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%%     Instantonen in der QCD  -    Marcus Hutter 28.10.95   %%
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\begin{document}
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%              T i t l e - P a g e                      %
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\begin{titlepage}
\begin{center}       \vspace*{3cm}
  {\Huge     INSTANTONS IN QCD                       } \\[5mm]
  {\LARGE    Theory and Application of the           } \\[5mm]
  {\LARGE    Instanton Liquid Model                  } \\[70mm]
  {\large\it Ph.D. Thesis of the Faculty for Physics } \\[2mm]
  {\large\it at the Ludwig Maximilians University Munich } \\[30mm]
  {\large    Presented by                            } \\[2mm]
  {\large    \bf Marcus Hutter                       } \\[2mm]
  {\large    Munich, 17$^{th}$ of October, 1995      } \\[20mm]
\end{center}
\end{titlepage}

%\footnotetext{E--Mail:hutter@hep.physik.uni--muenchen.de}

\begin{titlepage}

\begin{center}\large
Inofficial translation from the German Ph.D. thesis \\[6ex]\sl
INSTANTONEN IN DER QCD \\
Theorie und Anwendungen des \\
Instanton-Fl\"ussigkeit-Modells
\end{center}

\vspace*{14cm}
1. Advisor: Prof. H. Fritzsch \\
2. Advisor: Prof. S. Theisen \\
Defense: 9 February 1996
\end{titlepage}

\centerline{\bf Abstract}

The current theory of strong interactions, the quantum
chromodynamics (QCD), is a non-abelian gauge theory, based on the
gauge group SU(3). Despite its formal similarity to QED there are
significant differences. It was shown in 1973 already that the
coupling constant $g$ increases for large distances \cite{Gross}.
This gave hope for the possibility to explain quark and gluon
confinement. Soon after, in 1975, non-trivial solutions of the
Euclidian Yang Mills equations were found, nowadays called BPST
instantons \cite{BPST}, which significantly influence the low
energy structure of QCD. Many exact results are known for the
1-instanton vacuum \cite{Raj,Bro}, whereby an interesting
phenomenological result of it is the explicit breaking of the
axial $U(1)$ symmetry \cite{tHo}. On the other hand, a one
instanton approximation, similar to a tree approximation in
perturbation theory, cannot describe boundstates or spontaneous
symmetry breaking. The next step was the analysis of exact
\cite{Act} and approximate \cite{CDG} multi-instanton solutions.
There are two useful visualizations for these solutions. In
one of these, instantons are interpreted as tunneling processes between
different vacua. In the other interpretation, a solution describes
an ensemble of extended (pseudo) particles in 4 dimensions.

In conventional perturbation theory one computes fluctuations
around the trivial zero solution. The correct quantization process
is to consider all classical solutions of the field equations and
their fluctuations. In the path integral representation of QCD the
partition function is, hence, dominated by an ensemble of extended
particles (instantons) in 4 dimensions at temperature $g^2$. In
the simplest case the partition function describes a diluted ideal
gas of independent instantons. Unfortunately, this assumption
leads to an infinite instanton density caused by large instantons,
which obviously contradicts the assumption of a diluted gas. This
problem is known as the infrared problem.  The problem is avoided
by assuming a repulsive interaction \cite{Ilg} which prevents the
collapse. This is the model of a 4 dimensional liquid. Under
certain circumstances the interaction can be replaced by an
effective density. The \ILM\ in a narrow sense describes the QCD
vacuum as a sum of independent instantons with radius
$\rho=(600\mbox{MeV})^{-1}$ and effective density
$n=(200\mbox{MeV})^4$. The correctness of this model is still
being intensively investigated. So far the model is essentially
justified by its phenomenological success.

Numerical simulations of the \ILM\ allowed to determine a number of
hadronic quantities, especially meson masses, baryon masses,
hadron wave functions, and condensates \cite{Shu,ShV}.

For computing the quark propagators and the meson correlators
there are also analytical methods. The most important predictions
are probably the breaking of the chiral symmetry (SBCS) in the
axial triplet channel \cite{Dya} and the absence of Goldstone
bosons in the axial singlet channel.

The largest part of this thesis is devoted to extending the
analytical methods and to evaluating the results in (semi)analytical
form.

The meson correlators (also called polarization functions) will be
computed in the \ILM\ in zeromode and $1/N_c$ approximation,
whereby dynamic quark loops will be taken into account. A spectral
fit allows the computation of the masses of the $\sigma$, $\rho$,
$\omega$, $a_1$ and $f_1$ mesons in the chiral limit. A separate
consideration also allows computation of the $\eta'$ mass. The
results coincide on a 10\% level with the experimental values.
Furthermore, determining the axial form factors of the proton,
which are related to the proton spin (problem), will be attempted.
A gauge invariant gluon mass for small momentum will also be computed.

The thesis ends with several predictions which do not rely on the
\ILM. In the 1-instanton vacuum a gauge invariant quark propagator
will be computed and compared to the regular and singular
propagator. Rules for the choice of a suitable gauge, especially
between regular or singular, will be developed. A finite
relation between the quark condensate and the QCD scale $\Lambda$
will be derived, whereby neither an infrared cutoff, nor a specific
instanton model will be used.

{\parskip=0ex
 \tableofcontents
 \vspace{1cm}
 The chapters marked with * are more technical in nature.
 %\listoffigures
 \listoftables
}

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

%\chapter{Introduction}
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%\chapter{Theory of Instanton Liquid}
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%\chapter{Light Quark Propagator}
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%\chapter{Four \& Six Quark 't Hooft like Interaction}
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%\chapter{Correlators of Light Mesons}
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%\chapter{Axial Anomaly}
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%\chapter{Gluon Propagator}
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%\chapter{Gauge Invariant Quark Propagator}
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%\chapter{Conclusions}
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%\chapter{Private comments}
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\begin{appendix}
%\chapter{Appendices}
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%\chapter{Figures}
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%{References}
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\end{appendix}
\end{document}

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