Valerii I. Sanyuk

Absolute Minimum of the Energy Functional

We start with a remark on the importance of the absolute minimum state in a particular aspect to validate the interpretation of Skyrmions as baryons. Our claim is that if the Skyrme model admits the existence of a ground state with the least mass for isovector fields from the first homotopy class, that fact can be regarded as a valuable support to the Skyrme hypothesis. Remember that the proton, which is the longest-living hadron, has the least mass among all particles with a nontrivial baryonic charge. In what follows we demonstrate that in fact the Skyrmion is the absolute minimum energy configuration in the first homotopy class. To obtain this result first of all we use one of the direct minimization methods - the method of extending the original phase space of the model. Afterwards we apply the spherical rearrangement method to derive the structure of fields with the lowest energy (see Section 5.3).

Quantization of Skyrmions

This is the next and the inevitable step towards a practical implementation of the Skyrme approach. The quantization procedure, which was found acceptable to this problem in the pioneering paper (Adkins, Nappi and Witten 1983), was known long ago as Bogolubov’s method of collective coordinates. Note, that this method was originally developed and applied in (Bogolubov 1950) to the polaron problem, which has some close parallels to the description of a nucleon state in the soliton approach.

The Skyrme Model and QCD

This Part presents an attempt to bring the most evident applications of the Skyrme model to the attention of the reader, who is not exactly an expert in the field. This motivation gives us the right to concentrate our efforts mostly on background ideas and principle questions, neglecting some technical details of calculations, which can be found in the literature, if necessary. Therefore we will proceed whenever possible on rather elementary level. On the equal footing we are not going to collect all known facts and achievements in the field of the Skyrme model applications in hadron and condense matter physics obtained during the last decade. Such a task obviously requires an enormous increasement of the volume and the only way out we find in references to the original papers and review literature. We hope that the content of this Part provides sufficient background so that the unsatisfied reader is enabled to consult the original literature without undue difficulty.

The Principle of Symmetric Criticality

The theory of solitons along with the further development of the beautiful concept of complete integrability for (1+1)-models brought into focus the problem of analytical investigations in higher dimensional field theories. In spite of frequent efforts to extend the complete integrability or a similar concept to the realistic (3+1)-dimensional case, reported results are still far from successful. On the other hand the appearance of more powerful computers led to visible progress in the lattice approach in field theory. The latter caused a widespread belief that it is possible to solve almost any problem even in such complicated theories as QCD or the Standard Model, when using a ‘clever enough’ computerized lattice complex. There are, however, already some indications that the reality is not so straightforward. Definitely now it is not yet the time when solutions of physical problems may be reduced to an electronic engineering problem of the type of how to construct a needed lattice. Computer simulations can provide us with a reliable answer only in the case when a likelihood description (a “scenario”) of physical phenomena is available. From that point, attempts to extract nucleon observables from direct lattice calculations in the QCD are similar to ‘predictions of a match score on the basis of the rules of the game only.’ As an example in support of this opinion, we have a relative failure to get the correct value for the gluonic contribution in the “proton spin” puzzle directly from the lattice QCD calculations (Jaffe and Manohar 1990; Mandula 1990). From the general logic it is doubtful that any extreme position could ever be acceptable and both analytical and lattice approaches must supplement each other. Having this in mind, beginning with this chapter we try to give an answer as to what in particular one can expect from already available analytical methods in the study of multi-dimensional solitons.

Skyrmion as a Fermion

The fifth Skyrme’s suggestion, as listed in Section 1.5, without any doubt appears to be the most attractive one. Recall, that one of the central ideas of Skyrme’s approach was to discover a possible way to regard a bose-field theory originated soliton as a fermionic state. We have already mentioned that this idea entailed the new branch of study in nonlinear physics, called the Fermi-Bose transmutations. Localized structures with transmutations of spins and statistics has been studied in variety of nonlinear models, applied in gravitation (“geons”) (Friedman and Sorkin, 1980; Sorkin, 1986), in condensed matter physics for model description of such novel effects as high temperature superconductivity (“anyons”) (Arovas, Schrieffer, Wilczek and Zee, 1985; Frohlich and Marchetti, 1988), the fractional quantum Hall effect (“holons”) (Wilczek, 1982; Arovas, Wilczek and Schrieffer, 1984). We refer the reader to self-contained reviews and lecture notes on related questions (Mackenzie and Wilczek, 1988; Laughlin, 1988; Wilczek, 1990; Balachandran, Marmo, Skagerstam and Stern, 1991). Thus this Skyrme’s conjecture appeared to be not only valid, but also productive.

Quantized SU(3)Skyrmions and Their Interactions

Taking into account that among possible readers of this book, there would be people, mostly interested in practical implementations of Skyrme’s approach we add a brief account of relevant topics in this Chapter. It might rather be considered as a guide in innumerable literature, appeared on the subject during the last decade. In advance, we apologize to the authors, who will not find their contributions to the field in our references. We mostly refer to the papers, which help us to understand the situation with the Skyrme model applications, and hope that our own experience, exposed here, might be useful for some readers. Among other sources, we mostly learnt from (Adkins and Nappi, 1985; Balachandran, 1986; Balachandran, Marmo, Skagerstam and Stern, 1991; Yabu and Ando, 1988; Zahed and Brown, 1986)

Proofs of Stability and Existence Theorems

Let us start with an equivalent formulation of Theorem 3.2 from Chapter 3: Theorem. The second variation of an additive Lyapunov functional of the type (3.27) in a neighbourhood of a stationary soliton solution is a sign changing one for all dimensions D ≥ 2.

The Evolution of Skyrme’s Approach

It was one of Tony Skyrme’s characteristic features to work in close contact with experimental physicists, who interested him in different phenomenological problems. In this way Skyrme was involved in solving the experimental data puzzle, which arose as a result of the fast electron scattering experiments, conducted with the aim of studying the charge distribution within the nucleus (the results werepublished in 1953). As a consequence of those electromagnetic type experiments, as well as from μ-mesonic atoms spectroscopy data or from the isotopic shift data, the value for the radius of the charge distribution within the nucleus was suggested to be

The Existence of Skyrmions

\bar R \simeq 1.2A^{1/3} fm