Chiral Solitons

The following sections are included:

• PREFACE

• TABLE OF CONTENTS

The following sections are included:

• INTRODUCTION
  • D=2 Fermi-Bose Correspondence
  • D=4 Fermi-Bose Correspondence
  • Coda

• REFERENCES

The spontaneously broken chiral symmetry scenario of quantum chromo-dynamics (QCD) together with the Veltman theorem implies that the low energy effective Lagrangian, of QCD must be of the nonliear σ model type. The effective Lagrangians resulting from integrating the quark fields from the gauged or nongauged chiral quark model Lagrangian in a Skyrme-type scenario may be completely equivalent to QCD in the large N_c limit at low energy and therefore may serve as realistic models for low energy hadron dynamics. For this purpose, we present in detail the recent development of the effective action expansion, in particular the evaluation of the fermion determinant of these models. The implications on the low energy phenomenology associated with these effective Lagrangians will be discussed.

The following sections are included:

• INTRODUCTION

• CHIRAL FIELD THEORY
  • The Skyrme Lagrangian
  • Topological Solitons
  • Symmetries and Currents

• THE SKYRMION
  • The Hedgehog
  • The Shape Function
  • The Rotating Soliton

• QUANTIZATION
  • Quantization and Wave Functions
  • Matrix Elements

• STATIC PROPERTIES
  • Masses
  • Sizes and Magnetic Moments
  • Electric and Magnetic Transition Elements
  • Axial Coupling and the Goldberger-Treiman Relation
  • More Results and Predictions

• DISCUSSION AND CONCLUSIONS

• APPENDIX A: NOTATION

• APPENDIX B: PAULI MATRIX ALGEBRA

• APPENDIX C: THE ADJOINT REPRESENTATION OF SU(2)

• APPENDIX D: A FEW USEFUL FACTS

• ACKNOWLEDGEMENTS

• REFERENCES

The following sections are included:

• Introduction

• Large N

• Outline of Formalism

• Model-independent linear relations for πN → πN and πN → πΔ

• The Baryon Spectrum of the Skyrme Model

• The Big-Small-Small-Big Pattern

• Skyrmions and Vector Mesons

• Formalism for 3-Flavor Scattering

• The 3-Flavor Skyrme Model

• References

Methods for deriving a static baryon-baryon, nuc1eon-nucleon as well as nucleon-isobar and transition, interaction from the Skyrme model are presented. Results obtained with these methods are discussed. Various extensions of the Skyrme model and their predictions for the N-N interaction are also reported.

The following sections are included:

• INTRODUCTION

• A SIMPLE CHIRAL QUARK-MESON MODEL

• PROJECTING FROM THE HEDGEHOG

• CRANKING THE HEDGEHOG

• QCD AND THE QUARK-MESON THEORY

• VECTOR MESONS

• SMALL AMPLITUDE MOTION

• A MODEL OF CONFINEMENT WITH GLUEBALLS

• GLUEBALL AND THE BAG

• CONCLUSION

• ACKNOWLEDGEMENTS

• APPENDIX A

• REFERENCES

We present an effective Lagrangian for QCD and discuss dynamical mass generation for quarks and gluons. We describe applications made in the study of the structure of nucleons and mesons and also review our predictions of modified nucleon properties in nuclei.

A summary of recent and current research on the non-topological soliton bag model is presented. The unique feature of the model, namely dynamics, is emphasized, since this permits calculation of reactions within the framework of a covariant effective Lagrangian. One gluon exchange effects are included. Applications include hadronic spectra, recoil corrections, N-N interactions, Structure of the pion and dressing of the nucleon and delta by the pion with gross restoration of PCAC, many bag systems, and annihilation. One loop corrections to the Dirac and Bose fields are discussed.

Fermions appear naturally on quantizing a rotating Skyrme soliton made up of four classical boson fields arranged as a 2 × 2 matrix or quaternion in an isospin space which has been mapped topologically onto ordinary space.

The quantized Hamiltonian and currents for the skyrmion are derived in the curvelinear collective coordinates which describes the rotation of the soliton. The Euler-Lagrange equation for the rotated pion field is derived. It turns out to be the operator equations for the conservation of the axial-vector current in the massless pion case and the PCAC relation in the massive pion case. It admits stable rotating soliton solutions. The derived vector currents are applied to calculate the electromagnetic mass differences between the proton and the neutron and among the Δ’s. A down-up quark mass difference of −2 MeV is needed to explain the observed nucleon mass difference. The Goldberger-Treiman relation is derived in the massive pion case. The root mean square radii for the πNN vertex, the axial-vector form factor and the magnetic form factors are calculated in the Skyrme model.

We treat the rotational degrees of freedom in a constrained hamiltonian formalism. We investigate the coupling between rotational and vibrational degrees of freedom. For stationary states the radiation term is absent and time independent solutions with finite energy are possible.

We consider the Skyrme soliton from the viewpoint of realistic fluctuations about a quantum condensed phase. To study this, a systematic presentation of methods to obtain field-theory matrix elements for the Skyrmion and other extended states is given. The approach concentrates on maintaining covariance, translational invariance and other symmetries in a consistent quantum mechanical framework. The quantum kinematical features of Gaussian fluctuations and resulting matrix elements of local operators do not agree with common replacements such as the large F_π and mean-field prescriptions, but are investigated as a function of momentum transfer, parameters defining the field distributions, and the ultraviolet regulator. We emphasize the need for off-diagonal distribution functions, construct explicit momentum in-states, and illustrate methods of calculation with examples. Much of the presentation is designed to avoid sigma-model formalism, topological arguments etc., in order to highlight the quantum mechanics for students and others learning about solitons.

The ground state properties of an approximately chiral invariant bag-like Hamiltonian, which incorporates motion of the bag or nucleon core while it interacts dynamically with its pion field, is studied variationally and the quantitative effects of this motion on the static nucleon properties are presented with special emphasis placed on the effects of bag or nucleon core motion on values of the bag radius. The interaction of the moving bag or nucleon core with the pion field it generates is shown to result in a pionic nontopological soliton description of the nucleon. The computational methods employed to demonstrate the existence and nature of this soliton state are presented. The trial states used to describe the nucleon ground state have good total spin and isospin and are constructed without the use of projection techniques. The effects of the self-consistent motion of the core or bag on the static nucleon properties are computed, with the result that for fixed static nucleon properties the bag radius of the moving core with its pion field is significantly smaller than the corresponding radius of a static bag and the pion field it generates.

We discuss a method for including soliton recoil in calculations of meson-soliton scattering which is valid in any number of spatial dimensions. For the purpose of illustration we apply this method to the Φ⁴(x) kink field theory in 1+1 dimensions. In formulating the meson-soliton scattering problem we show how to construct free mesonsoliton states, i.e. states which describe a free meson and a free soliton each with good linear momentum. This is achieved without the use of projection techniques. We obtain analytic solutions for the bound state and scattering problems including soliton recoil by diagonalizing a specific subset of the 1-body parts of the phi-fourth Hamiltonian and by considering the nonrelativistic limit of both the soliton and meson. We show that these approximations are quite accurate for the bound states, i.e. the soliton mass including terms up to zeroth order in the meson coupling and the first excited soliton state which is a breather mode, when compared to the usual semiclassical calculations. We also present a comparison between the collective coordinate quantization methods of Christ and Lee and that of Tomboulis with emphasis on their differences. Future applications of the method are also discussed…