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Nonperturbative QCD

It is a notoriously difficult problem to derive the basic physical properties of hadrons and nuclei from the underlying fundamental theory of strong interactions, i.e., quantum chromodynamics (QCD). In the low-energy limit, QCD becomes a strongly interacting quantum field theory where the primary quantum quark and gluon fields are no longer proper degrees of freedom and the usual perturbative description (Feynman diagrams expansion) breaks down. Indeed, due to confinement they form colorless bound states, i.e., mesons and baryons (and perhaps glueballs). The complete description of these effective particles from first principles - that is, by direct computations in QCD - as well as the full understanding of the confinement phenomenon, and formation of color flux tubes, is a very complicated and still unsolved problem. The main reason for that is that at the strong coupling no QCD relevant analytical or exact frameworks are known. The are two main approaches which offer a way to resolve this problem, at least to some extent.

The first one consists in the introduction of low-energy effective field theories, where the fundamental fields of QCD, which are the relevant degrees of freedom at high energies, are replaced by some other degrees of freedom which are supposed to describe strong interaction physics at low energies more practically or more concisely. The main difficulty is that no derivation of such an effective theory of mesons in a rigorous way from the first principles is known.

The second completely independent way to describe strongly coupled regime of a nonabelian gauge theory is based on the famous Maldacena conjecture and AdS/CFT correspondence, where a quantum field (gauged) theory admits a formulation in a language of superstrings. In this approach the strongly coupled sector of the quantum theory is mapped into a weekly coupled (i.e., perturbative) string theory living in a curved gravity background. An obstacle, if one wants to use this approach to the low energy QCD, originates in an observation that the original duality was proposed for a very specific N = 4 supersymmetric Yang-Mills theory (in the large Nc limit). This quantum field theory is in fact a very special one - it is a conformal theory also on the quantum level. Due to that, it has a trivial running coupling constant (identically vanishing beta function), which results in the lack of a scale in the model. Even more important is the fact that there is no confinement! It is obvious, that a direct application of this duality to the nonperturbative sector of QCD (governed by the confinement phenomenon) is rather very disputable.


The Skyrme model was introduced by Skyrme as a field theoretic realization of his concept of a mesonic fluid model for nuclei, motivated by the substantially homogeneous nature of nuclear matter. The nucleons were supposed to result from a kind of local twist or vorticity in this mesonic fluid, that is, to be described by topological solitons in a more modern language. This original idea received further support and general acceptance once it was found that an effective theory of mesons may be derived from QCD in the limit of a large number of colors. So, the Skyrme model is a nonlinear field theory which is supposed to describe the low-energy limit of strong interaction physics in terms of hadrons. The primary fields of the Skyrme model are mesons, whereas baryons appear as collective excitations, that is, topological solitons (skyrmions). These solitons have the characteristic feature that they can be classified by an integer-valued topological degree or winding number. One important insight of Skyrme has been the proposal that this topological degree should be identified with the baryon number B, explaining in this way its conservation. For the specific case of two flavors, the field of the Skyrme model takes values in the group SU(2) (isospin). And as anticipated by Skyrme, one natural area of applications of the resulting theory is nuclear physics, providing in this manner a possible basis for a unified field theoretic description of nuclei and their properties. Indeed, shortly after the description of the nucleons (proton and neutron) in terms of the simplest skyrmion (the hedgehog solution), the Skyrme model was used to describe the deuteron and some additional light nuclei. More recently, nuclear excitation spectra have been studied within the Skyrme model with reasonable success.

BPS models

One common very serious problem in the application of the standard version of the Skyrme model to nuclei is that the resulting nuclear binding energies are too large. Concretely, although there exists a lower energy bound linear in the baryon number (Faddeev-Bogomolnyi bound), skyrmion solutions do not saturate this bound. The energy (mass) of the simplest B = 1 skyrmion (hedgehog) is about 23% above the bound, whereas for higher B this deviation is lowered, to less than 4% in the limit of very large B. As a consequence, the binding energies per baryon number of higher skyrmions (i.e., the energies needed to desintegrate higher skyrmions into their B = 1 constituents) are quite high, on the level of 10%, which is in striking contrast to the low binding energies of physical nuclei. A way of dealing with this problem in the standard Skyrme model is by a different renormalization of the coupling constants of the model for different nuclei. Another serious problem of the usual Skyrme model is the fact that the state of matter described by it is rather problematic. The matter behaves more like a crystal than a liquid, which is unacceptable for the nuclear matter. Also the fact that shell-like structures are preferred than the ball-like solitons seems to point in a wrong direction.

All these problems led to assumption that the correct low energy effective model for QCD should be a near BPS model with almost linear dependence of the total energy on the baryon charge.