q-r$.
The proof is simple: The kernel has the structure (\ref{e671}) with
$A_0=A_5=1$. The product of two vertices yields the sames structure:
\begin{center}
  \begin{picture}(90,18)
  \put(4,3){\line(1,0){7}}
  \put(7,3){\makebox(0,0)[cc]{$\prime$}}
  \put(11,15){\line(-1,0){7}}
  \put(7,15){\makebox(0,0)[cc]{$\prime$}}
  \put(11,0){\framebox(8,18)[cc]{$A$}}
  \put(2,15){\makebox(0,0)[cc]{$p$}}
  \put(46,15){\makebox(0,0)[cc]{$q$}}
  \put(46,3){\makebox(0,0)[cc]{$r$}}
  \put(2,3){\makebox(0,0)[cc]{$s$}}
  \put(52,9){\makebox(0,0)[cc]{$=$}}
  \put(29,0){\framebox(8,18)[cc]{$B$}}
  \put(37,3){\line(1,0){7}}
  \put(41,3){\makebox(0,0)[cc]{$\prime$}}
  \put(44,15){\line(-1,0){7}}
  \put(41,15){\makebox(0,0)[cc]{$\prime$}}
  \thicklines
  \put(29,15){\line(-1,0){10}}
  \put(24,15){\makebox(0,0)[cc]{$<$}}
  \put(19,3){\line(1,0){10}}
  \put(24,3){\makebox(0,0)[cc]{$>$}}
  \thinlines
  \put(60,3){\line(1,0){7}}
  \put(63,3){\makebox(0,0)[cc]{$\prime$}}
  \put(67,15){\line(-1,0){7}}
  \put(63,15){\makebox(0,0)[cc]{$\prime$}}
  \put(67,0){\framebox(8,18)[cc]{$AFB$}}
  \put(75,3){\line(1,0){7}}
  \put(79,3){\makebox(0,0)[cc]{$\prime$}}
  \put(82,15){\line(-1,0){7}}
  \put(79,15){\makebox(0,0)[cc]{$\prime$}}
  \put(58,15){\makebox(0,0)[cc]{$p$}}
  \put(84,15){\makebox(0,0)[cc]{$q$}}
  \put(84,3){\makebox(0,0)[cc]{$r$}}
  \put(58,3){\makebox(0,0)[cc]{$s$}}
  \put(90,9){\makebox(0,0)[cc]{$=$}}
  \end{picture}
 %\input ifig26.pic
\end{center}
\bqa\label{e673}
  &=& -{1\over n_R N_c}\sqrt{M_pM_qM_rM_s}\deltabar(p-s+r-q) \\
  & & \big( A_0F_0B_0(t)\delta^{i_p}_{\,\,i_s}\delta^{i_r}_{\,\,i_q}  +
        A_5F_5B_5(t)\gamma^{i_p}_{^5\,\,i_s}\gamma^{i_r}_{^5\,\,i_q} \big)
  \delta^{\alpha_p}_{\,\,\alpha_s}\delta^{\alpha_r}_{\,\,\alpha_q}
  \quad,\nonumber
%  &&  \Big[^{\nq\;p}_{\nq\;s}(G_0F_0H_0(t)1\!\!1_D\otimes 1\!\!1_D +
%     G_5F_5H_5(t)\gamma_5\otimes\gamma_5)
%       (1\!\!1_C\otimes 1\!\!1_C)\Big]^r_q \nonumber
\eqa
\bqa\label{e674}
  F_0(t) &=& -\int(dpds){1\over n_R}M_pM_s\mbox{tr}_D(S(p)S(s)) \quad,\\
  F_5(t) &=& -\int(dpds){1\over n_R}M_pM_s
    \mbox{tr}_D(S(p)\gamma_5 S(s)\gamma_5) \quad.\nonumber
\eqa

In other words, the vertices of structure (\ref{e671})
build a closed algebra.
The reason for this simple result is that the kernel is 
a simple product function up to the momentum conserving $\delta$.

Using (\ref{e670}), (\ref{e671}) and (\ref{e673})
the BS equations (\ref{e680}) reduce to primitive
algebraic equations for $G_{0/5}(t)$, $H_{0/5}(t)$ and $K_{0/5}(t)$:
\bqan
  G_{0/5}(t) &=&  1 + F_{0/5}(t)G_{0/5}(t) \quad,\\
  H_{0/5}(t) &=& -1 - F_{0/5}(t)K_{0/5}(t) \quad,\\
  K_{0/5}(t) &=& -F_{0/5}(t)H_{0/5}(t)     \quad,
\eqan
with the solution
\beq
  G =  {1\over 1-F}   \quad,\quad
  H = -{1\over 1-F^2} \quad,\quad
  K =  {F\over 1-F^2}
\eeq
where we have suppressed the index $0/5$ and the argument $t$.

%--------------------------------------------------------------
\paragraph{Triplet and Singlet Correlators} \hfill
%--------------------------------------------------------------

Because of isospin symmetry $SU(2)_f$, mesons form triplets and singlets.
Replacing $\bar\psi\psi$ in (\ref{e666})
by the triplet and singlet combinations (borrowing the notation
from the pseudoscalar correlator)
\beq
 \pi^0={1\over\sqrt{2}}(\bar uu-\bar dd) ,\quad
 \pi^+=\bar ud                           ,\quad
 \pi^-=\bar du                           ,\quad
 \eta ={1\over\sqrt{2}}(\bar uu+\bar dd) ,\quad
\eeq
one gets
\beq
  \makebox(150,10){
  \unitlength=0.85mm
  \begin{picture}(171,30)
  \put( 4, 5){\line(1,0){7}}
  \put( 7, 5){\makebox(0,0)[cc]{$\prime$}}
  \put(11,17){\line(-1,0){7}}
  \put( 7,17){\makebox(0,0)[cc]{$\prime$}}
  \put(11, 2){\framebox(8,18)[cc]{$C^t$}}
  \put(19, 5){\line(1,0){7}}
  \put(23, 5){\makebox(0,0)[cc]{$\prime$}}
  \put(26,17){\line(-1,0){7}}
  \put(23,17){\makebox(0,0)[cc]{$\prime$}}
  \put(34,11){\makebox(0,0)[lc]{$= {1 \over 2} \,\Bigg($}}
  \put( 1,11){\makebox(0,0)[cc]{$\pi^0$}}
  \put(29,11){\makebox(0,0)[cc]{$\pi^0$}}
  \put(49, 5){\line(1,0){7}}
  \put(52, 5){\makebox(0,0)[cc]{$\prime$}}
  \put(56,17){\line(-1,0){7}}
  \put(52,17){\makebox(0,0)[cc]{$\prime$}}
  \put(56, 2){\framebox(8,18)[cc]{$K$}}
  \put(64, 5){\line(1,0){7}}
  \put(68, 5){\makebox(0,0)[cc]{$\prime$}}
  \put(71,17){\line(-1,0){7}}
  \put(68,17){\makebox(0,0)[cc]{$\prime$}}
  \put(52,20){\makebox(0,0)[cc]{$u$}}
  \put(68,20){\makebox(0,0)[cc]{$u$}}
  \put(68, 2){\makebox(0,0)[cc]{$u$}}
  \put(52, 2){\makebox(0,0)[cc]{$u$}}
  \put(76,11){\makebox(0,0)[cc]{$-$}}
  \put(81, 5){\line(1,0){7}}
  \put(84, 5){\makebox(0,0)[cc]{$\prime$}}
  \put(88,17){\line(-1,0){7}}
  \put(84,17){\makebox(0,0)[cc]{$\prime$}}
  \put(88, 2){\framebox(8,18)[cc]{$H$}}
  \put(96, 5){\line(1,0){7}}
  \put(100,5){\makebox(0,0)[cc]{$\prime$}}
  \put(103,17){\line(-1,0){7}}
  \put(100,17){\makebox(0,0)[cc]{$\prime$}}
  \put( 84,20){\makebox(0,0)[cc]{$u$}}
  \put(100,20){\makebox(0,0)[cc]{$d$}}
  \put(100, 2){\makebox(0,0)[cc]{$d$}}
  \put( 84, 2){\makebox(0,0)[cc]{$u$}}
  \put(108,11){\makebox(0,0)[cc]{$-$}}
  \put(113, 5){\line(1,0){7}}
  \put(116, 5){\makebox(0,0)[cc]{$\prime$}}
  \put(120,17){\line(-1,0){7}}
  \put(116,17){\makebox(0,0)[cc]{$\prime$}}
  \put(120, 2){\framebox(8,18)[cc]{$H$}}
  \put(128, 5){\line(1,0){7}}
  \put(132, 5){\makebox(0,0)[cc]{$\prime$}}
  \put(135,17){\line(-1,0){7}}
  \put(132,17){\makebox(0,0)[cc]{$\prime$}}
  \put(116,20){\makebox(0,0)[cc]{$d$}}
  \put(132,20){\makebox(0,0)[cc]{$u$}}
  \put(132, 2){\makebox(0,0)[cc]{$u$}}
  \put(116, 2){\makebox(0,0)[cc]{$d$}}
  \put(140,11){\makebox(0,0)[cc]{$+$}}
  \put(145, 5){\line(1,0){7}}
  \put(148, 5){\makebox(0,0)[cc]{$\prime$}}
  \put(152,17){\line(-1,0){7}}
  \put(148,17){\makebox(0,0)[cc]{$\prime$}}
  \put(152, 2){\framebox(8,18)[cc]{$K$}}
  \put(160, 5){\line(1,0){7}}
  \put(164, 5){\makebox(0,0)[cc]{$\prime$}}
  \put(167,17){\line(-1,0){7}}
  \put(164,17){\makebox(0,0)[cc]{$\prime$}}
  \put(148,20){\makebox(0,0)[cc]{$d$}}
  \put(164,20){\makebox(0,0)[cc]{$d$}}
  \put(164, 2){\makebox(0,0)[cc]{$d$}}
  \put(148, 2){\makebox(0,0)[cc]{$d$}}
  \put(170,11){\makebox(0,0)[lc]{$\Bigg)$}}
  \end{picture}
  %\input ifig27.pic
  }
\eeq                    \\[0.3cm]
Therefore $C^t=K-H={1\over 1-F}$.
This coincides with $G={1\over 1-F}$ for the charged triplet
correlator $\langle(\pi^\pm)(\pi^\pm)\rangle$ as it should be.
In the singlet case we get $C^s=K+H=-{1\over 1+F}$.

When adding propagators in (\ref{e671}) to the external legs 
the final result for the connected 4 point function is
\begin{eqnarray}
  \Pi_{\Gamma\Gamma^\prime}^{conn}(p,s,q,r) =
  - {N_c\over n_R} \sqrt{M_pM_qM_rM_s}
  &\big [& \nq C_0(t)\mbox{tr}_D(S(s)\Gamma S(p))
              \mbox{tr}_D(S(q)\Gamma S(r)) +  \nonumber \\
    && \nq C_5(t)\mbox{tr}_D(S(s)\Gamma S(p)\gamma_5)
              \mbox{tr}_D(S(q)\Gamma S(r)\gamma_5)
  \big ]
\end{eqnarray}
\beq
  C^{s/t}_{0/5}(t) = - {1\over F_{0/5}(t)\pm 1} \quad
  {+\;\mbox{for the singlet correlator} \atop
   -\;\mbox{for the triplet correlator} }
\eeq
$F_{0/5}(t)$ are defined in (\ref{e674}).
The correlators of a singlet with a triplet current are zero
as expected.

The following graphs may contribute to the disconnected part:
\beq\label{e683}
 \makebox(120,24){
 \begin{picture}(120,62)
 \thicklines
 \put(44,8){\makebox(0,0)[lc]{$=-N_c^2 \mbox{tr}_D
  ( \Gamma S(p))\mbox{tr}_D ( \Gamma^\prime(q))
  \deltabar(p-s)\deltabar(q-r) $}}
 \put(1,8){\makebox(0,0)[cc]{$\Gamma$}}
 \put(39,8){\makebox(0,0)[cc]{$\Gamma^\prime$}}
 \put(4,14){\makebox(0,0)[cc]{$p$}}
 \put(36,14){\makebox(0,0)[cc]{$q$}}
 \put(36,2){\makebox(0,0)[cc]{$r$}}
 \put(4,2){\makebox(0,0)[cc]{$s$}}
 \put(44,32){\makebox(0,0)[lc]{$=N_c \mbox{tr}_D
    ( \Gamma S(p) \Gamma^\prime(s)) \deltabar(p-q) \deltabar(r-s) $}}
 \put(1,32){\makebox(0,0)[cc]{$\Gamma$}}
 \put(39,32){\makebox(0,0)[cc]{$\Gamma^\prime$}}
 \put(4,38){\makebox(0,0)[cc]{$p$}}
 \put(36,38){\makebox(0,0)[cc]{$q$}}
 \put(36,26){\makebox(0,0)[cc]{$r$}}
 \put(4,26){\makebox(0,0)[cc]{$s$}}
 \put(20,26){\makebox(0,0)[cc]{$>$}}
 \put(7,2){\line(1,0){5}}
 \put(12,2){\line(0,1){12}}
 \put(12,14){\line(-1,0){5}}
 \put(12,8){\makebox(0,0)[cc]{$\wedge$}}
 \put(33,14){\line(-1,0){5}}
 \put(28,14){\line(0,-1){12}}
 \put(28,2){\line(1,0){5}}
 \put(28,8){\makebox(0,0)[cc]{$\vee$}}
 \put(33,38){\line(-1,0){26}}
 \put(20,38){\makebox(0,0)[cc]{$<$}}
 \put(7,26){\line(1,0){26}}
 \thinlines
 \end{picture}
 }
%\input ifig28.pic
\eeq                   \\[1.3cm]
depending on the flavor structure of the correlator. Note
that the second two loop term is of the order $N_c^2$.
However, as discussed
above, in most applications it drops out or yields an uninteresting
constant or only the connected part is considered anyway.

For the triplet and singlet case one gets
\beq
  \deltabar(p-s+q-r)\Pi_{\Gamma\Gamma^\prime}^{disc}(psqr) =
  N_c\mbox{tr}_D(\Gamma S(p)\Gamma^\prime S(s))
  \deltabar(p-q)\deltabar(r-s)
  +
  \left\{\begin{array}{c@{\quad}l}
            0                  & \mbox{for triplet} \\
            2\cdot(\ref{e683}) & \mbox{for singlet}
         \end{array}\right.
\eeq
The 4 point functions obtained in this section will be discussed
in the following chapters.

%--------------------------------------------------------------
%\paragraph{Six Point Functions} \hfill
%--------------------------------------------------------------
\private{Six Point Functions}

%--------------------------------------------------------------

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Correlators of Light Mesons}\label{Ch7}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%--------------------------------------------------------------
%\paragraph{Introduction} \hfill
%--------------------------------------------------------------

%--------------------------------------------------------------
\paragraph{Analytical Expressions} \hfill
%--------------------------------------------------------------

In the last chapter we have calculated various quark 4 point functions
(\ref{e662}). The meson correlators or polarisation functions are just
local versions of these vertices and can be obtained by simply
setting $x=y$ and $z=w$. In momentum space the meson correlators
have the form

\beq
  \Pi_{\Gamma\Gamma^\prime}(t)  =
  \Pi^{disc}(t) + \Pi^{conn}(t) = \makebox(17,2)
  {\unitlength=0.5mm
   \begin{picture}(31,19)
   \put(14,1){\framebox(8,18)[cc]{$K$}}
   \thicklines
   \put(22,10){\oval(14,12)[r]}
   \put(14,10){\oval(14,12)[l]}
   \put(11,4){\makebox(0,0)[cc]{$\scriptstyle >$}}
   \put(11,15.14){\makebox(0,0)[cc]{$\scriptstyle <$}}
   \put(25,16){\makebox(0,0)[cc]{$\scriptstyle <$}}
   \put(25,4.86){\makebox(0,0)[cc]{$\scriptstyle >$}}
   \put(30,10){\makebox(0,0)[lc]{$\sim\!\!\!\!\prime$}}
   \put(6,10){\makebox(0,0)[rc]{$\sim\!\!\!\!\prime$}}
   \put(5,14.15){\makebox(0,0)[cc]{$\scriptstyle \Gamma$}}
   \put(31,14.15){\makebox(0,0)[cc]{$\scriptstyle \Gamma^\prime$}}
   \thinlines
   \end{picture}
   %\input ifig17.pic
   } =
\eeq
\beqn
  = \int\!(dpds)\int\!(dqdr)\Pi(p,s,q,r) =
  - \int\!dx\,e^{itx}
    \langle 0|{\cal T}j_\Gamma(x)j_{\Gamma^\prime}(0)|0\rangle \quad,
\eeqn
\beqn
  j_\Gamma(x)^{s/t} = {1\over\sqrt{2}}
    (\bar u\Gamma u(x)-\bar d\Gamma d(x)) \quad,\quad
  j_{\Gamma^\prime}(0)^{s/t} = {1\over\sqrt{2}}
    (\bar u\Gamma^\prime u(0)-\bar d\Gamma^\prime d(0)) \quad,
\eeqn
\beqn
  \int\!(dpds) = \int\!\dbar p\dbar s\,\deltabar(p-s-t) \quad,\quad
  \int\!(dqdr) = \int\!\dbar q\dbar r\,\deltabar(q-r-t) \quad,\quad
\eeqn
\beqn
  t=p-s=q-r \quad.
\eeqn

From the explicit expressions of the 4 point functions obtained in the
last chapter
one can get, up to integration, analytical expressions for
the meson correlators. The following list is a complete 
summary of all formulas needed to evaluate the meson correlators:
\bqa
  \Pi_{\Gamma\Gamma^\prime}^{disc}(t) &=&
  N_c\int\!(dpds)\,\mbox{tr}_D(\Gamma S(p)\Gamma^\prime S(s))
  \quad,\nonumber\\
  \Pi_{\Gamma\Gamma^\prime}^{conn}(t) &=&
  -N_c(C_0(t)\Gamma_\Gamma^0(t)\Gamma_{\Gamma^\prime}^0(t) +
       C_5(t)\Gamma_\Gamma^5(t)\Gamma_{\Gamma^\prime}^5(t))
  \quad,\label{e705}\\
  C_{0/5}(t) &=& - {1\over F_{0/5}(t)\pm 1} \quad
    {+\; \mbox{for singlet correlator} \atop 
     -\; \mbox{for triplet correlator} }
  \quad,\nonumber\\
  \Gamma_\Gamma^0(t) &=& {1\over\sqrt{n_R}}
    \int\!(dpds)\,\sqrt{M_pM_s}\mbox{tr}_D(S(p)\Gamma S(s))
  \quad,\nonumber\\
  \Gamma_\Gamma^5(t) &=& {1\over\sqrt{n_R}}
    \int\!(dpds)\,\sqrt{M_pM_s}\mbox{tr}_D(S(p)\Gamma S(s)\gamma_5)
  \quad,\nonumber\\
  F_0(t) &=& {-1\over n_R}
    \int\!(dpds)\,M_pM_s\mbox{tr}_D(S(p)S(s))
  \quad,\nonumber\\
  F_5(t) &=& {-1\over n_R}
    \int\!(dpds)\,M_pM_s\mbox{tr}_D(S(p)\gamma_5 S(s)\gamma_5)
  \quad,\nonumber\\
  S(p) &=& {1\over p\!\!/+i(m+M_p)} \quad,\quad
    M_p ={n_R\over 2\mu}p^2\varphi^{\prime 2}(p) \quad,
  \quad,\nonumber\\
  p\varphi^\prime(p) &=& 2\pi\rho z {\partial\over\partial z}
    [I_0(z)K_0(z)-I_1(z)K_1(z)]_{z=p\rho/2}
  \quad,\nonumber\\
  \mu &=& m + \int\dbar^4\!p\,
     {2\varphi^{\prime 2}(p)M_p \over p^2+(m+M_p)^2}
     (p^2+m(m+M_p))
  \quad,\label{e710}\\
  n_RN_c &=& (200\mbox{MeV})^4 \quad,\quad
    \rho = (600\mbox{MeV})^{-1}
  \quad.\nonumber
\eqa

%--------------------------------------------------------------
\paragraph{Analytical Results} \hfill
%--------------------------------------------------------------

Performing the Dirac traces leads to the following expressions:
\bqa
  F_{0/5}(t) &=& -{4\over n_R}\int\!(dpds)
    {M_pM_s(\pm (ps)-\tilde M_p\tilde M_s) \over
     (p^2+\tilde M_p^2)(s^2+\tilde M_s^2)}
  \quad,\nonumber\\
  \Gamma_{1/5}^{0/5}(t) &=& {4\over\sqrt{n_R}}\int\!(dpds)
    {\sqrt{M_pM_s}(\pm (ps)-\tilde M_p\tilde M_s) \over
     (p^2+\tilde M_p^2)(s^2+\tilde M_s^2)}
  \quad,\\
  \Gamma_{\mu 5}^5(t) &=& {4i\over\sqrt{n_R}}\int\!(dpds)
    {\sqrt{M_pM_s}(\tilde M_p s_\mu - \tilde M_s p_\mu) \over
     (p^2+\tilde M_p^2)(s^2+\tilde M_s^2)}
  \quad,\nonumber\\
  \tilde M_p &=& m + M_p
  \nonumber
\eqa
All other vertices $\Gamma$ are zero.
Consider the one instanton vertex (\ref{e670}) (the kernel).
It contributes
only to the scalar and pseudoscalar correlator. From this observation
one may have predicted that the connected part of all other channels is
small because a contribution has to be a multi-instanton effect.
Indeed, they are all zero as seen above except for the axial correlator.
Due to an extra factor $M\sim\sqrt{n_R}$ in the numerator
of $\Gamma_{\mu5}^5$ the connected part of the axial correlator is
suppressed by $O(n_R)$ therefore it is small as expected and will
be neglected in the following.
\private{ In addition it is purely longitudinal. }

Furthermore we will restrict
ourself to the {\it chiral limit}, taking $m=0$.
Using the selfconsistency equation (\ref{e710}) one can see that
\beq
  F_5(t=0)=1
\eeq
is leading to a pole at $t=0$ in the pseudoscalar triplet correlator
due to the $F_5(t)-1$ denominator in (\ref{e705}).
This is the massless Goldstone pion one expects in the chiral limit.
A more extensive discussion can be found in \cite{Dya}.
On the other side in the singlet correlator the minus sign is 
replaced by a positive sign and there is no Goldstone boson in this case.
Thus, the (two flavor) $\eta^\prime$ meson is massive ! Unfortunately we
cannot make any reliable prediction of the $\eta^\prime$ mass because
the kernal is very repulsive in this channel and no boundstate is
formed. There have to be other attractive forces, e.g.\ confinement
forces, to built an $\eta^\prime$ boundstate. Similar things
happen in the scalar triplet channel (compare figure \ref{fig73}
and \ref{fig76}). But the most important thing is
that there is no massless pseudoscalar singlet meson 
which is an important step towards discussing the $U(1)_A$-problem.

It is interesting to see that in leading order in the instanton
density $F_0(t)=-F_5(t)$, which leads to a massless pole in
the scalar singlet correlator. Numerically the $\sigma$-meson
indeed turns out to be very light. The experimental
situation is rather unclear. %\cite{...}

%--------------------------------------------------------------
\paragraph{Spectral Representation} \hfill
%--------------------------------------------------------------

To extract phenomenological information from the meson correlators
we make use of the spectral representation
\beq\label{e725}
  \Pi(p)= \int\!dx\,e^{ipx}
    \langle 0|{\cal T}j_\Gamma(x)j_{\Gamma^\prime}(0)|0\rangle =
    \int_0^\infty\!d\sigma^2\,D(\sigma,x)\rho(\sigma^2)
\eeq
where
\beq
  \rho(p^2) = (2\pi)^3\sum_n\delta(p-q_n)
    \langle 0|j_\Gamma(0)|n\rangle
    \langle n|j_\Gamma^\prime(0)|0\rangle
\eeq
is the spectral density and
\beq
  D(m,x) = \int\!\dbar^4 p{e^{-ipx}\over p^2+m^2}
         = {1\over 4\pi^2 x^2}(mx)K_1(mx)
\eeq
is the free propagator of mass $m$ in coordinate representation.
We have chosen the coordinate representation of $\Pi$ to be
able to compare the plots directly with lattice calculations \private{[??]}
and with numerical studies of the instanton liquid \cite{ShV}.

The spectrum consists of mesonic resonances and the continuum
contributions. If one is only interested in the properties
of the first resonance one might approximate the rest of the
spectrum by the perturbatively calculated continuum.

One might think that the disconnected part only contributes
to the continuum and the connected part will yield the boundstates.
But this is not the case. On one hand, Bethe Salpeter equations
have bound as well as continuum solutions. On the other hand
consider a theory with weak attraction between particles of mass m.
It is clear that there is only a cut above $2m$ and no
boundstate pole in the free loop. However in the exact polarization
function only a small portion of the continuum will be used
to form a pole just below the threshold because the attraction
is only weak. The Euclidian correlator will hardly be changed.
Therefore, assuming weak attraction, we can already estimate
the boundstate mass from the disconnected part. Of course in this
example we need not calculate anything because we know
that the bounstate mass is approximately $2m$ with errors of the
order of the strength of the interaction.

Assuming that all other forces neglected in QCD so far,
especially perturbative corrections, are small and attractive
in the vector and axialvector channel, we can obtain boundstate
masses although in these channels up to our approximation
there is no connected part. But things are less trivial than in
the example above because the quarks do not posses a definite mass
and we have to inspect the correlator to extract the meson masses.

Let us start with the scalar and pseudoscalar correlator.
The lowest resonance of mass $m_*$ is coupled to the current
with strength
\beq
  \lambda_*=\langle 0|j_{1/5}(0)|p\rangle \quad.
\eeq
The rest of the spectrum is approximated by the continuum
starting at the threshold $E_*$.
* means $\pi$, $\eta$, $\delta$ or $\sigma$ (see table \ref{tab3}).
{}
\begin{table}
  \begin{center}\begin{tabular}{|lcc|cc|} \hline
  Correlator & & $\Gamma=\Gamma^\prime$ & I=1 & I=0 \\
  \hline
  Pseudoscalar & $\Pi_5=\langle j_5j_5\rangle$
    & $i\gamma_5$         & $\pi$    & $\eta^\prime$              \\
  Scalar       & $\Pi_1=\langle j_1j_1\rangle$
    & $1\!\!1_D$          & $\delta$ & $\sigma$                   \\
  Vector       & $\Pi_ {\mu\mu}=\langle j_\mu j_\mu\rangle$
    & $\gamma_\mu$        & $\rho$   & $\omega$                   \\
  Axialvector  & $\Pi_{\mu\mu}^5=\langle j_\mu^5j_\mu^5\rangle$
    & $\gamma_\mu\gamma_5$ & $a_1$   & $f_1$                \\\hline
\end{tabular}\end{center}
\vspace{-3ex}
\caption{\label{tab3}
  Mesonic correlators
}\end{table}
{}
$E_*$ is typically of the order
$1.5$ GeV and therefore the continuum can be calculated perturbatively.
The spectrum thus has the form
\beq
  \rho_{1/5}(s) = \lambda^2_*\delta(s-m_*^2)+
    {3s\over 8\pi^2}\Theta(s-E_*^2) \quad.
\eeq
Inserting $\rho$ into (\ref{e725}) one gets
\beq
  \Pi^{fit}_{1/5}(x) = \lambda_*^2 D(m_*,x) + E(E_*,x) \quad,
\eeq
\beq
  E(E_*,x)= {3\over\pi^4x^6} {(E_*x)^3\over 16} (2K_3(E_*x)+(E_*x)K_2(E_*x))
  \stackrel{x\to 0}\longrightarrow {3\over\pi^4x^6} \quad.
\eeq
In the next section $m_*$, $\lambda_*$ and $E_*$ are obtained by
fitting the
phenomenological ansatz $\Pi^{fit}(x)$ to the theoretical curve
$\Pi^{sum}(x)$ in the Euclidian region where the theoretical
calculation is reliable.

Consider now the vector and axial vector correlator. The vector
current is conserved, thus the correlator is transverse and only
the vector meson can contribute. In the chiral limit the same
holds true for the axial current. In the singlet channel one
has to be careful because there are two currents.
A conserved one and a gauge invariant one which contains an anomaly.
Up to now we have only calculated the correlator of the conserved
current. Nevertheless to leading order in the instanton density
the two correlators coincide and should be both conserved.
\private{
  Current conservation and Ward identities will be discussed in the
  next chapter.}

For conserved vector and axial currents the spectral function is transverse:
\beq
  \rho_{\mu\nu}^{(5)}(p^2) =
    (-\delta_{\mu\nu}+{p_\mu p_\nu \over p^2}) \rho_T^{(5)}(p^2) \quad.
\eeq
The coupling of the vector and axial meson to the current is given by
\beq
  i\lambda_*\epsilon_\mu = \langle 0|j_\mu^{(5)}(0)|p\rangle
\eeq
where $\epsilon_\mu$ is the meson polarization.
The spectral and polarization functions have the form
\bqa
  -\rho_{\mu\mu}^{(5)}(s) &=& 3\lambda^2_*\delta(s-m_*^2)+
    {3s\over 4\pi^2}\Theta(s-E_*^2) \quad,\\
  -\Pi^{fit(5)}_{\mu\mu}(x) &=& 3\lambda_*^2 D(m_*,x) + 2E(x)
  \quad.\nonumber
\eqa
Here * means $\rho$, $\omega$, $a_1$ or $f_1$ (see table \ref{tab3}).

%--------------------------------------------------------------
\paragraph{Plot \& Fit of Meson Correlators} \hfill
%--------------------------------------------------------------

The meson correlators are shown in figure \ref{fig71} - \ref{fig76}.
The numerical evaluation of the integrals is discussed in appendix
\ref{Appb}.
The correlators are normalized to the free correlator
\beq
  \Pi^0_{1/5}(x) = {3\over\pi^4 x^6} \quad,\quad
  \Pi^{0(5)}_{\mu\mu} = - {6\over\pi^4 x^6} \quad.
\eeq
The diagrams therefore show the deviation from the perturbative
behaviour. The meson parameters obtained by fitting the parameter
ansatz to the theoretical curve are summarazied in table \ref{tab4}.
Shuryak \& Verbarshot \cite{ShV} have obtained the same parameters from
a numerical
investigation of the instanton liquid model. Their values are also
shown in table \ref{tab4}.
The parameters of the vector channel coincide extremely well with ours.
For the $\pi$ and $\sigma$ meson there is a large discrepancy in the masses
but this is not surprising:
We are working in the chiral limit thus the pion mass has to be zero.
A similar argument holds for the $\sigma$ meson as discussed above.
The couplings fit very well. The discrepancy in the axial channels
can have various origins which are under investigation.
Alternatively one may directly compare the graphs. They coincide
very well even in cases where a spectral fit does not work very well
like in the $\delta$ and $\eta^\prime$ channel.
The conclusion is that the terms neglected in our analytical treatment,
but included in the numerical study \cite{ShV}, are small and usually give
an correction less than 10\%. These are contributions from
nonzero modes and higher order corrections in $1/N_c$.
This is again an example for the surprisingly high accuracy of
the $1/N_c$ expansion. In the case of strange quarks the nonzero
mode contributions will become more important.

Finally, one should compare the numbers with
experiment. As far as known, these numbers are also listed in table
\ref{tab4}.
A general discussion of the meson correlators and comparision with
experimental results can be found in \cite{ShV}.
\begin{table}
  \begin{center}\begin{tabular}{|cl|lllc|} \hline
  Meson & $I^G(J^{PC})$ & $m_*$[MeV] & $\sqrt{\lambda_*}$[MeV]
                                                    & $E_*$[MeV] & source
  \\\hline\hline
        &         &  0         & 508$\pm$1  & 1276$\pm$33 & $1/N_c$      \\
  $\pi$ & $1^-(0^{-+})$ & 142$\pm$14 & 510$\pm$20 & 1360$\pm$100& simulation
\\
        &         & 138        & 480        & ---         & experiment\\
  \hline
        &         & $\neq$ 0   & ?          & ?           & $1/N_c$      \\
  $\eta^\prime$
        & $0^+(0^{-+})$ & $\neq$ 0   & ?          & ?           & simulation
\\
        &         & 960        & ?          & ---         & experiment\\
  \hline
        &         & $\neq 0$   & ?          & ?           & $1/N_c$      \\
$\delta$& $1^-(0^{++})$ & $\neq 0$   & ?          & ?           & simulation
\\
        &         & 970        & ?          & ---         & experiment\\
  \hline
        &         & 433$\pm$3  & 506$\pm$3  & 1446$\pm$20 & $1/N_c$      \\
$\sigma$& $0^+(0^{++})$ & 543        & 500        & 1160        & simulation
\\
        &         & ?          & ?          & ---         & experiment\\
  \hline
        &         & 930$\pm$5  & 408$\pm$4  & 1455$\pm$33 & $1/N_c$      \\
  $\rho$& $1^+(1^{--})$ & 950$\pm$100& 390$\pm$20 & 1500$\pm$100& simulation
\\
        &         & 780        & 409$\pm$5  & ---         & experiment\\
  \hline
        &         & 930$\pm$5  & 408$\pm$4  & 1455$\pm$33 & $1/N_c$      \\
$\omega$& $0^-(1^{--})$ & ?          & ?          & ?           & simulation
\\
        &         & 780        & 390$\pm$5  & ---         & experiment\\
  \hline
        &         &1350$\pm$200& 370$\pm$30 & 1050$\pm$80 & $1/N_c$ \\
  $a_1$ & $1^-(1^{++})$ & 1132$\pm$50& 305$\pm$20 & 1100$\pm$50 & simulation
\\
        &         & 1260       & 400        & ---         & experiment\\
  \hline
        &         &1350$\pm$200& 370$\pm$30 & 1050$\pm$80 & $1/N_c$ \\
  $f_1$ & $0^+(1^{++})$ & 1210$\pm$50& 293$\pm$20 & 1200$\pm$50 & simulation
\\
        &         & 1285       & ?          & ---         & experiment\\
  \hline
\end{tabular}\end{center}
\vspace{-3ex}
\caption{\label{tab4}
  Meson mass $m_*$, coupling constant $\lambda_*$ and continuum threshold
  $E_*$ obtained within the instanton liquid model in this work
  ($1/N_c$ expansion), from numerical simulation and from
  experiment.
}\end{table}

%--------------------------------------------------------------

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Zeromode Formulas}\label{Appa}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The covariant derivative $D\!\!\!\!/$ in the background of an
instanton has one zeromode
\beq
  (i\partial\!\!\!/-A\!\!\!/_I)\psi_I =0
\eeq
The solution for an instanton of topological charge $Q_I=\pm 1$
in singular gauge in coordinate and momentum space is given in
terms of modified Bessel functions $I_{0/1}$ and $K_{0/1}$
\bqa
  \psi_I(x+z_I) &=& \sqrt{2}\varphi(x)x\!\!\!/\chi_I^\pm \quad,\quad
  \varphi(x) = {\rho\over\pi|x|(x^2+\rho^2)^{3/2}}
 \nonumber\\
  \psi_I(p) &=& {\sqrt{2}\varphi^\prime(p)\over|p|}e^{ipz_I}\chi_I^\pm
  \quad,\quad \varphi(p) = \int\!d^4\!x\,e^{ipx}\varphi(x)
 \\
  \varphi^\prime(p) &=& \pi\rho^2 {d\over dz}
  [I_0(z)K_0(z)-I_1(z)K_1(z)]_{z=|p|\rho/2} = \left\{
  \begin{array}{c@{\quad:\quad}c}
    -{2\pi\rho\over|p|}     & p\rho\ll 1 \\
    -{12\pi\over p^4\rho^2} & p\rho\gg 1
  \end{array} \right.
 \nonumber
\eqa
\beqn
  \rho^2\int\!d^4\!x\,\varphi^2(x) =
  \int\!d^4\!x\,\varphi^2(x)x^2 =
  \int\!\dbar^4\!p\,\varphi^{\prime 2}(p) = {1\over 2}
\eeqn
\beqn
  \chi_I^\pm \sim \left. \gamma_\mu{1\mp\gamma_5\over 2}\tau_\mu^\mp
  \right|_{\mbox{\it some color \& Dirac column}} \quad,\quad
  \tau_\mu^\mp = (\underline{\tau},\pm i)
\eeqn
As usual only the projectors can be given in a
covariant way\footnote{In Euclidian space the bar operation
is the same as the hermitian conjugate: no $\gamma_0$ is introduced,
thus $\bar\chi=\chi^\dagger$.}:
\bqa\label{ea10}
  \chi_I^\pm\bar\chi_J^\pm &=& {1\over 16}(\gamma_\mu\gamma_\nu
  {1\pm\gamma_5\over 2})(U_I\tau_\mu^\mp\tau_\nu^\pm U_J^\dagger) \\
  \chi_I^\pm\bar\chi_J^\mp &=& \mp{i\over 4}(\gamma_\mu
  {1\mp\gamma_5\over 2})(U_I\tau_\mu^\mp U_J^\dagger)
\eqa
$U_{I/J}\in SU(N_c)$ is the orientation matrix of the instanton $I/J$.
Although multi-instanton effects are studied in this work formulas 
like (\ref{ea10}) concerning the overlap of different instantons are
not needed. (\ref{ea10}) is only needed for $I=J$ in the following
special cases:
\beq
  N_C\langle\chi_I^\pm\bar\chi_I^\pm\rangle_I =
  \mbox{tr}_D\chi_I^\pm\bar\chi_I^\pm =
  {1\over 2}({1\pm\gamma_5\over 2}) \nonumber
\eeq
\beq
  \bar\chi\chi = \mbox{tr}_{CD}\chi\bar\chi = 1
\eeq
Taking space, color and Dirac part together we get
\beq
  \psi_I^\dagger(p)\psi_I(p) = 2\varphi^{\prime 2}(p) \quad\ff\quad
  \int\!\dbar^4\!p\,\psi_I^\dagger(p)\psi_I(p) = 1
\eeq

%--------------------------------------------------------------

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Numerical Evaluation of Integrals}\label{Appb}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The integral expressions for the meson correlators have been
evaluated numerically. Two types of operations have to be performed:
{}
\begin{enumerate}\parskip=0ex\parsep=0ex\itemsep=0ex
  \item{Convolution of Lorentz covariant functions
    ($F_{0/5},\Gamma_\Gamma$)}
  \item{Fourier transformation (FT) of
    the correlators to coordinate space}
\end{enumerate}

%--------------------------------------------------------------
\paragraph{Fourier Transformation} \hfill
%--------------------------------------------------------------

Let us first consider the FT generalized to $d$ dimensions:
\beq
  \hat f_{\mu_1\ldots\mu_n}(x) =
  F_d\{f_{\mu_1\ldots\mu_n}\}(x) =
  \int\!\dbar^d\!p\,e^{-ipx}f_{\mu_1\ldots\mu_n}(p)
\eeq
For a scalar spherically symmetric function $f=f(|p|)$ the FT
reduces to a one dimensional integral
\beq\label{eb10}
  F_d\{f(|p|)\}(x) = \int_0^\infty \left({m\over 2\pi|x|}\right)^{d/2}
  f(m) J_{d/2-1}(m|x|)|x|\,dm
\eeq
where $J_\nu$ are Bessel functions.

If $x$ is not too large and $f$ decays rapidly, the integration
can be performed with gaussian (or other) integration methods.
If the decay is too slow one has to subtract the asymptotic
part from $f$ thus improving convergency. The FT of the asymptotic
part can be performed analytically and has to be added to the numerical
FT of the reduced function.

The FT of a general Lorentz covariant function can also be reduced to
(\ref{eb10}) with (formally) an increased dimension $d$:
\bqa
  F_d\{p_\mu f(|p|)\}(x) &=& 2\pi ix_\mu F_{d+2}\{f\}(x)
  \\
  F_d\{p_\mu p_\nu f(|p|)\}(x) &=&
    {1\over d-1}\Big[ (\delta_{\mu\nu}-{x_\mu x_\nu\over x^2})
      F_d\{p^2f(|p|)\}(x) -
  \nonumber\\
  & & (\delta_{\mu\nu}-d{x_\mu x_\nu\over x^2})
     (4\pi F_{d+2}\{f\}(x)-4\pi^2x^2F_{d+4}\{f\}(x)) \Big]
  \nonumber\\
  &\ldots&
  \nonumber
\eqa

%--------------------------------------------------------------
\paragraph{Convolution} \hfill
%--------------------------------------------------------------

Now we treat the convolution integrals
\beq
  f\!*\!g(p) = \int\!\dbar^d\!q\, f(q)\!\cdot\!g(p-q)
\eeq
This type of integral can be reduced to the FT discussed above:
\beq
  f\!*\!g(p) = F_d^{-1}\{F_d\{f\}\!\cdot\!F_d\{g\}\}
\eeq
This is a quick and easy method for evaluating convolution integrals.
For the disconnected part of the correlators it has the advantage
that in coordinate representation $F_d^{-1}$ can be dropped.
The disadvantage of this formula is that the FTs involve oscillating
integrals which are numerically problematic.
If the back-transformation $F_d^{-1}$ is needed as in the case of
$F_{0/5}$ and $\Gamma_\Gamma$ it is better to perform the convolution
directly.
Similar to the FT the convolution can be reduced to the scalar case. The
convolution
of two scalar functions can further be reduced to a two dimensional
integral:
\beq\label{eb20}
  f*g(p) = {(d/2-1)!\over 2\pi^{d/2+1}(d-2)!}
  \int_0^\infty\!dr\int_0^\pi\!d\theta\,
  f(r)g(\sqrt{p^2-2|p|r\cos\theta+r^2})(r\sin\theta)^{d-2}r
\eeq
It is again evaluated with gaussian integration methods. The second
advantage is that there are no problems with slowly decaying functions.
Sometimes there are large cancelations between different terms.
In this case it is essential to use nonadaptive integration methods
because they will not result in a loss of accuracy.

The explicit reduction of the various correlators to the basic
forms (\ref{eb10}) and (\ref{eb20}) is more or less
trivial. The selfconsistency equation has been solved by iteration.
The results are plotted in figure \ref{fig71} - \ref{fig76}.

%--------------------------------------------------------------

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Figures}\label{Chf}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[b]
   \vspace{80mm}
   \centerline{ifig13.pic}
   \vspace{80mm}
   \caption{$S$ is minimal at the river,
                   small in the valley and
                   large in the mountains,
            Therefore $Z=\int\!dx_1dx_2\,e^{-S[x_1,x_2]}$
              is dominated by the valley.
           }\label{fig31}
\end{figure}

\begin{figure}
   \epsfbox[85 0 480 550]{mass.out}
   \caption{Constituent quark mass M(p).}\label{fig51}
\end{figure}

\begin{figure}
  \epsfbox[85 0 480 550]{p55t.out}
  \caption{Pseudoscalar triplet correlator normalized to the
     free massless quark correlator. The pion coupling constant
     $\lambda_\pi$ and the continuum threshold $E_\pi$ are {\it fit}ted
     in order to match the spectral ansatz with the theoretical
     {\it sum} of the {\it free} and the {\it conn}ected part.
           }\label{fig71}
\end{figure}

\begin{figure}\label{fig72}
  \epsfbox[93 0 480 550]{p55s.out}
  \caption{Pseudoscalar singlet correlator normalized to the
     free massless quark correlator. There is a strong repulsion
     in this channel and no boundstate is formed.
     The theoretical curve is compared to a curve obtained from a
     pure continuum spectrum above $E_{\eta^\prime}$.
           }
\end{figure}

\begin{figure}\label{fig73}
  \epsfbox[93 0 480 550]{p11t.out}
  \caption{Scalar triplet correlator normalized to the
     free massless quark correlator. There is a strong repulsion
     in this channel and no boundstate is formed.
     The theoretical curve is compared to a curve obtained from a
     pure continuum spectrum above $E_\delta$.
           }
\end{figure}

\begin{figure}\label{fig74}
  \epsfbox[93 0 480 550]{p11s.out}
  \caption{Scalar singlet correlator normalized to the
     free massless quark correlator.
     The $\sigma$ mass $m_\sigma$ and coupling $\lambda_\sigma$
     and the threshold $E_\sigma$ are obtained from a
     spectral fit.
           }
\end{figure}

\begin{figure}\label{fig75}
  \epsfbox[93 0 480 550]{p66s.out}
  \caption{Axial vector correlator normalized to the
     free massless quark correlator. The triplet and singlet
     correlator are equal because the connected part has been neglegted.
     The $a_1$ and $f_1$ mass, coupling and threshold
     are obtained from a spectral fit.
           }
\end{figure}

\begin{figure}
  \epsfbox[93 0 480 550]{p22t.out}
   \caption{Vector correlator normalized to the
     free massless quark correlator. The triplet and singlet
     correlator are equal because the connected part is zero.
     The $\rho$ and $\omega$ mass, coupling and threshold
     are obtained from a spectral fit.
           }\label{fig76}
\end{figure}
\clearpage
%--------------------------------------------------------------

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                  B i b l i o g r a p h y                    %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\addcontentsline{toc}{section}{\arabic{section}$\;\;$ References}
\parskip=0ex plus 1ex minus 1ex
\begin{thebibliography}{99}\parskip=0ex\parsep=0ex\itemsep=0ex

\bibitem{xxx} \mbox{}

%--------------------------------------------------------------
\subparagraph{General References} \hfill
%--------------------------------------------------------------
\bibitem{Gli}
  {\bf J. Glimm, A. Jaffe}:
  {\it Quantum physics};
  {\rm New York, Springer (1981)}
\bibitem{LSZ}
  {\bf H. Lehman et al.}:
  {\it "LSZ theory"};
  {\rm Nuovo Cim. 1 (1955) 205}
\bibitem{War}
  {\bf J.C. Ward}:
  {\it }
  {\rm Phys.Rev. 78 (1950) 182}
\bibitem{CSZ}
  {\bf C.G. Callen}:
  {\it Broken scale invariance in scalar field theory};
  {\rm Phys.Rev. D2 (1970) 1541}
\bibitem{CSZ2}
  {\bf K. Symanzik}:
  {\it Small distance behaviour in field theory and power counting};
  {\rm Comm.Math.Phys. 18 (1970) 227}
\bibitem{CSZ3}
  {\bf W. Zimmermann}:
  {\it Composite operators, normal products and the short
       distance expansion in the perturbation theory of renormalizable
       interactions};
  {\rm  Ann.of Phys. 77 (1973) 536,570}
\bibitem{Fey}
  {\bf R.P. Feynman}:
  {\it Quantum mechanics and path integrals};
  {\rm Rev.Mod.Phys. 20 (1948) 367} 
\bibitem{Cre}
  {\bf M. Creutz}:
  {\it Quarks, gluons and lattices};
  {\rm Cambridge Univ. Press (1983)}

%--------------------------------------------------------------
\subparagraph{Instantons: Introduction} \hfill
%--------------------------------------------------------------

\bibitem{Raj}
  {\bf R. Rajaraman}:
  {\it Solitons and instantons};
  {\rm North-Holland, 1982}
\bibitem{SVZ}
  {\bf M.A. Shifman et al.}:
  {\it ABC of instantons};
  {\rm Fortschr.Phys. 32,11 (1984) 585}
\bibitem{CDG}
  {\bf G. Callen, R. Dashen, D. Gross}:
  {\it Toward a theory of strong interactions};
  {\rm Phys.Rev. D17 (1978) 2717}
  {\it A theory of hadronic structure};
  {\rm Phys.Rev. D19 (1979) 1826}

%--------------------------------------------------------------
\subparagraph{Instantons: General Formulas} \hfill
%--------------------------------------------------------------

\bibitem{Act}
  {\bf A. Actor}:
  {\it Classical solutions of SU(2) Yang-Mills theories};
  {\rm Rev.Mod.Phys. 51 (1979) 461}
\bibitem{tHo}
  {\bf G. 't Hooft}:
  {\it Computation of the quantum effects due to
       a four-dimensional pseudoparticle};
  {\rm Phys.Rev. D14 (1976) 3432}
\bibitem{Ber}
  {\bf C. Bernard}:
  {\it Gauge zero modes, instanton determinants,
       and QCD calculations};
  {\rm Phys.Rev. D19 (1979) 3013}
\bibitem{Bro}
  {\bf L.S. Brown et al.}:
  {\it Propagation functions in pseudoparticle fields};
  {\rm Phys.Rev. D17 (1978) 1583};
  {\it Massive propagators in instanton fields};
  {\rm Phys.Rev. D18 (1978) 2180}
\bibitem{Car}
  {\bf R.D. Carlitz}:
  {\it Bound states from instantons};
  {\rm Phys.Rev. D17 (1978) 3225}

%--------------------------------------------------------------
\subparagraph{Instantons: The Liquid Model} \hfill
%--------------------------------------------------------------

\bibitem{Ilg}
  {\bf E. Ilgenfritz et al.}:
  {\it Hard-core Model};
  {\rm Nucl.Phys. B184 (1981) 443}
\bibitem{Dya}
  {\bf D.I. Dyakonov, V.Yu. Petrov}:
  {\it Instanton-based vacuum from the feynman variational principle};
  {\rm Nucl.Phys. B245 (1984) 259};
  {\it A theory of light quarks in the instanton vacuum};
  {\rm B272 (1985B) 457}

%--------------------------------------------------------------
\subparagraph{Instantons: Numerical Studies} \hfill
%--------------------------------------------------------------

\bibitem{Shu}
  {\bf E.V. Shuryak}:
  {\it The role of instantons in QCD (A1-A4)};
  {\rm Nucl.Phys B203 (1982) 93,116,140; B214 (1982) 237}
  {\it Toward the quantitative theory of the instanton liquid (B1-B4)};
  {\rm Nucl.Phys B302 (1988) 559,574,599,621}
  {\it Instantons in QCD (C1-C4)};
  {\rm Nucl.Phys B319 (1989) 521,541; B328 (1989) 85,102}
\bibitem{ShV}
 {\bf E.V. Shuryak, J.J.M. Verbaarschot}:
 {\it Quark propagation in the random instanton vacuum};
 {\rm Nucl.Phys. B410 (1993) 37}
 {\it Mesonic correlation functions in the random instanton vacuum};
 {\rm Nucl.Phys. B410 (1993) 55}

%--------------------------------------------------------------
\subparagraph{Gluon Mass} \hfill
%--------------------------------------------------------------

\bibitem{Cor}
  {\bf J.M. Cornwall}:
  {\rm Phys.Rev. D26 (1982) 1453}
\bibitem{Hal}
  {\bf F. Halzen}:
  {\it Relating the QCD pomeron to an effective gluon mass};
  {\rm preprint MAD/PH/702 (1992) }
\bibitem{Hut1}
 {\bf M. Hutter}:
 {\it Gluon mass from instantons};
 {\rm preprint LMU-Muenchen HEP/18 (1993)}

%--------------------------------------------------------------
%\subparagraph{Proton Spin} \hfill
%--------------------------------------------------------------
%
%\bibitem{Baa}
%  {\bf S.D. Baas, A.W. Thomas}:
%  {\it The EMC spin effect};
%  {\rm preprint SMC Report SMC/92/25 (1992)}
%\bibitem{Ell}
%  {\bf J. Ellis, M. Karliner}:
%  {\it }
%  {\rm Phys.Lett. B213 (1988) 73}
%\bibitem{Ash}
%  {\bf J. Ashman et al.}:
%  {\it An investigation of the spin structure of the proton in deep
%       inelastic scattering of polarised muons on polarised protons};
%  {\rm Nucl.Phys. B328 (1989) 1}
%\bibitem{For}:
%  {\bf S. Forte, E.V. Shuryak}:
%  {\it Instanton-induced supression of the
%       singlet axial charge of the proton};
%  {\rm Nucl.Phys. B357 (1991) 153}
%\bibitem{Jaf}
%  {\bf R.L. Jaffe}:
%  {\rm Phys.Lett. B193 (1987) 101}
%\bibitem{Fri}
%  {\bf H. Fritzsch}:
%  {\rm Phys.Lett B256 (1991) 75}

%--------------------------------------------------------------
\subparagraph{Operator Product Expansion} \hfill
%--------------------------------------------------------------

\bibitem{SVZ2}
  {\bf M.A. Shifman et al.}:
  {\rm Nucl.Phys. B147 (1979) 385,448,519}
\bibitem{NSVZ}
  {\bf V.A. Novikov et al.}:
  {\it Calculation in external fields in QCD; technical review};
  {\rm Fort.Phys. 32 (1984) 585};
  {\it Wilson's OPE: can it fail ?};
  {\rm Nucl.Phys. B249 (1985) 445}

%--------------------------------------------------------------
\subparagraph{Miscellaneous} \hfill
%--------------------------------------------------------------

\bibitem{Wit}
  {\bf E. Witten}:
  {\it Current algebra theorems for the U(1) "Goldstone boson"}
  {\rm Nucl.Phys. B156 (1979) 269}
\bibitem{Wit2}
  {\bf E. Witten}:
  {\it $1/N_c$ expansion in QCD};
  {\rm Nucl.Phys. B160 (1979) 57}
\bibitem{Shi}
  {\bf M.A. Shifman}:
  {\it Anomalies in gauge theories};
  {\rm Phys.Rep. 209 (1991) 341}
\bibitem{Shu2}
  {\bf E.V. Shuryak}:
  {\it Correlation functions in the QCD vacuum}
  {\rm Rev.Mod.Phys. 65 (1993) 1}
\bibitem{Wil}
  {\bf K.G. Wilson}:
  {\it Confinement of quarks};
  {\rm Phys.Rev. D10 (1974) 2445}
\bibitem{Gas}
  {\bf J. Gasser, H. Leutwyler}:
  {\it Quark masses};
  {\rm Phys.Rep. 87 (1982) 77}
%\bibitem{MH}
%  {\bf M. Hutter}
%  {\it unpublished}
%\bibitem{xx}
%  {\bf }
%  {\it }
%  {\rm }
% ...
\end{thebibliography}

