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.

Table of Contents

1. Introduction (Slides: 0 a b c)
   1. Methods to Solve QCD
   2. Contents
2. Theory of Instanton Liquid (Slides: 1 a b c d e f g 3 a b c d)
   1. Separating Gaussian from non-Gaussian Degrees of Freedom
   2. Effective QCD Lagrangian in a Background Field
   3. The Semiclassical Limit
   4. Instantons in QCD
   5. Quarks
   6. The Instanton Liquid Model
3. Light Quark Propagator (Slides: 5 a b c d e f g)
   1. Perturbation Theory in the Multi-instanton Background
   2. Exact Scattering Amplitude in the one Instanton Background
   3. Zero Mode Approximation
   4. Effective Vertex in the Multi-instanton Background
   5. A Nice Cancellation
   6. Renormalization of the Instanton-Density
   7. Selfconsistency Equation for the Quark Propagator
   8. Some Phenomenological Results
   9. Large N_c expansion
   10. Summary
4. Four Point Functions (Slides: 6 a b c d e)
   1. Introduction
   2. Large N_c approximation
   3. Solution of the Bethe-Salpeter Equations
   4. Triplet and Singlet Correlators
   5. Summary
5. Correlators of Light Mesons (Slides: 7 a b c d)
   1. Analytical Expressions
   2. Analytical Results
   3. Spectral Representation
   4. Plot & Fit of Meson Correlators
6. The Axial Anomaly (Slides: 10 a b c d e f g 9 a b c d e f g h i)
   1. The Mass of the eta' Meson
   2. Measurement of the Axial Form Factors
   3. Axial Singlet Currents & Anomaly
   4. The Proton Spin and its Interpretation
   5. Reduction of the Proton Form Factors to Vacuum Correlators
   6. The Axial Form Factors G_1/2^GI(q)
   7. The Anomaly Form Factor A(q)
   8. The Gluonic Form Factors K_1/2^GI(q)
   9. Discussion
7. Gluon Mass (Slides: 2 a b c)
   1. Introduction
   2. Gluon Propagator
   3. Propagator in Statistical Background
   4. A Naive Estimate of the Gluon Mass
   5. Expansion in the Instanton Density
   6. QCD Propagators
   7. Propagators for Small Momentum
   8. Zeromodes
   9. Conclusions and further developments
8. Gauge Invariant Quark Propagator
   1. Generalities On the Choice of Gauge
   2. A Natural Gauge
   3. On the Gauge in Instanton Physics
   4. The Quark Propagator in Axial Gauge
   5. Effective Quark Mass
   6. The Quark Condensate
   7. Summary
9. Conclusions (Slides: 0 d)
   1. New Results
   2. Outlook
   3. Acknowledgement
10. Appendices
    1. Notations
    2. Instantons in Singular, Regular and Axial Gauge
    3. Averaging over the Instanton Parameter
    4. Numerical Evaluation of Integrals
    5. Numerical Evaluation of the Convolution
11. Figures (Slides: 8 a b c d e f g h)
    1. Panorama Function
    2. Constituent Quark Mass in Regular, Singular and Axial Gauge
    3. Pseudoscalar Triplet Correlator (pi)
    4. Pseudoscalar Singlet Correlator (eta')
    5. Scalar Triplet Correlator (delta)
    6. Scalar Singlet Correlator (sigma)
    7. Axial Vector Correlator (a₁,f₁)
    8. Vector Correlator (rho,omega)
12. References

BibTeX Entry

@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",
}
      

Extended 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 (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, a₁ and f₁ 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 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.