Skyrmions from SU(3) harmonic maps and their quantization

Abstract

Static properties of SU (3) multiskyrmions with baryon numbers up to 6 are estimated. The calculations are based on the recently suggested generalization of the SU (2) rational map ansätze applied to the SU (3) model. Both SU (2) embedded skyrmions and genuine SU (3) solutions are considered and it is shown that although, at the classical level, the energy of the embeddings is lower, the quantum corrections can alter these conclusions. This correction to the energy of the lowest state, depending on the Wess-Zumino term, is presented in the most general case.1. Topological soliton models, and the Skyrme model among them [1], have recently generated a fair amount of interest because they may be able to describe various properties of low energy baryons. So far, most of such studies have involved the SU (2) Skyrme model, whose solutions were then embedded into the SU (3) model. This is justified as the solutions of the SU (2) model are also solutions of SU (N ) models. However, there exist solutions of the SU (N ) models which are not embeddings of SU (2) fields and it is important to assess their contribution.As with solutions of the SU (2) Skyrme models, the solutions of the SU (3) model, with very few exceptions, can only be determined numerically. Like for the SU (2) case [2] one starts with a harmonic map ansatz which gives fields "close" to the genuine solutions [3,4]. Then one can use these fields as starting configurations of various numerical minimization schemes. For the SU (2) fields such an approach was carried out in [2] where it reproduced the results of the earlier numerical approaches [5,6] and so showed that the harmonic map approximations are very close to the final fields. The real solutions have energies only few hundreds of M eV lower than the harmonic approximations and the baryonic charge and energy distributions do not look very different etc. This has justified the use of harmonic field approximants in calculating various quantities when estimating quantum corrections to the classical results.In this paper we take a look at the nonembedding solutions of the SU (3) model. First we recall the results of [3] and use them in a SU (3) numerical minimization program to estimate the values of energies of true solutions. The SU (3) variational minimization program [7] has been rearranged for this purpose to allow the consideration of quite general field ansätze. This is discussed in the next section.In the following section we discuss various quantum corrections and compare our results with the similar results for the embeddings. The paper ends with a short section presenting our conclusions.2. The Lagrangian of the SU (3) Skyrme model, in its well known form, depends on the parameters F π and e and is given by [8,9]:

The Interaction Energy of Well-Separated Skyrme Solitons

“…In the following we use the term Skyrme solitons for minimal energy solutions of the Skyrme equation with non-zero degree k. There is overwhelming numerical evidence that the minimum in C 2 is attained by a configuration of toroidal symmetry [7]. For higher degree, too, much is known numerically about the minima of Skyrme's energy functional in the sector C k .…”

The α particle as a canonically quantized multiskyrmion

“…The identification of the topological number with baryon number also leads to solitonic fullerene structures in light atomic nuclei [8]. The RM ansatz has been generalized to the SU(3) Skyrme model as well [9].…”