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## Instantons in QCD: Theory and Application of the Instanton Liquid Model

Author:Marcus Hutter (1996) Comments:100 pages, 8 figures, translated from the German original. Reference:Ph.D. Thesis of the Faculty for Physics at the Ludwig Maximilians University Munich (1996) Report-no:hep-ph/0107098 Paper:PostScript (1505kb) - PDF (629kb) Slides:See Table of Contents below

Keywords:Instanton liquid model, non-perturbative QCD, meson correlators, meson masses, gluon mass, gauge invariant quark propagator, axial anomaly, eta' mass, proton spin.

Abstract:Numerical and analytical studies of the instanton liquid model have allowed the determination of many hadronic parameters during the last 13 years. Most part of this thesis is devoted to the extension of the analytical methods. The meson correlation (polarization) functions are calculated in the instanton liquid model including dynamical quark loops. The correlators are plotted and masses and couplings of the sigma, rho, omega, a1 and f1 are obtained from a spectral fit. A separated analysis allows the determination of the eta' mass too. The results agree with the experimental values on a 10% level. Further I give some predictions for the proton form factors, which are related to the proton spin (problem). A gauge invariant gluon mass for small momenta is also calculated. At the end of the work some predictions are given, which do not rely on the instanton liquid model. A gauge invariant quark propagator is calculated in the one instanton background and is compared to the regular and singular propagator. An introduction to the skill of choosing a suitable gauge, especially a criterion for choosing regular or singular gauge, is given. An application is the derivation of a finite relation between the quark condensate and the QCD scale Lambda, where neither an infrared cutoff nor a specific instanton model has been used. In general the instanton liquid model exhibits an astonishing internal consistency and a good agreement with the experimental data.

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## Table of Contents

Introduction(Slides: 0 a b c)

- Methods to Solve QCD
- Contents
Theory of Instanton Liquid(Slides: 1 a b c d e f g 3 a b c d)

- Separating Gaussian from non-Gaussian Degrees of Freedom
- Effective QCD Lagrangian in a Background Field
- The Semiclassical Limit
- Instantons in QCD
- Quarks
- The Instanton Liquid Model
Light Quark Propagator(Slides: 5 a b c d e f g)

- Perturbation Theory in the Multi-instanton Background
- Exact Scattering Amplitude in the one Instanton Background
- Zero Mode Approximation
- Effective Vertex in the Multi-instanton Background
- A Nice Cancellation
- Renormalization of the Instanton-Density
- Selfconsistency Equation for the Quark Propagator
- Some Phenomenological Results
- Large
Nexpansion_{c}- Summary
Four Point Functions(Slides: 6 a b c d e)

- Introduction
- Large
Napproximation_{c}- Solution of the Bethe-Salpeter Equations
- Triplet and Singlet Correlators
- Summary
Correlators of Light Mesons(Slides: 7 a b c d)

- Analytical Expressions
- Analytical Results
- Spectral Representation
- Plot & Fit of Meson Correlators
The Axial Anomaly(Slides: 10 a b c d e f g 9 a b c d e f g h i)

- The Mass of the eta' Meson
- Measurement of the Axial Form Factors
- Axial Singlet Currents & Anomaly
- The Proton Spin and its Interpretation
- Reduction of the Proton Form Factors to Vacuum Correlators
- The Axial Form Factors
G(_{1/2}^{GI}q)- The Anomaly Form Factor A(q)
- The Gluonic Form Factors
K(_{1/2}^{GI}q)- Discussion
Gluon Mass(Slides: 2 a b c)

- Introduction
- Gluon Propagator
- Propagator in Statistical Background
- A Naive Estimate of the Gluon Mass
- Expansion in the Instanton Density
- QCD Propagators
- Propagators for Small Momentum
- Zeromodes
- Conclusions and further developments
Gauge Invariant Quark Propagator

- Generalities On the Choice of Gauge
- A Natural Gauge
- On the Gauge in Instanton Physics
- The Quark Propagator in Axial Gauge
- Effective Quark Mass
- The Quark Condensate
- Summary
Conclusions(Slides: 0 d)

- New Results
- Outlook
- Acknowledgement
Appendices

- Notations
- Instantons in Singular, Regular and Axial Gauge
- Averaging over the Instanton Parameter
- Numerical Evaluation of Integrals
- Numerical Evaluation of the Convolution
Figures(Slides: 8 a b c d e f g h)

- Panorama Function
- Constituent Quark Mass in Regular, Singular and Axial Gauge
- Pseudoscalar Triplet Correlator (pi)
- Pseudoscalar Singlet Correlator (eta')
- Scalar Triplet Correlator (delta)
- Scalar Singlet Correlator (sigma)
- Axial Vector Correlator (a
_{1},f_{1})- Vector Correlator (rho,omega)
- References

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@PhdThesis{Hutter:96thesis, author = "Marcus Hutter", institution = "Faculty for Theoretical Physics, LMU Munich", title = "Instantons in QCD: Theory and application of the instanton liquid model", year = "1996", pages = "1--100", url = "http://arxiv.org/abs/hep-ph/0107098 ", url2 = "http://www.hutter1.net/physics/pdise.htm", abstract = "Numerical and analytical studies of the instanton liquid model have allowed the determination of many hadronic parameters during the last 13 years. Most part of this thesis is devoted to the extension of the analytical methods. The meson correlation (polarization) functions are calculated in the instanton liquid model including dynamical quark loops. The correlators are plotted and masses and couplings of the sigma, rho, omega, a1 and f1 are obtained from a spectral fit. A separated analysis allows the determination of the eta' mass too. The results agree with the experimental values on a 10% level. Further I give some predictions for the proton form factors, which are related to the proton spin (problem). A gauge invariant gluon mass for small momenta is also calculated. At the end of the work some predictions are given, which do not rely on the instanton liquid model. A gauge invariant quark propagator is calculated in the one instanton background and is compared to the regular and singular propagator. An introduction to the skill of choosing a suitable gauge, especially a criterion for choosing regular or singular gauge, is given. An application is the derivation of a finite relation between the quark condensate and the QCD scale Lambda, where neither an infrared cutoff nor a specific instanton model has been used. In general the instanton liquid model exhibits an astonishing internal consistency and a good agreement with the experimental data.", note = "Translated from the German original http://www.hutter1.net/physics/pdiss.htm", }

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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 (Gross
1973). 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 (Belavin et al. 1975), which significantly
influence the low energy structure of QCD. Many exact results are
known for the 1-instanton vacuum (Rajaraman 1982; Brown 1978),
whereby an interesting phenomenological result of it is the
explicit breaking of the axial U(1) symmetry ('t Hooft 1976). 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 (Actor 1979) and approximate (Callen Dashen Gross 1979)
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.

Numerical simulations of the Instanton Liquid Model allowed to determine a number of hadronic quantities, especially meson masses, baryon masses, hadron wave functions, and condensates (Shuryak et al. 1982..1994).

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 (Dyakonov and Pedrov 1984,1985) 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 Instanton Liquid Model 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

The thesis ends with several predictions which do not rely on the Instanton Liquid Model. 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.

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